The Spline

Embedding a Computable Document in a Blogger Post

2011-07-22T01:26:00.001Z

Peeking at the source code in the Wolfram blog post here has revealed to me how to embed CDF documents in Blogger posts, and here is the proof:

To view this content, please install Wolfram CDF Player. You can install the free CDF Player here.

Computable Document Format

2011-07-21T14:39:00.002Z

A post Launching the Computable Document Format (CDF): Don’t Compress the Idea, Expand the Medium at the Wolfram Blog explains why CDF is so important, and it points to some nice examples of its use. If you find that mapping your high-dimensional thoughts onto a 1-dimensional line (i.e. a traditional static document) destroys most of the information, then CDF is for you (i.e. an interactive dynamic document). I have always found that using Mathematica - the engine under CDF's hood - has given me an enormous advantage over my peers who used "A N Other Product", because it allows me to do literate programming and lots more in a unified way. However, I always ended up rewriting everything in traditional 1-dimensional style to communicate with other people - usually not very successfully. CDF changes the game because it allows me to present material in the natural form in which it was created in the first place. Not only that, the interactivity of CDF it makes it much easier for the reader to understand what you are saying/doing. My previous post contains some practise runs at using CDF, though I'm sure I will do better after I have studied the Wolfram's examples of CDF use.

What about long-term archiving of material? Will CDF be around in 10 (or 100) years? As far as I know, the only "complete" and "open" document format with a long track record is TeX/LaTeX, so that is my preferred choice to ensure my place on the "dusty shelf" in perpetuity. Sometimes, I even print things out on paper!

Interactive Demonstrations - Computable Document Format

2011-07-19T18:20:00.003Z

Wolfram Research supplies a free plug-in for viewing and interacting with online documents saved from Mathematica in Computable Document Format (CDF) - the plug-in can be downloaded from here. I thought that I would try it out on a few of my interactive Mathematica demonstrations, and here are the (draft) results for you to enjoy:

Topographic String: This grows a 1-dimensional self-organising map, starting from 3 nodes and progressively inserting additional nodes. It is an implementation of the SOM training method that I published in 1988: "Self-organising multilayer topographic mappings", Proceedings of 2nd International Conference on Neural Networks (San Diego, USA), pp. I/93-I/100 - an online version is available here.
Ising Model: This simulates a 2-dimensional Ising model, and it allows you to dynamically vary the clique factors for the 4 distinct types of 2-clique (i.e. N/S, E/W, NE/SW, SE/NW) to see how the Ising model behaves. It is an implementation of the Ising model simulations described in a report that I wrote in 1985: "The implications of Boltzmann-type machines for SAR data processing: a preliminary survey", RSRE technical report, 3815 - an online version is available here.
BZ Reaction: This simulates the Belousov–Zhabotinsky reaction-diffusion system - it's very pretty.
Current Algebra: This interactively computes commutators of products of current operators - it's a bit inflexible, but it shows some interesting Mathematica techniques in action.
IMO 2011: This is hot off the press in response to Terence Tao's Minipolymath3 project: 2011 IMO, which plans to work on Q6 of the 2011 International Mathematics Olympiad. I have implemented an interactive version of Q6, so you can get an intuitive feel for the geometry involved. Update: I see that TT eventually decided to go with Q2 rather than Q6. Oh well, my interactive Q6 is fun to play with anyway.

Some More Unpublished Work

2010-12-22T09:18:00.003Z

Here are some more unpublished papers that I have uploaded to the arXiv:

Self-Organising Stochastic Encoders

The processing of mega-dimensional data, such as images, scales linearly with image size only if fixed size processing windows are used. It would be very useful to be able to automate the process of sizing and interconnecting the processing windows. A stochastic encoder that is an extension of the standard Linde-Buzo-Gray vector quantiser, called a stochastic vector quantiser (SVQ), includes this required behaviour amongst its emergent properties, because it automatically splits the input space into statistically independent subspaces, which it then separately encodes. Various optimal SVQs have been obtained, both analytically and numerically. Analytic solutions which demonstrate how the input space is split into independent subspaces may be obtained when an SVQ is used to encode data that lives on a 2-torus (e.g. the superposition of a pair of uncorrelated sinusoids). Many numerical solutions have also been obtained, using both SVQs and chains of linked SVQs: (1) images of multiple independent targets (encoders for single targets emerge), (2) images of multiple correlated targets (various types of encoder for single and multiple targets emerge), (3) superpositions of various waveforms (encoders for the separate waveforms emerge - this is a type of independent component analysis (ICA)), (4) maternal and foetal ECGs (another example of ICA), (5) images of textures (orientation maps and dominance stripes emerge). Overall, SVQs exhibit a rich variety of self-organising behaviour, which effectively discovers the internal structure of the training data. This should have an immediate impact on "intelligent" computation, because it reduces the need for expert human intervention in the design of data processing algorithms.

A Self-Organising Neural Network for Processing Data from Multiple Sensors

This paper shows how a folded Markov chain network can be applied to the problem of processing data from multiple sensors, with an emphasis on the special case of 2 sensors. It is necessary to design the network so that it can transform a high dimensional input vector into a posterior probability, for which purpose the partitioned mixture distribution network is ideally suited. The underlying theory is presented in detail, and a simple numerical simulation is given that shows the emergence of ocular dominance stripes.

Some Unpublished Work

2010-12-17T01:20:00.005Z

I have decided to upload some of my unpublished work to the arXiv:

Adaptive Cluster Expansion (ACE): A Multilayer Network for Estimating Probability Density Functions

We derive an adaptive hierarchical method of estimating high dimensional probability density functions. We call this method of density estimation the "adaptive cluster expansion" or ACE for short. We present an application of this approach, based on a multilayer topographic mapping network, that adaptively estimates the joint probability density function of the pixel values of an image, and presents this result as a "probability image". We apply this to the problem of identifying statistically anomalous regions in otherwise statistically homogeneous images.

Stochastic Vector Quantisers

In this paper a stochastic generalisation of the standard Linde-Buzo-Gray (LBG) approach to vector quantiser (VQ) design is presented, in which the encoder is implemented as the sampling of a vector of code indices from a probability distribution derived from the input vector, and the decoder is implemented as a superposition of reconstruction vectors, and the stochastic VQ is optimised using a minimum mean Euclidean reconstruction distortion criterion, as in the LBG case. Numerical simulations are used to demonstrate how this leads to self-organisation of the stochastic VQ, where different stochastically sampled code indices become associated with different input subspaces. This property may be used to automate the process of splitting high-dimensional input vectors into low-dimensional blocks before encoding them.

The Development of Dominance Stripes and Orientation Maps in a Self-Organising Visual Cortex Network (VICON)

A self-organising neural network is presented that is based on a rigorous Bayesian analysis of the information contained in individual neural firing events. This leads to a visual cortex network (VICON) that has many of the properties emerge when a mammalian visual cortex is exposed to data arriving from two imaging sensors (i.e. the two retinae), such as dominance stripes and orientation maps.

Yawn, stretch, ...

2010-12-08T02:15:00.001Z

It's time to wake things up again ...

Second Life for Virtual Conferences

2008-11-28T18:55:00.006Z

Continuing the theme I blogged about recently (see Second Life for Science and Scholarship), here is an example of a virtual conference that will be held in Second Life:

Virtual Conference on Climate Change and CO₂ Storage

The organisers of this particular conference have an interest in getting the conference delegates to the "venue" with the minimum of travelling, so organising a virtual conference is the obvious choice.

The trend towards having virtual conferences is in its early stages, but there will be a lot more of this sort of thing in the future. There are many conferences that you would like to attend in person, but which would involve a lot of travelling/expense/fatigue/etc so you don't bother going. I can think of many annual conferences that fall into this category for me, but then I hate travelling. Perhaps there could be some sort of hybrid real/virtual conference to allow such people to attend conferences that would otherwise be difficult to attend. Of course, a purely virtual conference would be much easier to organise, and would present a very low barrier to attendance.

Currently, the main inhibiting factors working against the adoption of virtual conferences are unfamiliarity with the possibilities offered by the virtual medium, low quality of virtual reality compared to real reality, lack of communication cues that are only available in interactions between real humans, and so on. I would have thought that all of these inhibitors would reduce with time, so virtual conferencing will inevitably take off sooner or later.

The so-called "conference call", where multiple participants connect their telephones to have a multiway conversation (if it works at all!), will seem positively archaic in comparison with virtual reality.

Mathematica 7

2008-11-19T11:16:00.013Z

Mathematica 7 is released today, and its new features are summarised here. Hang on! I haven't yet mastered all of the new features that were added in Mathematica 6 (see here).

The Mathematica "universe" is growing so large that I find that there is a dynamic equilibrium between the things that I learn about it and the things that I forget, so I can never hold it all simultaneously in my head. I wonder if anybody understands it all.

Anyway, for those of you who don't already know, Mathematica is a "tool of thought" that raises your consciousness to levels that you didn't think were possible. But it does require a lot of practise to become a master of this art.

Update (20 November 2008): Something that caught my eye in the list of new features of Mathematica 7 was "Multicore parallelism standard with zero configuration on all versions of Mathematica" (see here). What this means is that when you run Mathematica 7 on a multicore computer (these days, all new computers are multicore) it can parallelise across the cores. In the basic version of Mathematica 7 you can have a maximum of 4 cores running in parallel (see here), which allows you to have 1 master and 3 slave processes, which gives a useful degree of parallelism straight out of the box. This parallel processing capability will be very useful when applied to the image processing capabilities of Mathematica 7 (see here).

Update (24 November 2008): It just keeps getting better! Running Mathematica on your own personal super-computer (for a reasonable cost, that is) will be reality not that far in the future judging by the following announcements:

Update (2 December 2008): Some gratuitous showing off of the image processing capabilities of Mathematica 7 has been published at The Incredible Convenience of Mathematica Image Processing on the Wolfram blog.

The Multiverse

2008-11-13T15:07:00.002Z

Discover Magazine has published a very useful article on the multiverse entitled Science's Alternative to an Intelligent Creator: the Multiverse Theory. The article is entirely non-technical, but it is well written and it shows how the various aspects of physics which are relevant to cosmology are interrelated. It is a good read that I would recommend to anyone who is interested in the big picture.

Adopt a Book

2008-11-08T11:09:00.003Z

The British Library has set up an Adopt a Book scheme in which you select a book to "adopt", provided you make a donation in support of the British Library's book conservation programme.

The benefits of adopting a book are tied to the size of your donation, and a cumulative list is as follows:

£25+: An attractive personalised certificate recording the beneficiary’s name and details of the book
£75+: A voucher for a public tour of the British Library for two people
£150+: A bookplate containing your personal dedication added to the book
£250+: An invitation for two people for special behind-the-scenes tour of the conservation studios, including the chance to ‘meet’ your book
£500+: The addition of the your name on the Adopt a Book Benefactor List in the British Library, and acknowledgement in the Annual Report
£1,000: If you would like to adopt a book which doesn’t appear on the list, we can offer a ‘choose your own book’ option for gifts of £1,000 or more. You will also enjoy all of the benefits listed above.

Amongst the 200 books that are currently available for adoption are some of your favourites, ranging from the profound "Philosopiae Naturalis Principia Mathematica" (3rd edition, 1739) by Sir Isaac Newton, to the tedious "A Law Dictionary" (1839) by John Bouvier. They even offer a list of gift ideas for Christmas which includes (for the children) "Alice's Adventures in Wonderland" (1908) by Lewis Carroll and "Aesop's Fables" (1666) by Aesop, and the venerable (take one average-sized cow, and stew it for a week) "Mrs Beeton’s Family Cookery and Housekeeping Book" (1907) by Mrs Beeton.

Proof by Computer

2008-11-07T17:00:00.006Z

The Notices of the American Mathematical Society has published A Special Issue on Formal Proof in mathematics, which is freely available online. There is a report on this by PhysOrg at Proof by computer: Harnessing the power of computers to verify mathematical proofs.

There are 4 articles in the Special Issue:

Formal Proof, by Thomas Hales
Formal Proof - The Four-Colour Theorem, by Georges Gonthier
Formal Proof - Theory and Practice, by John Harrison
Formal Proof - Getting Started, by Freek Wiedijk

The last paragraph of "Formal Proof - Getting Started" reads as:

However, having mathematics become utterly reliable might not be the primary reason that eventually formal mathematics will be used by most mathematicians. Formalisation of mathematics can be a very rewarding activity in its own right. It combines the pleasure of computer programming (craftsmanship, and the computer doing things for you), with that of mathematics (pure mind, and absolute certainty). People who do not like programming or who do not like mathematics probably will not like formalisation. However, for people who like both, formalisation is the best thing there is.

Clearly, formalisation is "geek heaven"!

The Wikipedia page on Formal Proof is a useful place to start learning the basic concepts. Informally, the idea of "formal proof" is that you replace error-prone human mathematicians by error-free computers, which are then used to expand each step of a (human-generated) proof all the way down to the fundamental axioms of mathematics. Naturally, this leads to extremely verbose formal proofs, but computers are ideally suited to handling this verbosity, and the advantage for us humans is that we can ensure that our proofs are error-free, because they have been checked by computer in every detail. Of course, we might have neither the time nor the inclination to fully "understand" the details of these proofs.

The following is a verbatim copy of the main part of a posting of mine Burden of Proof that I wrote over 2 years ago on my ACEnetica blog. It is very relevant to the issue of "formal proof" which is why I have included it here.

My own view on this issue is that a computer generated proof has exactly the same status as a human generated proof. The difference is only one of the degree of assistance provided to the brain of the human to help with the generation of the proof. A totally unassisted human would have to somehow do the whole proof mentally, which severely limits the length of proofs that are accessible. A human with the typical assistance that is allowed in an examination room (i.e. pen and paper) has the luxury of at least being able to write things down, which allows much longer proofs to be reliably generated. The mechanics of generating a proof then reduce to using well-defined rules to manipulate symbolic expressions, where pen and paper are used as a medium for representing these symbols, and the rules are implemented in the human brain.

The degree of assistence in generating a proof can be taken one stage further by using a computer to implement some or all of the rules for manipulating the symbolic expressions, rather than implementing all of the rules in the human brain. This seems to be a fairly radical step to take, because hitherto the only part of the proof that was "outside" the human brain was its "dumb" representation using pen and paper, whereas the "clever" bit involving the implementation of rules to manipulate this representation was "inside" the human brain.

Let us consider what these rules of manipulation actually are. Effectively, they define a procedure for taking an initial expression constructed out of symbols, and repeatedly operating on it using the rules to eventually generate the required final expression. The cleverness is in the construction of the set of rules, which is where a human is the best source of the cleverness needed to create the rules. There is no cleverness in the repeated application of these rules; all that is required is that their application is done reliably, which is where a computer is the best approach, especially if the proof has many steps.

Use a human to define the rules of manipulation, and use a computer to implement these rules. This approach seems to me to be entirely uncontroversial, and it is exactly how computer generated proofs are done. Note that software bugs in the computer part of the proof are dealt with in an analogous way to "software" bugs in human part of the proof, i.e. try a variety of approaches on a variety of platforms.

Second Life for Science and Scholarship

2008-11-03T18:45:00.004Z

Demonstration of a Lorenz Attractor in Second Life.

I created the little video above as a simple example of how you can implement a dynamical 3D model in Second Life. All you need to do make a rudimentary demonstration of a Lorenz attractor is to create a set of particles in SL, and to embed a script inside each of the particles to tell it how to move according to the equations that govern the Lorenz attractor. The simulation itself is then automatically carried out by the SL virtual reality engine, whilst you move your virtual camera around the simulated Lorenz attractor in order to film a demonstration. That's all there is to creating the rather basic video that I posted above.

Some additional points:

Each particle's motion leaves behind it a trail of "hot embers" that gradually cools off yellow/orange/red until it vanishes. This traces out the Lorenz attractor so we can easily see it.
In this example I moved the camera manually rather than by scripting its motion, so the camera motion is rather clumsy.
I had planned to include a voice-over commentary, but found that all my attention was needed just to operate my mouse and keyboard, so all you can hear is the occasional mouse-click.
The background scenery is not actually relevant to this demonstration, which I performed in a small corner of my cliff-top land holding in Second Life. But maybe you can see a few objects of interest in the background.
The almost invisible translucent motion in the background is an animated movie that I am displaying on a large screen I built in SL. More to come later on this...

That leads me onto the main subject of this posting.

George Djorgovski, Professor of Astronomy at Caltech, has a guest post at Cosmic Variance in which he vividly describes his experiences in using the virtual world Second Life for science and scholarship. To those who think that SL is just a game he offers the following advice:

Judging by my own experience, there is no way that you can really understand all this just by reading or listening; you have to try it. It is a fundamentally visceral, as well as an intellectual experience. It is as if you have never seen a bicycle, let alone ridden one, and someone was showing you pictures of people having a good time biking around, and telling you what a fun it is. Please keep that in mind. You gotta try it, then judge for yourself.

On the quality of the virtual experience he writes:

What really surprised me; knocked my virtual socks off, so to speak; is the subjective quality of the interpersonal interaction. Even with the still relatively primitive graphics, the same old flat screen and keyboard, and a limited avatar functionality, it is almost as viscerally convincing as a real life interaction and conversation. Somehow, our minds and perceptive systems interpolate over all of the imperfections, and it really clicks. I cannot explain it; it has to be experienced; it is not a rational, but a subjective phenomenon. It is much better than any video- or teleconferencing system I have tried, and like most of you, I have suffered through many of those. As a communication device, this is already a killer app. Going back to the good old email and Web feels flat and lame.

On the use of SL for science and scholarship he writes:

So the first major scholarly use of [virtual worlds] is as a communication, interaction, and collaboration venue. This includes individual, group, or collaboration meetings, seminars, or even full-blown conferences. You can interact with your colleagues as if they were in the same room, and yet they may be half way around the world.

And he writes much more about how virtual worlds in general (and Second Life in particular) are a key technology in the future of science and scholarship. Commentary, such as this by George Djorgovski, on the serious (rather than gaming) use of Second Life is to be welcomed.

I never have travelled well, typically arriving at conferences totally knackered and not recovering for days, so I look forward to virtual meetings becoming the norm, at least for short meetings, that is. Also, I have a highly visual way of explaining science (to myself and to others), so I look forward to building illustrative 3D dynamical models in SL. I think a key technology that is missing here is ready access to a higher-level set of tools for building and scripting such models in SL, at least that is what I see as being the main thing that is slowing down my progress in using SL.

This is only the start of what is to come...

Martin Gardner Mathematical Library

2008-10-25T15:14:00.007Z

I have just received a flyer from the Cambridge University Press advertising The New Martin Gardner Mathematical Library. Of course, the name Martin Gardner immediately attracted my attention (isn't it so useful to have a widely recognised name?), because I immediately thought of his excellent Mathematical Games column that used to appear in Scientific American. It says here that his column stopped being published in 1981 - was it that long ago?

Anyway I clicked through to The New Martin Gardner Mathematical Library to discover that it is exactly what I thought it might be, i.e. an updated version of his Mathematical Games column. The library is described thus:

The books based on Martin Gardner's enormously popular Scientific American columns and puzzles continue to challenge and fascinate readers. In these new editions, the author, in consultation with experts, has written updates to all the chapters, including new game variations, new mathematical proofs, and connections to recent developments and discoveries. New diagrams and illustrations have been added and old ones improved, and the bibliographies have been greatly expanded throughout.

The web page looks unfinished, but it gives at least some of the titles that will be in the library, which I list below with links that I have added for convenience:

1. Hexaflexagons, Probability Paradoxes [I have linked to the Monty Hall problem as an example of this genre], and the Tower of Hanoi
2. Origami, Eleusis, and the Soma Cube
3. Sphere Packing, Lewis Carroll, and Reversi

It looks like the sort of good stuff that will provoke those familiar mental gymnastics of yore.

Many Worlds Theory

2008-10-23T13:15:00.012Z

Nova has a nice collection of information here about Hugh Everett's so-called Many Worlds Theory of quantum mechanics. The package includes a letter from Everett to Bryce DeWitt explaining the basic concepts underlying his theory, and the published version of Everett's PhD dissertation. It's fascinating stuff that I highly recommend.

For a long time I have had an affinity for Everett's theory, but I didn't find out about Everett's work until long after I had discovered "Many Worlds Theory" for myself whilst doing my PhD work (circa 1980) in high energy particle physics. The reasoning that led me to this theory was to try to see the world from the "point of view" of a simple QM system (e.g. a fundamental particle), and to then work upwards in complexity towards ever larger QM systems.

The only way a fundamental particle can "see" the world is to exchange particles with it, and QM does this by progressively applying (the infinitesimal version of) the evolution operator exp(i H t), which is a unitary operator that rotates the system state (e.g. scattering/creating/annihilating particles) in a norm-preserving way (i.e. probability conserving). This leads to a QM description of the world in which there are physical processes going on "in parallel", where all the alternative processes that can be generated by exp(i H t) actually do occur simultaneously. QM (unlike classical physics) automatically does parallel processing at each and every point of space-time, which is where the processing power of a quantum computer comes from.

Working upwards towards larger QM systems involves no change in the theory (that we know of, that is) because the evolution operator exp(i H t) can be applied to any state no matter how complicated it is. There is no system "size" above which the physics is fundamentally different from what is already known to be correct at the level of elementary particles. This includes the use of effective degrees of freedom, because these are still governed by the underlying exp(i H t) although many of the details are usually hidden from view; I reserve the right to revise my opinion here having now seen the paper More Really is Different.

Carried on to physically large system sizes (e.g. human brains), this line of reasoning inevitably leads to a "Many Worlds Theory" point of view, where it is QM all the way up from the bottom to the top. We are inside a QM universe, not outside it looking in.

I need direct experimental evidence for "non-QM physics" (i.e. evidence that exp(i H t) is not the whole story) in order to discard my assumption that it is QM all the way from bottom to top. Isn't that the way science should normally be done (I innocently ask), where you preserve the status quo until experimental evidence contradicts it?

Circa 1980 I was on the receiving end of a lot of criticism from physicists around me, but in the interests of self-preservation I then decided to keep quiet about my contrarian thoughts on QM. It was only many years after completing my PhD (and moving to another research field outside QM, but continuing to think about QM) that I finally realised that I had not been the first person to think of these ideas. Duh!

Virtual Forbidden City

2008-10-13T15:01:00.009Z

IBM and Palace Museum announce the opening of the Forbidden City Virtual World celebrating 600 years of Chinese culture (see here). The Virtual Forbidden City website says:

The Virtual Forbidden City is a 3-dimensional virtual world where visitors from around the world can experience the Forbidden City in Beijing. You can explore the magnificient palace as it was during the Qing dynasty, which ruled from 1644 until 1912, the end of the Imperial period in China.

The image above shows a location that I "photographed" on my first visit to the Virtual Forbidden City. There is much more than can be seen in this single photograph.

This is not a fully featured virtual world (e.g. Second Life), but it is good enough for visiting and familiarising yourself with the Forbidden City. This virtual reconstruction of the Forbidden City has been done quite carefully. The in-world objects have been "painted" with textures that appear to have been derived from photographs of their real-world counterparts, which adds to the realism. This is quite hard work to do properly, especially for irregularly shaped objects, as I have found when creating virtual copies of real-world objects in Second Life.

Sustainable Energy - without the hot air

2008-09-26T11:05:00.007Z

David MacKay (Professor of Natural Philosophy, Department of Physics, University of Cambridge) has just finished writing his book Sustainable Energy - without the hot air. The online version of the book is free.

You can learn what the purpose of the book is from this extract quoted from the book's preface:

I’m concerned about cutting UK emissions of twaddle – twaddle about sustainable energy. Everyone says getting off fossil fuels is important, and we’re all encouraged to “make a difference,” but many of the things that allegedly make a difference don’t add up.

Twaddle emissions are high at the moment because people get emotional (for example about wind farms or nuclear power) and no-one talks about numbers. Or if they do mention numbers, they select them to sound big, to make an impression, and to score points in arguments, rather than to aid thoughtful discussion.

This is a straight-talking book about the numbers. The aim is to guide the reader around the claptrap to actions that really make a difference and to policies that add up.

Nice one! I would recommend this book to anyone who wants to base their knowledge about sustainable energy on science rather than hot air.

Methane Bubbles in the Arctic Bathtub

2008-09-24T18:53:00.005Z

The Independent has an alarming report on a potential methane time bomb. The sub-sea deposits of methane beneath the Arctic are beginning to bubble to the surface as the region warms up and the ice retreats. Methane is a very potent greenhouse gas, and its past release from deposits has been suggested as the cause of abrupt changes in the past global climate. If these observations are confirmed, and if there is found to be a positive feedback loop driving the effect, then it would be rather bad news for the projected rate of climate change.

2008 Dirac Medal - Institute of Physics

2008-09-18T17:07:00.009Z

The Institute of Physics has awarded its 2008 Dirac medal to Bryan Webber who was my PhD supervisor at the Cavendish Laboratory circa 1980.

Congratulations!

The brief version of the citation is:

For his pioneering work in understanding and applying quantum chromodynamics (QCD), the theory of the strong interaction which is one of the three fundamental forces of Nature.

The full citation is:

The Dirac medal of the Institute of Physics for outstanding contributions to theoretical, mathematical and computational physics has been awarded to Professor Bryan R. Webber, Professor of Theoretical Physics at the University of Cambridge, for his pioneering work in understanding and applying quantum chromodynamics (QCD), the theory of the strong interaction which is one of the three fundamental forces of Nature.

The strong force is felt by quarks, the constituents of protons and neutrons, and is carried by gluons which themselves interact via the strong force. To verify that the theory is correct requires being able to make accurate predictions of its consequences in particle physics experiments. Since the interactions are complex, this represents a formidable challenge.

Professor Webber is recognised worldwide as having a profound understanding of QCD - from which he has derived key practical numerical tools for extracting quantitative information from high-precision experimental data. Over the past 20 years, these tools have been used in high-energy experiments around the world, for example, in the Large Electron-Positron Collider at CERN.

Webber proposed a number of successful models that show what happens during high-energy particle collisions, for example, the break-up of quarks into jets of other particles. He developed powerful algorithmic approaches that not only allow much more accurate interpretation of particle events but also provide theoretical insights into the complexities of QCD. His work led to the theoretical consolidation of QCD, as recognised by the ensuing award of the Nobel Prize to the originators of the theory.

Recently, Webber performed ground-breaking work on the phenomenology associated with the kind of physics that will be explored in the very high energy proton-proton collisions shortly to begin at the Large Hadron Collider at CERN. Professor Webber’s contributions to our understanding of the fundamental properties of matter have been invaluable, as revealed by the large number of citations of his published research.

I notice that the winner of the 2008 Dirac Medal (Institute of Physics) appeared in Wikipedia on 8th October 2007, so this blog posting of mine brings year-old news to you. My apologies for this oversight.

I also notice on Wikipedia that the winner of the 1987 Dirac Medal (Institute of Physics) was Stephen Hawking, who was my brother Julian's PhD supervisor, so we are now both "descended" from Dirac Medallists.

Luttrell Psalter

2008-09-09T19:30:00.007Z

I've just got around to looking at a book that I bought in early 2007. The photo shows the front cover of the book, which immediately suggests the reason that I bought it. It is 36cm high by 25cm wide, it is 7.5cm thick, it weighs over 5kg, and it is by far the largest book that I possess.

It is a facsimile copy of the Luttrell Psalter, which was written and illuminated during the second quarter of the 14th century, and is famed as a source of pictorial information about everyday life during the Middle Ages. A small sample of this can be seen in the photo above.

The original project to create the Luttrell Psalter was very expensive in both time and money. It was commissioned by Sir Geoffrey Luttrell who ensured that an image of him and his family appeared in the book, which guaranteed that his name would never be forgotten, as no doubt he intended.

The Luttrell Psalter will be a great source of pictures for me to write about in this blog.

Scientific Linux - navigation blocked

2008-09-04T13:13:00.007Z

I was browsing the Scientific Computing World article A Universe of Data on the computing resources at CERN, when my attention was caught by mention of a version of Linux that I had not heard of before called Scientific Linux. What's so special about that? So I duly Googled the string "Scientific Linux" and the top hit was www.scientificlinux.org described as:

Scientific Linux - Welcome to Scientific Linux (SL) - 13:58Is a Linux release put together by Fermilab, CERN, and various other labs and universities around the world ready tuned for experimenters.
www.scientificlinux.org/ - 26k - Cached - Similar pages - Note this

So I then followed the www.scientificlinux.org link to get this:

I have never seen this sort of warning before, and I do a lot of internet browsing. I didn't explore any further because unexpected things on the internet always spook me, and by playing very safe I have managed to avoid all the nasty problems that I regularly hear about from other people.

I am reasonably sure that the problem is a trivial misconfiguration of the www.scientificlinux.org site, rather than a malicious attempt by Internet Explorer to try to prevent people from visiting this particular site. Surely, it couldn't be the case that a www.scientificlinux.org is so self-righteous that they shoo away Internet Explorer users? Or am I just being paranoid?

Dropping the Baton

2008-08-24T14:51:00.004Z

There seem to have been rather a lot of dropped batons in the relay races at the Olympics (e.g. see here).

It set me thinking about where I might have seen this sort of thing happening elsewhere, and I realised that dropping the baton is like annihilating the vacuum state.

How so?

The simplest possible algebra that one can use to model the process of baton-passing goes like this:

a^† increments (by 1) the number of hands holding the baton
a decrements (by 1) the number of hands holding the baton

|0> is the "vacuum" state where the baton has one hand holding it
a^†|0> is the state where the baton has two hands holding it

a|0> = 0 is the annihilation of the vacuum where the baton has zero hands holding it, i.e. a state from which there is no way to recover

Note that it is important to define the vacuum state as corresponding to one (rather than zero) hand holding the baton, otherwise the algebra (i.e. annihilation of the vacuum) doesn't correctly model the dropping of the baton. Thus the counting of hands holding the baton is really a measure of how many excess hands are holding the baton, because the case of one hand is actually the ground (or vacuum) state in a relay race.

Most of the time the state is |0>, and during a successful handover of the baton it passes through the transition state a^†|0>, after which it returns to the state |0>. However, during an unsuccessful handover of the baton it goes to the state a|0> which is 0, where the vacuum has been annihilated.

Successful handover: a a^†|0> = |0>
Unsuccessful handover: a^†a|0> = 0

The order in which the a and a^† operations are applied is important, and is neatly summarised by how their commutator a a^† - a^†a acts on |0> (take the difference of the above equations).

(a a^† - a^†a)|0> = |0>

A stronger form of this result is the operator relation

a a^† - a^†a = 1

This relation takes note of the fact that there are n ways of applying a to the state (a^†)ⁿ |0> (i.e. choose from 1 of n excess hands to decrement by 1 the number of excess hands holding the baton), but there is only 1 way of applying a^† to the state (a^†)ⁿ |0>. The case n=0 is when the vacuum gets annihilated by application of a.

The Olympic athletes who dropped the baton were the victims of a a^† - a^†a = 1 (rather than 0). I wonder whether they saw it that way.

Ethel the Aardvark Goes Quantity Surveying

2008-08-24T13:42:00.005Z

Here is some Sunday afternoon entertainment.

There was a "cheese shop sketch" before the famous Cheese Shop Sketch that we all remember. I was reminded about it whilst browsing the Wikipedia entry for Marty Feldman.

Here it is (text copied from here). The customer is played by Marty Feldman and the shop assistant by John Cleese.

Assistant: Good morning, sir.
Customer: Good morning. Can you help me? Do you have a copy of 'Thirty Days In the Samarkand Desert with a Spoon' by A.E.J. Elliott?
Assistant: Um ... well, we haven't got it in stock, sir.
Customer: Never mind. How about 'A Hundred and One Ways to Start a Monsoon'?
Assistant: ... By ... ?
Customer: An Indian gentleman whose name eludes me for the moment.
Assistant: I'm sorry, I don't know the book, sir.
Customer: Not to worry, not to worry. Can you help me with 'David Copperfield'?
Assistant: Ah, yes. Dickens ...
Customer: No.
Assistant: ... I beg your pardon?
Customer: No, Edmund Wells.
Assistant: ... I think you'll find Charles Dickens wrote 'David Copperfield', sir.
Customer: No, Charles Dickens wrote 'David Copperfield' with two 'p's. This is 'David Coperfield' with one 'p' by Edmund Wells.
Assistant: (a little sharply) Well in that case we don't have it.
Customer: Funny, you've got a lot of books here.
Assistant: We do have quite a lot of books here, yes, but we don't have David Coperfield' with one 'p' by Edmund Wells. We only have 'David Copperfield' with two 'p's by Charles Dickens.
Customer: Pity - it's more thorough than the Dickens.
Assistant: More thorough?
Customer: Yes ... I wonder if it's worth having a look through all your 'David Copperfields'...
Assistant: I'm quite sure all our 'David Copperfields' have two 'p's.
Customer: Probably, but the first edition by Edmund Wells also had two 'p's. It was after that they ran into copyright difficulties.
Assistant: No, I can assure you that all our 'David Copperfields' with two 'p's are by Charles Dickens.
Customer: How about 'Grate Expectations?
Assistant: Ah yes, we have that ...
He goes to fetch it and returns to the counter.
Customer: ... That's 'G-r-a-t-e Expectations', also by Edmund Wells.
Assistant: I see. In that case, we don't have it. We don't have anything by Edmund Wells, actually - he's not very popular.
Customer: Not 'Knickerless Nickleby'? That's K-n-i-c-k-e-r
Assistant: No!
Customer: Or 'Quristmas Quarol 'with a Q?
Assistant: No, definitely ... not.
Customer: Sorry to trouble you.
Assistant: Not at all.
Customer: I wonder if you have a copy of 'Rarnaby Budge'?
Assistant: (rather loudly) No, as I say, we're right out of Edmund Wells.
Customer: No, not Edmund Wells - Charles Dikkens.
Assistant: Charles Dickens?
Customer: Yes.
Assistant: You mean 'Barnaby Rudge'.
Customer: No, 'Rarnaby Budge' by Charles Dikkens ... that's Dikkens with two 'k's, the well-known Dutch author.
Assistant: No, no - we don't have 'Rarnaby Budge' by Charles Dikkens with two 'k's the well-known Dutch author, and perhaps to save time I should add right away that we don't have 'Carnaby Fudge' by Daries Tikkens, nor 'Stickwick Stapers' by Miles Pikkens with four Ms and a silent Q, why don't you try the chemist?
Customer: I did. They sent me here.
Assistant: (making a mental note) ... Did they?
Customer: I wonder if you have ... 'The Amazing Adventures of Captain Gladys Stoat-Pamphlet and her Intrepid Spaniel Stig among the Giant Pygmies of Corsica', Volume Two.
Assistant: No, we don't have that one. Well, I mustn't keep you standing around all day ..
Customer: I wonder if ...
Assistant: No, no, we haven't got it. I'm closing for lunch now anyway.
The assistant moves rapidly away from the counter.
Customer: ... But I thought I saw it over there.
The assistant checks and turns slowly.
Assistant: ... What?
Customer: Over there.
He indicates a bookshelf.
Customer: 'Olsen's Standard Book of British Birds'.
Assistant: (very suspiciously) 'Olsen's Standard Book of British Birds'?
Customer: Yes.
Assistant: ... 0-l-s-e-n?
Customer: Yes!
Assistant: B-i-r-d-s?
Customer: Yes!
Assistant: Well, we do have that one, yes.
He goes and takes the book off a shelf.
Customer: ... The expurgated version, of course.
Assistant: ... I'm sorry, I didn't quite catch that.
Customer: The expurgated version.
Assistant: The expurgated version of 'Olsen's Standard Book of British Birds'?
Customer: Yes. The one without the gannet.
Assistant: The one without the gannet?! They've all got the gannet it's a standard bird, the gannet, it's in all the books.
Customer: Well I don't like them. They've got long nasty beaks! And they wet their nests.
Assistant: But ... but you can't expect them to produce a special edition for gannet-haters!
Customer: I'm sorry, I specially want the one without the gannet.
The assistant is speechless.
Assistant: All right!
He suddenly tears out the relevant page.
Assistant: Anything else?
Customer: Well, I'm not too keen on robins.
Assistant: Right! Robins, robins ...
He tears that one out too and slams the book on the counter.
Assistant: No gannets, no robins - there's your book!
Customer: I can't buy that. It's torn.
Assistant: ... So it is! He tosses it into the bin.
Customer: I wonder if you've got ...
Assistant: Go on! Ask me another.
Customer: How about 'Biggles Combs his Hair'?
Assistant: No, no, we haven't got that one, funny. Try me again.
Customer: 'The Gospel According to Charlie Drake'?
Assistant: No ...
Customer: Have you got 'Ethel the Aardvark Goes Quantity-Surveying'?
Assistant: No, no, we haven't ... which one?
Customer: 'Ethel the Aardvark Goes Quantity-Surveying'.
Assistant: 'Ethel the Aardvark'?! I've seen it! We've got it!!
He dashes to a bookshelf, finds it, and holds it up triumphantly.
Assistant: Here! Here!!! 'Ethel the Aardvark Goes Quantity Surveying'. Now - buy it!
He slams it on the desk. The customer stares in horror!
Customer: ... I haven't got enough money on me.
Assistant: (quickly) I'll take a deposit!
Customer: I haven't got any money on me.
Assistant: I'll take a cheque!
Customer: I haven't got a cheque book!
Assistant: It's all right, I've got a blank one!
Customer: I don't have a bank account!!
Assistant: ... All right!! I'll buy it for You!
He rings the purchase up and pays for it himself. He gives the change to the customer.
Assistant: There we are, there's your change - that's for the taxi home ...
Customer: Wait! Wait! Wait!
Assistant: What? What? What?!!!
Customer: ... I can't read ...
Assistant: Right! Sit!! ...
He sits the customer down on his knees and starts to read aloud.
Assistant: 'Ethel the Aardvark was trotting down the lane one lovely summer day, trottety-trottety-trot, when she saw a nice Quantity-Surveyor ...

Twerp Bollickagh - the website

2008-08-24T11:10:00.008Z

Twerp Bollickagh

2008-08-15T18:02:00.006Z

It could be the title of a journal!

I suspect that many of the experimentally accurate theoretical "predictions" given in a "Grand Unified Theory" that is available online (search for the string "The calculated relations between the lepton masses") were arrived at by exhaustive numerology, i.e. by searching through a large number of simple expressions to find the ones that gave the required results. Then a "proof" of each of these results was reverse-engineered using pseudo-physical explanations rather than using rigorous maths.

Let me show you an example of what I mean.

This "GUT" gives some simple expressions for various mass ratios. There is even an expression for the ratio of the neutron mass to the electron mass, which depends only on the electomagnetic coupling strength (i.e. fine structure constant) and not on the strong interaction strength. How do the quarks and gluons in the neutron know how to interact in order to give this amazing result?

The expression given by this "GUT" for the muon to electron mass ratio (i.e. mμ/me) is

(α^(-2) / 2π)^(2/3) (1 + 2π α^2 / 2) / (1 + α/2)

which produces a value 206.76828 that closely corresponds to the experimentally observed value 206.76827.

Let's see whether it is possible to "derive" this result by an exhaustive search of all simple expressions of this general type. The parameterisation that I will use is the most general form that is suggested by the mass ratio quoted above

f[{a1, a2, a3, a4}, {b1, b2, b3, b4}, {c1, c2, c3, c4}, {d1, d2, d3, d4}, α]] =
(a1/a2)^(a3/a4) (α)^(b1/b2) (2π α^2)^(b3/b4) (1 + (c1/c2)(α) + (c3/c4)(2π α^2)) / (1+ (d1/d2)(α) + (d3/d4)(2π α^2))

where all of the parameters are integers which are grouped in pairs to form rational fractions. To compute numerical results I inserted specific values for these parameters (avoiding singular cases), I use α=0.00729735, and a target mass ratio mμ/me=206.76827.

I then computed f[{a1, a2, a3, a4}, {b1, b2, b3, b4}, {c1, c2, c3, c4}, {d1, d2, d3,
d4}, α]] for all parameter values in the following small ranges (I have been rather cavalier and restricted the ranges to save time):
{a1, 1, 2}, {a2, 1, 2}, {a3, 1, 3}, {a4, 1, 3},
{b1, 0, -3, -1}, {b2, 1, 3}, {b3, 0, -3, -1}, {b4, 1, 3},
{c1, 0, 2}, {c2, 1, 2}, {c3, 0, 2}, {c4, 1, 2},
{d1, 0, 2}, {d2, 1, 2}, {d3, 0, 2}, {d4, 1, 2}
There is some repetition of trial solutions here, but this doesn't matter.

I then selected from this large set of trial solutions all of the cases that predicted a value for mμ/me that lay within 0.01 of the target value, and here they are (in decreasing order of goodness of fit) with the prediction errors shown in square brackets:

(1/(2π α^2))^(2/3) (1 + π α^2) / (1 + α/2) [0.0000110213]
(1/(2π α^2))^(2/3) (1 + 2π α^2) / (1 + α/2 + π α^2) [0.000130968]
(1/(2π α^2))^(2/3) (1 + α/2 + π α^2) / (1 + α) [0.00261802]
(1/(2π α^2))^(2/3) (1 + α/2 + 2π α^2) / (1 + α + π α^2) [0.0027371]

The best fit solution at the top of this list is the same as the one given by the "GUT".

What do we conclude from this little exercise?

It is really easy to do exhaustive searches to find best-fit solutions. The above fit works as well as it does because it starts with two different quantities (α) and (2π α^2) (where π is not a rational fraction), and combines them in various ways using lots of rational fractions to tailor the combination, which then leads to a dense set of candidate solutions from which the best-fit solution can then be picked.

Unless you happened to pick the physically correct parametric form to search over (Balmer got lucky with atomic spectra, but that is not to be used as a justification for this approach), then there is no physical significance to solutions that are obtained in this way. If you hedge your bets by searching over a large set of parametric forms, then you will almost certainly find many solutions that have a good fit to the target value, but this doesn't guarantee that any of them is physically significant. Interestingly, a related problem occurs in the context of the Landscape.

The approach used in this "GUT" is numerology, pure and simple. Of course, I only suspect that this is the way that the above expression for mμ/me was "derived"; I can't prove that this is the case.

Holiday in South West Cornwall

2008-08-06T14:49:00.014Z

I have been neglecting this blog; my previous posting was more than half a year ago. Not only that, but the broken basin that I described in my previous posting has still not been fixed!

I have just returned from a few weeks camping in South West Cornwall (the area known as West Penwith), which is an area that I enjoy visiting because it is like leaving your "baggage" at home and going away to "the edge of the world". Here is a small sample of some of the photographs that I took whilst I was there.

The West Penwith area is rather exposed to the Atlantic weather systems, so you have to pitch your tent away from bushes that look like this:

Much of the local granite is beautifully weathered. I wonder how long it takes for granite exposed to the elements to begun to look like this:

As usual, I tried to do a bit of maths to exercise my mind but I kept losing factors of π, so I ended up just banging the rocks together:

To satisfy my curiosity, I visited the famous tin mine engine houses at the Botallack Crowns Mine, which are much more precariously located than they seem in this photograph (someone needs to invent a simple photographic system that gives the viewer a full 3D spatial awareness, like a feel for the yawning chasm just in front of them):

I watched a bit of Cornish cricket at the Lafrowda Festival in St Just, which seems to be much more fun than the activity called "cricket" that I have seen on TV:

I did quite a bit of moorland walking, but I was never quite sure whether I was inside or outside the areas of open moorland, as this gate in the middle of nowhere illustrates:

There appears to be only a loose correspondence between the moorland paths marked on the Ordnance Survey map and the actual paths on the ground (I double checked my position using my GPS locator), so I sometimes found myself wading through a sea of gorse and heather that was up to waist high in places. This scenery was very pretty to look at, but it tore my legs to shreds.

There is a move to enclose the moors and to graze cattle, which has caused uproar amongst some of the local inhabitants who have started a Save Penwith Moors campaign. My preference is for open moorland scenery unspoilt by fencing and cattle, and I hope yours is too.

Update (21 December 2008): I have just noticed that the 9 Maidens Common has had a reprieve from the cattle grazing plans (see here). Excellent news!

Butterflys in my bathroom

2008-01-20T14:12:00.000Z

It has not been a particularly successful weekend for me plumbing-wise.

The butterfly flapped its wings.

My bathroom basin had a dripping tap, so I surmised that the tap needed a new washer. So, off I went to the local hardware store to buy said item, and returned to unscrew the top of the tap to fit the new washer. Unfortunately, the tap thread was completely frozen and no amount of fiddling about would shift it; I recall my plumber telling me that this tap had a problem several years ago. Never mind, I didn't like the taps anyway, so I decided to replace both taps on the basin, which would involve unscrewing the taps from below. So I applied some penetrating oil, and then went off to a large out-of-town hardware store where they had a good selection of taps, found what I wanted, and returned to fit them to my basin. I turned off the water at the main supply, and proceeded to use my under-the-sink wrench to loosen the nut connecting the water pipe to the tap, but rapidly discovered that the wrench's handle was far too short (and thin!) to apply sufficient torque to do the job, nor was it possible to attach anything to the handle to increase the torque. Off I went to the large hardware store again to buy a better under-the-sink wrench, which did the job once I inserted my hammer in its jaws to apply sufficient torque. Now all I needed to do was to detach the tap itself from the basin, so I applied the wrench again but found that the nut attaching the tap to the basin was frozen in place. To unfreeze it I reasoned that I could jiggle the tap backwards and forwards, using wrenches simultaneously above and below the basin. Jiggle, jiggle, wrench, curse, WRENCH ... crack!! The basin was now in several pieces, with cracks radiating from the tap that I had been working on. WTF happened? Oh well, the basin would now need to be replaced as well so I started to pull away the loose section behind the tap, and immediately cut my thumb by trapping it between the edges of two broken basin pieces. It was a deep cut about 2cm long and there was blood everywhere, so I had to retire for a while to mend my thumb. Later on, I returned to the basin to discover that the reason it had broken was that the part of the tap that passed through the basin had a square cross section, and the hole in the basin that it passed through was also square. Great!! Now I know that wrenching the tap around backwards and forwards was guaranteed to break the basin. Anyway, the tap was now free of the basin because I had smashed the basin, but there was enough of the basin left intact that I could still use it pending its replacement, so I put a new tap on the free end of the water pipe and gracefully dangled it over the broken edge of the basin. I turned the main water supply back on, so now I had a tap that didn't drip, but a basin that needed to be replaced.

The butterfly flapped its wings, and the puff of air developed into a gust of wind.

OK, so now I needed a new bashroom basin. I picked up a piece of the broken basin in order to use it as a colour swatch to match its nice pale blue colour to a new basin in the large out-of-town hardware store. Off I went to the store to find that all of their basins were white. Oh no! At first I assumed that they put the white basins on display and held a selection of coloured ones in the store room, but I rapidly discovered that they had only white. This was a large store, so this was as good as things would get for me. I realised with dawning horror that if I wanted a colour matched bathroom suite then I would need to replace the entire suite. I looked around nervously at the prices of whole suites, and realising that I would have to have it fitted professionally I mentally added in that cost as well. The total cost would not be less than £1000 in round figures. Perhaps I'll get used to having a broken bathroom basin! Maybe I could pass it off as an interesting new art form, and put a First Aid kit by the side of it for people to staunch their wounds when they cut themselves! No, that won't work. I'll have to spend loadsamoney on fixing this problem.

The butterfly flapped its wings, the puff of air developed into a gust of wind, and the gust of wind developed into a howling storm.

I should have got a plumber in to fix my dripping tap, but personal pride took over and made me attempt to do the job myself.

By the way, I have now found some specialist suppliers who sell discontinued coloured bathroom suites, so maybe I could replace only the basin, but I'll have a think about things to decide whether I might as well buy a whole suite anyway

NanoArt 2007 voting

2008-01-09T19:54:00.000Z

I blogged a couple of times before about NanoArt 2007, see here (background information on the competition) and here (my 5-frame animation competition entry).

The NanoArt 2007 voting is now open here (I don't know why the year 2006 appears in this link!) until 31 March 2008.

Here are the voting steps (you can go straight to vote for my entry here):

Click on the album thumbnail to open the album.
Click on the image thumbnail to view the image.
Click on the number of stars you would like to rate this image.

My entry is different from the others because it is an animated GIF that you view here; this link is quoted beneath the static GIF that is displayed on the NanoArt 2007 competition web page. Unfortunately, the static GIF looks really boring, so I don't have much hope that many people will discover that my entry is actually an animation. Never mind, at least I tried.

Update (17 January 2008):

It is fascinating to watch the votes accumulate. At each stage you can see the number of votes cast thus far and their average rating, so if you take a peek often enough you can deduce each vote as it is cast, provided that the total number of votes is not so large that there are rounding errors in the average rating.

There is obviously someone who is trying to "spike" my entry by voting with a rating 0/5, whilst all the other votes that I have received have a high rating. How pathetic is that?!

Although I am curious to observe the pattern of voting, it is not that important to me what score I get because I have already achieved my goal which was to create an animated nanoartform. I would be interested to hear of any prior examples of this artform.

Interpreting mathematics

2008-01-09T14:14:00.000Z

In the 5th January 2008 issue of New Scientist there is a short article by Mark Buchanan entitled "When it's time to sit back and think again", which discusses some of the results reported in an arXiv paper entitled "Symbolic manipulators affect mathematical mindsets" at http://arxiv.org/abs/0712.1187.

The phenomenon that is discussed in the paper is the tendency for people to switch off their brain when they use symbolic algebra programs (the paper specifically singles out Mathematica), but the problem is more widespread than this because it occurs with basic 4-function calculators (e.g. do I divide by 1.1 or multiply by 0.9?) or with advanced numerical software (e.g. why do the eigenvectors come out completely different for trivially different data?). This causes people to drop down into a calculational mode where they act merely as operators of the software/hardware, whilst not bothering to form a higher-level interpretation of what they are calculating (e.g. its physical interpretation).

This is like the difference between a worm's eye-view (e.g. low-level calculational mode) and a bird's eye-view (e.g. high-level physical interpretation mode). It is like the difference between having a local serialised view of each part of the problem that you are solving or a global parallelised view of the whole problem. It is like the difference between being a calculator or a visionary.

I know of people who are calculators but who can't see the grand picture, and who usually cannot communicate with anyone other than like-minded calculators. I know of people who are visionaries but who can't express their ideas in enough detail to carry them out, and who are highly articulate but whose apparent lack of rigour really annoys the ace calculators. I know of very few people who are both calculators and visionaries, but these people are really interesting to know.

The education system trains people to produce standard solutions to problems, so that everyone calculates using the same language. It is relatively easy to teach people to rote-learn standard procedures, and to then test them on this knowledge in exams. It is much less easy to teach people the skills that are needed to relate these calculations to the rest of the world, or so one would think.

My approach to counteracting the tendency to drop down into a calculational mode of thinking is to visualise what I am calculating; I try to avoid doing calculations that I can't visualise. When I draw a picture of what I want to calculate then the calculation itself follows almost automatically, and calculational subtleties (e.g. the epsilons and deltas) are easily resolved by referring back to the picture. I would go so far as to say that if I can't draw a picture then I don't understand what I am calculating.

I was amused to see that the principal example cited in the paper http://arxiv.org/abs/0712.1187 was one in which several students struggled to use Mathematica to evaluate the following integral (I have omitted various constants):

Integrate[x^2 Sin[x]^2, {x, -Infinity, Infinity}]

This is a clear example of students in calculational mode, who have adopted the worm's eye view of the problem as just being a calculation. They tried feeding the integral to Mathematica in various different ways, but without success. What they do not do was to diagnose their problem by simply visualising what they were calculating; it is not even necessary to know the physics behind this integral.

Using the same tool (i.e. Mathematica) that the students were using in their attempts to evaluate the integral, here is a plot of the integrand over a finite interval.

Normally, an integrand that is simple as this would not need to be plotted out explicitly because its behaviour is obvious from its structure, i.e. an x^2 factor that diverges times a Sin[x]^2 factor that oscillates between 0 and 1. Nevertheless, in this case I did plot it out as part of my ingrained habit of visualising calculations using Mathematica. The students should have been doing this as well, so I presume that they had not been very well tutored in their use of Mathematica. Had the students attempted even a rudimentary visualisation then they would have immediately realised that the limits of the integral they were trying to evaluate could not possibly be infinite.

As the visualisation habit becomes part of your way of working, you eventually reach a point where the solution of some problems comprises visualisation followed by calculation. There always remains a set of "difficult" problems for which your current set of visualisation techniques is inadequate, in which case you have to use pure calculation to get to a solution. But then you should be on the lookout for ways to capture the essence of your solution in a new visualisation technique.

Wouldn't it be nice if there was a standard set of visualisation techniques that you could use alongside the existing set of calculational techniques? If this set of visualisation techniques was carefully designed then it would be just as rigorous (and teachable) as standard calculational techniques. It would be a very interesting exercise to reformulate existing material using such a visual language; for instance, I made an attempt to do this sort of thing for the topology of the SO(3) rotation group here.

Nano flower

2007-12-29T20:54:00.000Z

A while ago I blogged about NanoArt 2007 , which is an online competition to create nanoart. Well, it turns out that I had completely forgotten about this competition, so I have done nothing more than form a few vague ideas about what I planned to create as a submission. The closing date for submissions is 31 December 2007, which leaves very little free time to create a worthwhile submission, so I wondered how I could create something using the bits and pieces that I happen to have lying around already.

One of the 3 electron microscope pictures that are supplied for you to start from is this

which they fittingly call "Nano Flower", and you are allowed to submit up to 5 "artistic" images for the competition.

I think I will choose to submit a 5-frame animation generated using a variant of the Belousov-Zhabotinsky reaction simulation software that I described here, where I use the "Nano Flower" image as an external input to steer the the reaction dynamics. This means that I am simulating a modified version of the BZ reaction, where the surface on which the reaction occurs is weakly contaminated so that the BZ reaction dynamics vary across the surface.

Here is a 5-frame animation that I quickly created, which fortuitously turns out to be cyclic because the period of the BZ dynamics is 5 frames.

Well, that's one frame of the animation. I gave up trying to get the animation to upload successfully to blogger.com, so I have put it on my web site here.

I have called this work "Fizzix" for fairly obvious reasons. Now I will wait to see whether I have broken the submission rules for the NanoArt 2007 competition, which don't mention animation as being an acceptable artform, not even 5-frame animations.

Update (7 January 2008):

All of the NanoArt 2007 competition entries can be viewed here, and the album containing my single competition entry is here (with the animation hosted here). My competition entry is a lot less colourful than other peoples' entries, so it is unlikely to attract much attention. I have to hope that people click down to my animation to discover the true nature of my competition entry.

Molecular manufacturing

2007-12-20T09:12:00.000Z

The Center for Responsible Nanotechnology has a nice overview of its current findings on molecular manufacturing here, which summarises the pros and cons of being able to manipulate matter in a controlled way at the molecular scale.

Molecular manufacturing is fundamentally different from the "heat and stir" approach to building things with molecules, because it aims to directly control the placement and interaction of individual molecules. This opens up lots of new possibilities for building things (beneficial or dangerous), and the world will not be the same when the transformation to molecular manufacturing has occurred.

This revolution in manufacturing will make the Industrial Revolution seem trivial in comparison, and it will happen within a small number of decades (CRN says 2 decades). This means that it is likely to have a big impact on the lives of most people alive today, which gives you an incentive to read all about it here.

Merry Yuletide

2007-12-15T17:27:00.000Z

Here is a movie in which I have created an animation of a festive texture on the surface of a trefoil knot. Merry Yuletide!

How did I create this movie?

I have been playing around with the Belousov-Zhabotinsky reaction simulation that I described here.

One thing that you can do is to find limit cycles of the simulation, where the state of the array of cells returns to a state that it visited earlier in the simulation. The whole sequence of states between two such repeats (including one of the end points) is then a limit cycle of the BZ simulation. Such limit cycles must exist because the state space is finite in size, so it it inevitable that the simulation must eventually revisit states that it visited earlier, though starting from a random initial state the likelihood that this occurs in a given timescale decreases rapidly as the size of the array increases.

One example that I particularly like is a cycle of length 10 that I found on a toroidal 13 by 13 array (using the same parameter values and colour scheme as here), and I show 2 of these cycles in the movie below:

This cycle is unusual because it has a low symmetry, and because it is pretty to look at despite its short length.

Using this cyclic BZ solution on a torus it is relatively easy to create the movie at the start of this posting, by using it to texture the toroidal surface of a trefoil knot, and choosing a colour scheme that maximises its festive feel.

Nerd test

2007-12-11T10:57:00.000Z

I happened upon Nerdiometer, which describes an on-line test for assessing how nerdy you are. You can find the test here.

I couldn't resist trying this one. I suspect that the results of the test are biassed by a selection effect where the nerdier you are the more likely you are to take the test in the first place.

Anyway, here is my result:

17% of people score higher than me, and 82% score lower. I was disappointed to score so highly, because as I filled in the multiple-choice questionnaire I could see that the extremely nerdy answers were off the scale compared to where I stood. Maybe that was a trick to let me make the choices that I did without being too embarrassed about them.

The questionnaire was extremely selective in the areas it covered, because it focussed on asking about nerdy things to the almost complete exclusion of asking you about anything else. So, if you spend only small fraction of your time being a nerd, you will register highly on the nerd-scale as defined by this test.

I wish there was a way to find out how each answer was weighted to produce the overall result. Oh no! Wishing for that must in itself be a nerdy thing to do! Aargh! This is renormalisation gone mad!

Update (12 December 2007):

The result I got when I tried version 2 of the test (see here) is

OK, so now I am a "cool nerd" which sounds pretty good to me, but I am annoyed that my Science/Math and Technology/Computer scores are deemed to be so low! Who are the people who designed this test anyway? I'm not playing any more, maybe...

Belousov-Zhabotinsky reaction

2007-12-06T12:33:00.000Z

Whilst exploring Wikipedia I came across the Belousov-Zhabotinsky reaction, and the phenomenology of its reaction dynamics such as the development of propagating waves of reactants is described here. Although I previously knew about the BZ reaction I had never implemented a simulation of it. It turns out to be remarkably easy to implement in Mathematica, and the BZ reaction dynamics can readily be displayed as a very pretty movie.

Although the original purpose of this type of simulation was to model the BZ reaction dynamics, you could imagine using variants of this type of simulation for generating interesting dynamic art forms.

A slightly modified form of an algorithm to simulate the BZ reaction due to Professor A K Dewdney is listed here.

(i) Select an integer q in the range 2 through 255. Cells may be in any of the states 1 through q.
(ii) Select two integers k1 and k2 in the range 1 through 8 and an integer g in the range 0 through 100.
(iii) In the transition from one "step" to the next the state of each cell is changed once according to rules (iv) - (vii).
(iv) A cell in state q changes to state 1.
(v) A cell in state 1 changes to state a/k1 + b/k2 + 1 where a is the number of neighbors of the cell which are in states 2 through q-1 and b is the number of neighbors in state q.
(vi) A cell in any of states 2 through q-1 changes to S/(9 - c) + g, where S is the sum of the states of the cell and its neighbors and c is the number of neighbors in state 1.
(vii) If the application of rule (v) or rule (vi) would result in a cell having a state > q then the state of that cell becomes q.

After some experimentation to find the fastest (and clearest) implementation in Mathematica, I created a function for implementing a BZ update step with toroidal boundary conditions:

update[state_, q_, k1_, k2_, g_] :=
Module[
{ones, qs, kernel, total, totalones, totalqs},
ones = Map[If[#==1, 1, 0]&, state, {2}];
qs = Map[If[#==q, 1, 0]&, state, {2}];
kernel = Table[1, {3}, {3}];
total = ListCorrelate[kernel, state, {{2,2},{2,2}}];
totalones = ListCorrelate[kernel, ones, {{2,2},{2,2}}] - ones;
totalqs = ListCorrelate[kernel, qs, {{2,2},{2,2}}] - qs;
MapThread[If[#>q, q, #]&[Switch[#1, 1, Floor[#4/k1+#3/k2+1], q, 1, _, Floor[#5/(9-#2)+g]]]&, {state, totalones, totalqs, 8-totalones-totalqs, total}, 2]
];

I used this BZ update function to generate the above movie of the BZ reaction dynamics, using suitable parameter values that I copied from here.

statesequence =
With[
{n=64, q=200, k1=2, k2=3, g=70},
NestList[update[#, q, k1, k2, g]&, RandomInteger[{1,q}, {n,Floor[4/3n]}], 250]
];
movie = Map[ArrayPlot[#, ColorFunction->"Rainbow"]&, statesequence];
ListAnimate[movie]

The poor image quality in the movie is due to the high degree of image compression, rather than due to any limitations of the above algorithm. Also, each frame is separately scaled and mapped onto the colour lookup table, which is probably not the best way of creating the movie.

Enigmatic comments

2007-11-29T16:02:00.000Z

This is a placeholder for comments on postings at Enigmatics.

Please remember to identify which Enigma problem you are commenting on.

Enigmatic movements

2007-11-29T13:16:00.000Z

I have decided that my Mathematica solutions to the New Scientist Enigma problems are cluttering up this blog, so I will move them to a separate blog called Enigmatics which I will generate using the incredibly useful WorkLife FrameWork; believe me, using WLFW is like having a 24/7 personal assistant organising all of your Mathematica notebooks. This allows me to present my solutions in a form that is much closer to the original Mathematica, but it does not allow for comments on blog entries.

I have looked into the possibility of managing the comments myself, but the effort needed to counter the inevitable spam is too great. I will instead provide a link in each Enigmatics blog entry back to this blog, which will allow comments to be made using the Blogger commenting system.

NanoArt 2007

2007-11-28T12:17:00.000Z

Nanoart is defined here as:

NanoArt is a new art discipline related to micro/nanosculptures created by artists/scientists through chemical/physical processes and/or natural micro/nanostructures that are visualized with powerful research tools like Scanning Electron Microscope and Atomic Force Microscope.

The process of creating nanoart is described here as:

I bring the small world in front of my audience through high resolution electron microscope scans of natural micro or nanostructures and sculptures I create at micro or nano scale by physical or/and chemical processing. I take further steps by mixing the realistic images of this structures with abstract colors, digitally painting and manipulating the monochromatic electron scans, and finally printing them with long-lasting inks on canvas or fine art paper (giclee prints). This way, the scientific images become artworks and could be showcased for a large audience to educate the public with creative images that are appealing and acceptable.

In a nutshell, the electron microscope provides the raw image, and the artist colours it in to produce an enhanced (i.e. artistic) result. This could range from a trivial colour tinting of the raw image, through to an artistic rendition that is based only very loosely on the raw image. Of course, I favour the more artistic style of nanoart, and I have some ideas on new ways of creating such artwork, but there isn't room in this tiny blog posting to tell you all about it!

Anyway, back to the title of this posting: NanoArt 2007. This is an online competition to create nanoart, and the submission deadline is 31 December 2007. For people without ready access to an electron microscope to create their own raw images there are 3 high resolution monochromatic electron scans provided here (nano-flower), here (micro & nano), and here (nano-crystals). I think it would be fun to enter this competition.

Dresden art gallery

2007-11-24T11:41:00.000Z

Here is a nice photo of a very small part of the world famous Dresden Art Gallery:

Here is a snapshot that I took of the same scene (with some more context) in Second Life:

The exact location of this duplicate of the Dresden Art Gallery in Second Life is http://slurl.com/secondlife/Dresden%20Gallery/77/121/26. Visitors to SL do not need to pay anything just to explore.

I won't show you inside the Second Life duplicate of the art gallery, so you will have to visit it yourself to enjoy it. You can have a look at the WIRED report here if you want more details. Set aside a spare half-day for this adventure in SL. You won't be disappointed.

Sand painting

2007-11-21T11:09:00.000Z

I found this video of sand painting here at Backreaction:

I think it is a wonderful art form, and it has given me some ideas ...

Enigma 1469

2007-11-16T21:57:00.000Z

Here is how Mathematica can be used to solve New Scientist Enigma number 1469 in the 17 November 2007 issue.

WARNING: See the comments for details. There is an error in the solution given below that I need to fix. In working on a fix I discovered a bug in the HamiltonianCycle function in Mathematica's Combinatorica package (this bug has been acknowledged by the author of the Combinatorica package) which destroyed any chance of my fix working fully correctly, although the correct solution to Enigma 1469 could be seen lurking in amongst the erroneous clutter. I will abandon my solution to Enigma 1469 because it has absorbed too much of my time already, and the discovery of the HamiltonianCycle bug can stand as a testimonial to the cleverness of Enigma 1469. I suppose that I ought to leave the rest of this posting intact otherwise the comments would be orphaned, and also there are some useful techniques illustrated in my erroneous solution.

Load the Combinatorica package.
Needs["Combinatorica`"];

Derive the formula for triangular numbers.
triangularformula=Sum[i,{i,n}]
1/2 n (1 + n)

Compute a list of all triangular numbers less than 1000.
triangularnumbers = Table[triangularformula, {n,44}]
{1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990}

Convert this to the corresponding list of lists of digits.
triangulardigits = IntegerDigits[triangularnumbers]
{{1}, {3}, {6}, {1, 0}, {1, 5}, {2, 1}, {2, 8}, {3, 6}, {4, 5}, {5, 5}, {6, 6}, {7, 8}, {9, 1}, {1, 0, 5}, {1, 2, 0}, {1, 3, 6}, {1, 5, 3}, {1, 7, 1}, {1, 9, 0}, {2, 1, 0}, {2, 3, 1}, {2, 5, 3}, {2, 7, 6}, {3, 0, 0}, {3, 2, 5}, {3, 5, 1}, {3, 7, 8}, {4, 0, 6}, {4, 3, 5}, {4, 6, 5}, {4, 9, 6}, {5, 2, 8}, {5, 6, 1}, {5, 9, 5}, {6, 3, 0}, {6, 6, 6}, {7, 0, 3}, {7, 4, 1}, {7, 8, 0}, {8, 2, 0}, {8, 6, 1}, {9, 0, 3}, {9, 4, 6}, {9, 9, 0}}

Construct the adjacency matrix of allowed adjacencies between triangular numbers.
adjacencies = Outer[(If[Last[#1]==First[#2], 1, 0])&, triangulardigits, triangulardigits, 1];

Convert the adjacency matrix to a set of rules that allow me to add a "tooltip" to each node of the corresponding the graph so that it can be explored by hovering the mouse over the graph: (1) extract the sparse array form of the adjacency matrix, (2) convert the sparse array into a list of rules for which graph nodes are linked by edges, (3) add a tooltip to each graph node.
adjacencies2 = SparseArray[adjacencies];
adjacencies3 = ((adjacencies2//ArrayRules//Most) /. HoldPattern[{i_,j_}->1] :> i->j);
adjacencies4 = adjacencies3 /. x_?IntegerQ :> Tooltip[x, triangularnumbers[[x]]];

Display the graph with tooltips.
GraphPlot[adjacencies4, DirectedEdges->True]
[Output omitted]

Convert the adjacency matrix to a graph. I need this representation in order to use graph manipulation algorithms.
g = FromAdjacencyMatrix[adjacencies, Type->Directed];

Extract the longest cycle in the graph, and extract the edges within this cycle.
longestcyclenodes = ExtractCycles[g]//Sort//Last//Most;
longestcycleedges = Partition[longestcyclenodes,2,1,{1,1}];

Display the graph with the longest cycle highlighted.
GraphPlot[adjacencies4, EdgeRenderingFunction -> (If[MemberQ[longestcycleedges,#2], {Red,Arrow[#1]}, {Opacity[0.3],LightGray,Arrow[#1]}]&)]

Convert the longest cycle to the corresponding a list of triangular numbers, and thence the sequence of digits encountered when going around the cycle.
longestcyclenumbers = triangularnumbers[[longestcyclenodes]];
longestcycledigits = longestcyclenumbers//IntegerDigits//Flatten
[Output omitted]

Compute the number of digits encountered when going around the longest cycle.
longestcycledigits//Length
[Solution omitted]

The Goldstone theorem

2007-11-14T23:28:00.000Z

There is an interesting posting on Tommaso Dorigo's blog A Quantum Diaries Survivor entitled The Goldstone Theorem for Real Dummies. Quoting from his posting the Goldstone theorem is: "The spontaneous breaking of a continuous symmetry of the lagrangian generates massless scalars". Despite its claim to be for real dummies his derivation of the Goldstone theorem assumes familiarity with a lot of theory, which ensures that the "dumb" reader will not develop an intuitive feel for the masslessness of the particles (the Goldstone bosons, that is).

I am personally at ease with the level of theory that he uses, but I am in a very small minority in this respect. Wouldn't it be nice to write an intuitive description that does not assume that the reader has prior knowledge of much theory, but where the writer ensures that the intuitive description faithfully conforms to the underlying theory that is known to him. The Goldstone theorem can be understood intuitively without having to follow the detailed algebraic steps of its proof.

Here I will attempt to "prove" the Goldstone theorem using only words (i.e. no pictures, and no maths). Pictures would greatly improve the presentation, and maths would very succinctly summarise and generalise what is going on, but it is interesting to see what you can do with words alone.

Mass is defined as the propensity of the vector-valued field value alone (i.e. not including any field derivatives, or equivalently the field values at neighbouring points) to contribute to a restoring force that pushes the field towards an equilibrium value of the field. If within a local neighbourhood of field space there are direction(s) that have zero restoring force then each of these directions corresponds to a zero mass excitation.

It suffices to visualise a potential in field space whose (negative) gradient defines the field restoring force, and to ask ourselves what sort of shapes this potential could have in field space. We need to find places in field space where this potential has zero gradient, and then enquire whether the gradient of the potential is actually zero over the whole of a local neighbourhood rather than just at a single point, so that displacement of the field value within such a neighbourhood would encounter zero restoring force, and would thus correspond to a zero mass excitation.

The visualisation problem reduces to the classification of the stationarities of a scalar potential function in field space, which is something that is very easy to visualise (e.g. minima, maxima, saddle points, etc). The particular case that is relevant to the Goldstone theorem is when the potential function is constrained to be symmetric, e.g. under the continuous group of rotations about the origin of the vector-valued field. What sort of stationarities are allowed under such symmetry constraints? For a rotation symmetry the potential must be constant on each origin-centred spherical shell in field space, but different spherical shells have independent values of the potential. The potential is therefore fully described by its variation as a function of distance from the origin in field space (i.e. a 1-dimensional potential).

To find an equilibrium point you need to find a stationary point of the 1-dimensional version of the potential, then constancy of the full version of the potential over each spherical shell gives us zero gradient of the potential over the entire spherical shell that intersects this stationary point, which thus gives as many zero-mass degrees of freedom of the vector-valued field as there are directions within the spherical shell (i.e. one less than the dimensionality of the field). This is the Goldstone theorem.

Rereading what I have written above I am now uncertain whether it is of much use to anyone other than me. It is just the internal chatter (minus the pictures) that goes on inside my head when I think "Goldstone theorem", so it isn't guaranteed to be as intuitive for other people as it is for me.

Enigma 1468

2007-11-13T17:02:00.000Z

I have had to skip a couple of Enigmas because there hasn't been time to do them. I may come back to them later, but don't hold your breath.

Here is how Mathematica can be used to solve New Scientist Enigma number 1468 in the 10 November 2007 issue. This solution draws heavily on the graph manipulation facilities in the Combinatorica package. Most of the work involves setting up an appropriate graph to represent the problem, after which the solution to the problem drops out by simply calling the HamiltonianCycle function.

Load the Combinatorica package for doing operations on graphs.

Needs["Combinatorica`"];

Define a function for computing 2D indexing from 1D indexing on a 6 by 6 array.

indices[i_] := {Quotient[i+5,6], Mod[i,6,1]};

Define a function for computing if a pair of nodes is adjacent in a Manhattan metric. You could use GridGraph[6,6] instead, but I want to show how to use an explicit adjacency function here.

manhattan[{i1_,j1_}, {i2_,j2_}] := Abs[i1-i2]==1&&j1==j2 || Abs[j1-j2]==1&&i1==i2;

Define a function for testing if adjacency between a pair of nodes is excluded.

exception[{i1_,j1_}, {i2_,j2_}] := Apply[Or, Map[{i1,j1}==#[[1]] && {i2,j2}!=#[[2]]&, {{{2,3},{1,3}}, {{3,4},{4,4}}, {{5,3},{6,3}}, {{5,6},{5,5}}}]];

Use these adjacency functions to define an adjacency matrix.

adjacency = SparseArray[{{_, _}?((manhattan[#1,#2] && Not[exception[#1,#2]])&[indices[#[[1]]], indices[#[[2]]]]&)->1}, {36,36}];

Use this adjacency matrix to define an adjacency function.

edge[from_, to_] := adjacency[[from, to]]==1;

Define the layout (i.e. embedding) of the graph.

coordinates = Table[index->{{0,1},{-1,0}}.indices[index], {index,36}];

Display the graph.

GraphPlot[g=MakeGraph[Range[36],edge], DirectedEdges->True, VertexLabeling->True, VertexCoordinateRules->coordinates]

[Output omitted]

Compute all of the Hamiltonian cycles in the graph. There is only one, as expected.

cycles = HamiltonianCycle[g,All]

[Output omitted]

Extract the edges in the Hamiltonian cycle.

cycleedges = Partition[cycles[[1]],2,1];

Define a function for drawing only the edges that lie on the Hamiltonian cycle.

cycleedge[coords_, vertices_, label_] := If[MemberQ[cycleedges,vertices], {Red,Thickness[0.01],Arrow[coords]}, {}];

Display the Hamiltonian cycle.

GraphPlot[MakeGraph[Range[36],edge], VertexLabeling->True, VertexCoordinateRules->coordinates, EdgeRenderingFunction->cycleedge]

[Output omitted]

Rotate the Hamiltonian cycle so that it starts at the correct node.

cycles2 = RotateLeft[#, Position[#,11][[1,1]]-1]&[Most[cycles[[1]]]]

[Output omitted]

Compute which positions in the Hamiltonian cycle fall on the nodes labelled "ENIGMA".

Map[Position[cycles2,#][[1,1]]&, Range[19,24]]

[Solution omitted]

Notebook computer woes

2007-10-24T19:37:00.000Z

This is a very sad and nerdy posting, but those who have spent hours or days tracking down computer faults will know the state of mind that it creates, and the consequent need to drink large amounts of beer and write drivel!

I bought myself a new notebook computer (HP Pavilion dv9340ea) three months ago, and would you believe that a fault has already developed? Luckily, the fault turns out to be fairly benign so I can continue to use the computer, but the fault continues my unbroken record of always managing to buy notebook computers that go wrong.

What is the fault? The graphics display developed a habit of locking up with no warning, and the computer would then reboot showing various regular patterns on the boot screen, and then go to a "blue screen of death" (BSOD) informing me that nvlddmkm.sys had thrown error 116. The computer would then reboot again offering me the choice of booting into a "safe mode" of Windows Vista, which I duly selected, and I was then allowed back into the machine. At least I could get at any files that I wanted to extract from the computer.

So, off I went onto the internet searching for information about nvlddmkm.sys and error 116. In turns out that you can find lots of complaints about nvlddmkm.sys, which I found perversely gratifying because it meant that I had company. Various wise individuals offered solutions to this problem, which mainly focussed on manually uninstalling the old and buggy graphics driver (i.e. nvlddmkm.sys) and installing the latest version of this driver. I did this but it didn't fix the problem. I repeated this process several times, steadily enlarging the scope of my manual deletion of old files before installing new files, but I completely failed to fix the problem.

Maybe my nvlddmkm.sys problem was not in fact the same as the one that could be fixed by updating the graphics driver. Apparently, error 116 is quite a general fault that could be thrown by almost any error condition in nvlddmkm.sys.

I then did a full virus scan but it found nothing unusual.

I then decided to pay more attention to how my computer behaved without any special graphics driver installed. It showed the same regular patterns on the boot screen that preceded the BSOD that I mentioned at the start of this posting, but the BSOD itself did not occur, and I could boot into normal (i.e. not "safe mode") Windows Vista, so I had all Windows services available to me, but the screen resolution was relatively low because there was no special graphics driver installed.

The only other sign that something was wrong was a strange coloured granularity that showed up in what should have been featureless regions of colour on the screen. On rebooting, this fault occasionally disappeared, so I tried various experiments to check whether it was a thermal problem that went away temporarily if I switched off for a while. No luck there. I could not find a systematic way of influencing the fault.

In desperation I tried tapping the computer case with my finger, moving around all over the case, both on top and underneath. I noticed that when I tapped in one particular place under the case the strange coloured granularity disappeared at exactly the moment that I tapped the case. Jackpot! Although the fault quickly returned, and after a few repetitions of this cycle I eventually couldn't get the fault to go away at all, I had now absolutely convinced myself that my computer had a hardware fault that was probably in the graphics subsystem.

A loose memory chip maybe? The only memory that you could fiddle with was system RAM not video RAM, so reseating the system RAM modules didn't fix the problem, nor did I really expect it to.

However, I managed to find some photos of the system board of my notebook computer, and I noticed that immediately under the area of the case where I did my few successful taps was ... drum roll ... the graphics subsystem. I think this fairly convincingly demonstrates that something has gone wrong with the graphics hardware in my computer.

Now, all I have to do is to convince whoever it is that actually fixes my computer that they don't need to reformat my hard disk to check whether doing so cures the fault. In fact, it would be really nice if they would simply bring out a new system board to swap with my faulty one. What's the betting that things will not go as smoothly as this?

Update: After multiple emails and phone calls to HP they have agreed to service my computer under warranty, but they insist on restoring my hard disk to its factory settings. Resistance is useless! Since I don't relish going through the whole process of reinstalling all of my software, especially since there are lots of fiddly little things you have to do to get some of my old Windows XP software to work correctly under Windows Vista, I have decided to buy some disk mirroring software (i.e. Paragon Drive Copy 8.5). This software spent 2 1/2 hours mirroring my notebook computer's hard disk, and then announced that it had successfully done the job. Do I believe this? Do I believe that if it has done this part successfully, then it will also do the reverse mirroring successfully when I get my computer back from HP? I don't know what to think, but at least I have tried, and the Paragon software didn't cost too much (i.e. £30).

Update: Finally, after further phone calls to HP, they have agreed on specific dates to collect & return my notebook PC for its servicing-under-warranty. My main criticism of this process is not so much the large number of phone calls that I had to make, but the fact that HP uses call centres staffed by people who speak fluent English but with an impossible-to-understand accent. I had to ask several times for things to be repeated, and only with a certain amount of ingenuity and prior knowledge on my part could I understand what was being said. On more than one occasion I had to abort the phone call because I couldn't understand the accent at all, and on one occasion they aborted the phone call because of my inability to understand them!

Update: Finally! My notebook computer is back from the HP repair centre. The repair notes state that it had exactly the problem that I had originally diagnosed (i.e. a video hardware fault), and that the repair was a straight replacement of the motherboard. I hope that this fault was merely an "infant mortality", and that now I am likely to have years of trouble-free operation.

Enigma 1465

2007-10-19T18:17:00.000Z

Here is how Mathematica can be used to solve New Scientist Enigma number 1465 in the 20 October 2007 issue.

This Enigma problem can be solved by using a very simple analytic approach which I give at the end of this posting. But first I give a much more flexible approach to illustrate the use of one of the Mathematica packages.

Solution 1: Brute force approach using the Combinatorica package.

Load the Combinatorica package.
Needs["Combinatorica`"]

Define a large enough grid graph size that the solution is contained within it.
n=11;

Build a directed grid graph which allows movement only rightwards and upwards. Note that the starting node is at the bottom-left hand corner. This type of directed structure forces all paths from the bottom left node to each other node to also be shortest paths.
g = MakeGraph[Range[n^2], Function[{x,y}, If[y==x+1 && Mod[x,n]>0 y==x+n, True, False]]]
-Graph:<220,121,directed>-

Compute the shortest path length from the bottom left corner node to each of the nodes in the graph. Because of the directed structure of the graph all paths are also shortest paths.
pathlengths = AllPairsShortestPath[g][[1]];

Generate a list of distinct shortest path lengths.
pathlengths2 = pathlengths//Union;

For each distinct shortest path length generate a list containing the number of distinct shortest paths to each node that has this shortest path length.
pathnumbers = Map[NumberOfKPaths[g,1,#]&, pathlengths2];

Extract the two cases where the same number of paths occurs 4 times. Since the solution lies in the quadrant opposite the starting node it must have a relatively large number of paths, so it can be selected by inspection of this result.
pathnumbers2 = Transpose[Cases[pathnumbers//Flatten//Tally, {_,4}]][[1]]
{10, 3003}

Solution 2: Quick approach using the known number of paths in a directed grid graph.

Compute a table each of whose entries is the number of paths from the top-left node of a directed (rightwards or downwards) grid graph.
With[{n=11}, Table[Binomial[i+j,j], {i,0,n-1}, {j,0,n-1}] // TableForm]
{
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11},
{1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66},
{1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286},
{1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001},
{1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003},
{1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008},
{1, 8, 36, 120, 330, 792, 1716, 3432, 6435, 11440, 19448},
{1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, 43758},
{1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620, 92378},
{1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184756}
}

The three 6-fold paths mentioned in the first part of the problem can be seen in the correct positions in the table (i.e. (1,5), (5,1) and (2,2), where the top-left corner is (0,0)). The 4-times repeated solution(s) found above can be seen elsewhere in the table, and the larger solution lies in the lower right quadrant.

Ising model simulation - online software

2007-10-19T11:08:00.000Z

I showed here how to use Mathematica (version 6) to create an interactive simulation of a 2-dimensional Ising model, but the software could be used only by people who owned their own copy of Mathematica. I reported here that Wolfram had now made available the Mathematica Player online conversion system for transforming Mathematica notebooks into a form that can be run by the Mathematica Player.

I have submitted my Ising model notebook for online conversion, and I have uploaded the converted notebook here (855 kB) plus a zipped version here (26 kB).

The Mathematica Player can be downloaded from here.

I would be interested to hear from anybody who does not have Mathematica (version 6) whether they can successfully get this converted Ising model notebook to work. I would particularly like comments on the ease of use of the zipped version, because this is by far the more disk-space-efficient way of doing things.

Update: Since no-one has responded yet to say that that they have succeeded in getting my converted Ising model notebook to work with Mathematica Player, I decided to install Mathematica Player on my old notebook PC (which didn't have Mathematica already installed) to check things out for myself. I downloaded the zipped version of the converted Ising model notebook, and fired it up in Mathematica Player. The contents of the converted notebook look perfectly OK, but seem to be completely inactive, so you can't run the Ising model simulation. I haven't got the faintest idea what is wrong here, so I'll now go away to try to sort this problem out.

Mathematica Player online conversion

2007-10-15T22:56:00.000Z

At last! We can now use Mathematica to write software that we can publish on our websites and which can be run by anybody, i.e. this is not limited to end-users who own their own copy of Mathematica. This development was quietly announced in a Wolfram blog posting entitled The Day that Documents and Applications Merged; at least, that's where I learnt about it. It takes the form of a Mathematica Player online conversion system for transforming Mathematica notebooks into a form that can be run by the Mathematica Player. The blog posting summarises this as:

But the new online conversion service opens up almost any notebook. If you're a Mathematica user, that means you don't have to share your work just with other users anymore. Create your dynamic applications and share them with anyone you'd like. You can even post them on your website and link people to the Player download - a great way to publish research papers, books, or interesting computational models.

This changes everything! Long-time users of Mathematica such as me have been gnashing their teeth for 20 years at the fact that, although they have a uniquely powerful tool in Mathematica, any software that they write has been limited to being used by only those people who have their own copy of Mathematica. Unfortunately, there is almost zero chance that any randomly selected person owns Mathematica.

Many years ago I made the decision to use Mathematica because it was my "secret weapon" that allowed me to do things that would be much more cumbersome to do with other software tools such as C++, Matlab, etc. It allowed me to do a lot more research than would otherwise have been the case. In fact, I covered ground so fast that people couldn't keep up with me; that's praise for Mathematica rather than an inflated view of my ability to do research. I have received almost universal criticism from those around me (who use C++, Matlab, etc), because the fruits of my Mathematica labours are not available for them to use, at least not without a lot of effort to translate the software into a form that satisfies them (i.e. into C++, Matlab, etc). My use of Mathematica is regarded by these people as a selfish act, because it makes life easy for me and difficult for them. I can see their point! However, I am not one to be swayed by other peoples' misguided opinions, so I carried on regardless.

Of course, the Mathematica Player online conversion system comes with some restrictions, which limit its use to one of the following categories:

Educational purposes in a not-for-profit, educational institution.
Publicly accessible or published research content.
Demonstration purposes, including at commercial or governmental organizations.
None of the above. I am interested in commercial distribution.

This set of alternatives seems remarkably generous to me, after 20 years of not being able to use Mathematica in this way. The category above that particularly interests me is (2), which allows me to include working software with research papers that I publish. In fact, I wonder whether I will move to a model in which I seamlessly combine software and research paper in the Mathematica notebook that I feed to the Mathematica Player online conversion system (I assume this is possible).

Thank you so much Wolfram! I hope there aren't hidden caveats that break what appears to be a fully working model.

Random sculptures

2007-10-14T00:32:00.000Z

Here is a piece of Mathematica code for generating interesting random sculptures by generating 2-dimensional manifolds embedded in 3-dimensional space.

Define a function that computes a random sum of sinewaves, where the maximum frequency is controlled by the n parameter. Differently seeded versions of this function will be used to determine the displacement of the 2-dimensional manifold in each of the directions in 3-dimensional space.

f[x_,n_] := Sum[RandomReal[{-1,1}]Cos[i x+RandomReal[2\[Pi]]], {i,0,n}];

Display a random sculpture. The n parameter controls the complexity of the sculpture.

With[{n=1}, ParametricPlot3D[Table[f[u,n]f[v,n],{3}]//Evaluate, {u,0,2\[Pi]}, {v,0,2\[Pi]}, PlotStyle->Opacity[0.25], Mesh->False, Boxed->False, Axes->False]]

Here are 3 examples of the sort of result that you can obtain. Not bad for a simple piece of code!

Enigmatics

2007-10-13T23:11:00.000Z

I thought that I should explain the purpose of me posting Mathematica solutions to the New Scientist Enigma problems.

The main purpose is to show how one can quickly get a solution to each problem (usually by brute force attack) using the unique processing abilities of Mathematica. Each posting shows the first line of reasoning that led me to the solution, so my reasoning is usually not elegant and is not intended to be elegant; the goal is to find the correct solution and to do it quickly. Sometimes this means that I deliberately use an unusual sequence of manipulations in order to illustrate a particular style of programming. Occasionally this means that I use a peculiar Mathematica idiom that I just happen to know.

Actually, I do worry about the elegance of solutions, but optimising the elegance of a solution is completely different from simply finding the correct solution to a problem. In the case of the New Scientist Enigma problems I strongly suspect that most of them are constructed in a brute force way (i.e. start with a large set of potential solutions, then impose various perverse constraints, until there is only one member of the set remaining), so there is no guarantee that there are slick and elegant solutions to them.

Enigma 1464

2007-10-12T20:11:00.000Z

Here is how Mathematica can be used to solve New Scientist Enigma number 1464 in the 13 October 2007 issue.

Generate a list of leap years in the 20th century during which the births can occur. The year 1900 is excluded because it was not a leap year.
lpyrs = Table[i, {i, 1904, 2000, 4}]
{1904, 1908, 1912, 1916, 1920, 1924, 1928, 1932, 1936, 1940, 1944, 1948, 1952, 1956, 1960, 1964, 1968, 1972, 1976, 1980, 1984, 1988, 1992, 1996, 2000}

Generate a list of all days during the above leap years in {yyyy, mm, dd} format.
(days = Flatten[Map[Table[Take[DateList[t],3], Evaluate[{t,#,#+365*86400,86400}&[AbsoluteTime[{#}]]]]&, lpyrs], 1]) // Length
9150

Convert this list of days into DDMMYY format.
(days2 = Map[(10000#[[3]] + 100#[[2]] + If[#[[1]]>=2000, #[[1]]-2000, #[[1]]-1900])&, days]) // Length
9150

Select the cases which are square days.
sqrdays=Select[days2, IntegerQ[Sqrt[#]]&]
{300304, 10404, 200704, 40804, 121104, 91204, 60516, 10816, 110224, 101124, 20736, 11236, 150544, 131044, 80656, 70756, 20164, 50176, 30276, 30976, 51076, 240100, 260100, 230400, 270400, 220900, 280900}

Split this list into sublists for each year.
sqrdays2 = Split[sqrdays, Take[IntegerDigits[#1],-2]==Take[IntegerDigits[#2],-2]&]
{{300304, 10404, 200704, 40804, 121104, 91204}, {60516, 10816}, {110224, 101124}, {20736, 11236}, {150544, 131044}, {80656, 70756}, {20164}, {50176, 30276, 30976, 51076}, {240100, 260100, 230400, 270400, 220900, 280900}}

Select the cases that have 6 or more different square days.
sqrdays3 = Select[sqrdays2, Length[#]>=6&]
{{300304, 10404, 200704, 40804, 121104, 91204}, {240100, 260100, 230400, 270400, 220900, 280900}}

Split the list of square days into sublists for each month.
sqrdays4 = Table[Select[sqrdays, FromDigits[Take[IntegerDigits[#],{-4,-3}]]==i&], {i,12}]
{{20164, 50176, 240100, 260100}, {110224, 30276}, {300304}, {10404, 230400, 270400}, {60516, 150544}, {80656}, {200704, 20736, 70756}, {40804, 10816}, {30976, 220900, 280900}, {131044, 51076}, {121104, 101124}, {91204, 11236}}

Select from all of the cases with 6 or more square days only those for which there is but a single square day in that month. These are the cases where the birth and death days are forced to lie in different months. There is one case that satisfies this criterion, which is thus the grandfather's date of birth.
Select[sqrdays3//Flatten, Length[sqrdays4[[Position[sqrdays4,#1][[1,1]]]]]==1&]
{300304}

Geodesic dome construction

2007-10-07T16:31:00.000Z

I have been thinking about how to automate the design of geodesic domes using Mathematica. I started off by looking at what Google Earth could see of the geodesic domes at the Eden Project (see 50.3612°N, -4.74483°W in Google Earth). Here is a clear view of the structure of one of the domes.

There is a central pentagon at the highest point of the dome, which is surrounded by 5 hexagons each of which is divided into 6 small triangles, which are in turn surrounded by hexagons (which are not divided into triangles). If you look very carefully you will find other pentagons lower down the dome.

How could this geodesic dome be designed in a semi-automatic way? I have worked out a way of doing this using Mathematica which goes as follows.

Load the library of polyhedron operations so that you can derive tessellated versions of polyhedra.

Needs["PolyhedronOperations`"];

Fix the order of the tessellation. A larger order will eventually give you a geodesic dome with smaller faces.

n=6;

Display an icosahedron and its tessellation, and extract the faces of the tessellation for later use.

GraphicsRow[Map[Graphics3D,{#, faces=Geodesate[#,n]}&[PolyhedronData["Icosahedron", "Faces"]]]]

The main problem now is to find a way of extracting pentagons and hexagons from all of the triangles that appear in the above tessellation. This is not trivial because it has to be done in such a way that the extracted polygons neither overlap nor have any gaps between them. The strategy that I use to achieve this is to gradually build up a set of polygon centres in such a way that each new centre is exactly far enough away from all of the centres found thus far that it defines a new polygon that neither overlaps nor leaves a gap with the polygons found thus far.

Define a function for computing all of the triangle face edges surrounding a set of vertices.

faceedges[perimeters_, centres_] := Flatten[Map[Cases[perimeters, {x___,#,y___} :> {x, y}]&, centres], 1] // Union;

Define a function for computing all of the vertices surrounding a set of triangle face edges.

facecentres[perimeters_, edges_] := Map[Cases[perimeters, {x___,#[[1]],y___,#[[2]],z___} | {x___,#[[2]],y___,#[[1]],z___} :> {x,y,z}]&, edges] // Flatten // Union;

Iterating the above pair of functions causes the set of vertices to grow until it covers the whole of the original set with pentagon and hexagon centres. As long as the order of the tessellation is chosen appropriately, then this algorithm will give the centres needed to construct a geodesic dome. Have a play to see what the algorithm does for you, and how it might be improved.

Fix the starting vertex to be the one at the top of the tessellated icosahedron.

startingcentre = {Position[faces[[1]], {0,0,1}][[1,1]]};

Derive the pentagon and hexagon centres and edges of the geodesic dome.

centres = Nest[
(
faceedgesold = faceedges[faces[[2,1]], #];
centresnew = facecentres[faces[[2,1]], faceedgesold]
)&,
startingcentre,
5
];

edges = faceedges[faces[[2,1]], centres];

Display the geodesic dome.

Graphics3D[
{
{Opacity[0.75], faces},
{PointSize[0.02], Red, Map[Point[faces[[1,#]]]&, centres]},
{Thickness[0.01], Blue, Map[Line[{faces[[1,#[[1]]]], faces[[1,#[[2]]]]}]&, edges]}
}
]

If you compare this result with the photo of the geodesic dome at the Eden Project then you can see that it has all of the essential features correct. There is a pentagon at the top of the dome, which is surrounded by hexagons, and lower down the dome there are more pentagons.

Enigma 1463

2007-10-06T22:51:00.000Z

Here is how Mathematica can be used to solve New Scientist Enigma number 1463 in the 6 October 2007 issue. Because this problem is so easy to solve I don't bother to make use of symmetry to simplify the search for a solution.

Generate a list of permutations of the digits 1, 2, ..., 9.
(digitperms = Permutations[Range[9]]) // Length
362880

Arrange each of these sets of 9 digits into a 3 by 3 array and generate the corresponding 12 3-digit numbers by reading rows and columns both forwards and backwards.
(numbers = Map[Flatten[Map[{FromDigits[#], FromDigits[Reverse[#]]}&, Join[Partition[#,3], Transpose[Partition[#,3]]]], 1]&, digitperms]) // Length
362880

Select the cases which satisfy the required divisibility conditions.
(numbers2 = Select[numbers, Apply[And, Table[Mod[Count[Map[Mod[#,i]&, #], 0], i]==0, {i,9}]]&]) // Length
8

Extract the minimum and maximum 3-digit numbers from these (symmetrically related) results.
{Min[#], Max[#]}&[numbers2]
{164, 971}

Rotation group topology

2007-10-05T17:49:00.000Z

Here is a video animation that illustrates the topology of the rotation group SO(3), which is the group of rotations in 3-dimensional space. My apologies for the small size and poor visual quality of the video - I am still experimenting with how best to generate animations and to upload them to blogger.

A rotation in 3-dimensional space can be done about an axis pointing in any direction, so there is a whole spherical surface of possible axis directions, which is represented by the sphere in the video. Each point on this spherical surface represents one possible orientation of the rotation axis, and the whole spherical surface is a 2-dimensional manifold representing all of the possible rotation axes. For the purpose of this blog posting only the topology of this manifold is important (i.e. it has a spherical topology); the detailed geometry of the surface is not important.

Once you have decided where the rotation axis points you then rotate about this axis, which generates a 1-dimensional angle subspace that lives outside the 2 angle dimensions that already live within the spherical manifold. Thus far the overall manifold is the original 2-dimensional spherical manifold (generated by the choice of axis direction) times a 1-dimensional manifold (generated by the choice of rotation). If you rotate by one full circuit (i.e. the rotation angle varies from 0 to 2Pi) then you appear to return to the starting point in the 1-dimensional angle subspace, so it seems that this subspace has a standard circular topology. But not all is what it seems to be.

The spherical manifold of possible rotation axes has a hidden degeneracy, because each rotation axis appears twice over. Each point on the sphere and its opposite point (i.e. its antipode) define rotation axes that are parallel, except that they have the opposite physical effect when a given rotation angle is applied about each of these two axes, i.e. the rotation axes are anti-parallel. So a rotation of x about one axis produces the same physical effect as a rotation of -x about its antipodal axis.

How does this degeneracy manifest itself topologically? The additional 1-dimensional angle subspace that is attached to each point on the spherical manifold must be identified with (i.e. glued to) the corresponding 1-dimensional angle subspace that is attached to the antipodal point on the spherical manifold.

This gluing-together of what were hitherto separate parts of the overall manifold imposes the constraint that a rotation of x about one axis produces the same physical effect to a rotation of -x about its antipodal axis. These constraints impose additional topological structure on the original 2-dimensional spherical manifold (generated by axis directions) times a 1-dimensional manifold (generated by rotations), because now antipodal points on the 2-dimensional spherical manifold are glued together via their attached 1-dimensional rotation angle subspaces.

This final glued version of the overall manifold captures all of the topological structure of the group of rotations in 3-dimensional space (i.e. SO(3)). It does not have the same topology as the unglued version because the gluing causes what were hitherto separate points on the manifold to become identified.

This gluing process is illustrated in the video, which shows how the gluing can be implemented by gradually morphing the 1-dimensional manifolds living at antipodal points on the 2-dimensional spherical manifold, until they eventually lie alongside each other ready to be glued together, and then they are finally coalesced into a single 1-dimensional manifold. Colour helps to identify corresponding points on the pair of yet-to-be-glued 1-dimensional manifolds, so that when the colours align with each other then the gluing operation can take place. It is clear from the video that this forces the pair of 1-dimensional manifolds to lie along a diameter of the sphere. Remember that the interior of the sphere is hitherto unused, so it is free to be used to represent rotation angle. Following the original 1-dimensional circular path from a point on the sphere back to itself again now corresponds to following a 1-dimensional path along a diameter of the sphere to its antipodal point, and then instantaneously returning to the starting point which has been glued to its antipodal point.

What consequences does this non-trivial topology have for rotations in 3-dimensional space? A rotation through 2Pi now corresponds to tracing a path along the diameter of the sphere from a point on the surface of the sphere to its antipodal point, and then returning instantaneously to the starting point. There is a non-trivial loop here because the path has clearly travelled somewhere else (i.e. the antipodal point) before returning home, and the loop can't be made to go away by moving it around inside (or on the surface of) the sphere, because the constraint that a pair of points on the path are antipodes is locked into the definition of the path.

So rotating by 2Pi in 3-dimensional space does not return you to where you started, because the path that you follow has a persistence that is enforced by the above antipodal locking-in effect. Of course, the physical object that you rotate has to have an appropriate physical sensitivity to rotation in order to observe this effect. A rotational scalar (e.g. the temperature of an object) will not show this effect, nor will a rotational vector (e.g. directions in 3-dimensional space) show this effect. An example of an object that does show this and other related effects is the Dirac Belt, where an object (i.e. the belt buckle) is rotated whilst the belt itself is used to retain a memory of what rotations have been applied to the buckle. After 2Pi rotation the belt is twisted so the overall state of the system is not the same as before the rotation was applied. The key to making an object that is physically sensitive to a rotation of 2Pi is that it must retain a memory of what rotations have been applied, which is equivalent to the object knowing about the path through the sphere discussed above. Obviously, there are many types of objects that exhibit this 2Pi sensitivity

The best bit of all is that the locking-in effect of the pair of antipodal points can easily be neutralised by making an additional rotation of 2Pi, so that the overall rotation is 4Pi. This is illustrated in the case of the Dirac Belt, where the twist of the belt is zero after a 4Pi rotation, so the belt is not physically sensitive to rotating its buckle by 4Pi. The reason that the antipodal locking-in effect goes away when you rotate by 4Pi (i.e. make two circuits along a diameter through the sphere in the video) is that there are now two pairs of locked-in antipodal points, which can separately be moved around inside (or on the surface of) the sphere in such a way that the overall length of the loop can be made to go to zero, i.e. equivalent to no path at all. This interesting manipulation is not illustrated in the video.

So the topology of the rotation group SO(3) (i.e. the group of rotations in 3-dimensional space) is non-trivial as shown in the video, and it has highly non-trivial physical consequences such as those illustrated by the Dirac Belt.

Ig Nobel Prizes 2007

2007-10-05T11:12:00.000Z

The Ig Nobel Prizes 2007 have now been awarded. Quoting from News24 here this year's winners include:

Chemistry: Mayu Yamamoto of the International Medical Centre of Japan, for developing a way to extract vanillin, or vanilla fragrance and flavouring, from cow dung.
"She seems to claim if companies start using this method it might help with global warming because some of all the cow dung that causes problems in the atmosphere will start getting used," Abrahams said in an interview.
Linguistics: Juan Manuel Toro, Josep B. Trobalon and Nuria Sebastian-Galles, of Universitat de Barcelona - for a study showing rats sometimes fail to distinguish between a person speaking Japanese backwards and a person speaking Dutch backwards.
Peace Prize: The Air Force Wright Laboratory, Dayton, Ohio for instigating research and development on a chemical weapon, the so-called "gay bomb", that "will make enemy soldiers become sexually irresistible to each other".
Biology: Dr Johanna EMH van Bronswijk of Eindhoven University of Technology, The Netherlands, for their census of all the mites, insects, spiders, pseudoscorpions, crustaceans, bacteria, algae, ferns and fungi that share our beds at night.
Economics: Kuo Cheng Hsieh, of Taichung, Taiwan, for patenting a device in 2001 that catches bank robbers by dropping a net over them, known as the "net trapping system for capturing a robber immediately".
The inventor, however, could not be found by Ig Nobel representatives in Taiwan "We had people in Taiwan looking for him. He's vanished. Somebody suggested to us the possibility that maybe the poor man was trapped inside his own machine," Abrahams said.

Enigma 1462

2007-09-28T15:59:00.000Z

Here is how Mathematica can be used to solve New Scientist Enigma number 1462 in the 29 September 2007 issue.

List of all 3-tuples of side lengths of red, blue, and yellow cubes. The upper limit of each of these is constrained by the known number of digits in the cubes of these side lengths in the red and yellow cases, and by the known number of digits in the square of the side length in the blue case.
(sizelist = Flatten[Table[{r,b,y}, {r,0,9}, {b,0,9}, {y,0,21}], 2]) // Length
2200

Select the cases where red volume is a 3-digit number, and the blue area is a 2-digit number, and the yellow volume is a 4-digit number, and the first digit of the red volume is the same as the first digit of the blue area, and the last digit of the blue area is the same as the last digit of the yellow volume.
sizelist2 = Select[sizelist, 100<=#[[1]]^3<=999 && 10<=#[[2]]^2<=99 && 1000<=#[[3]]^3<=9999 && IntegerDigits[#[[1]]^3][[1]] == IntegerDigits[#[[2]]^2][[1]] && IntegerDigits[#[[2]]^2][[-1]] == IntegerDigits[#[[3]]^3][[-1]]&]
{{5,4,16}, {6,5,15}, {7,6,16}}

Check the digits corresponding to the letters in "nil", "no", and "zero". Only the first case is admissable because it has all digits distinct for all these letters.
Map[
(
{volumered, areablue, volumeyellow} = {#[[1]]^3, #[[2]]^2, #[[3]]^3};
nilnozero = Flatten[Map[IntegerDigits[#]&, {volumered, areablue, volumeyellow}]];
nilozer = Drop[Drop[nilnozero, {-1}], {4}]
)&,
sizelist2
]
{{1,2,5,6,4,0,9}, {2,1,6,5,3,3,7}, {3,4,3,6,4,0,9}}

1. Choose the first case above which thus fixes the values of {n, i, l, o, z, e, r}.
2. Compute the volumes of the 3 coloured cubes.
3. Generate a list of permutations of these volumes, because which pile is which is unknown.
4. List all the possible alternative cases of {t, h, g}, i.e. permutations of one base case.
5. For each of these cases of {t, h, g} compute the total volume "nothing".
6. Generate a list of the numbers of cubes in each of the 3 piles, where the 3rd pile is the one with the unknown number.
{n, i, l, o, z, e, r} = %[[1]]
{volumered, volumeblue, volumeyellow} = {#[[1]]^3, #[[2]]^3, #[[3]]^3}&[sizelist2[[1]]]
volumelist = Permutations[{volumered, volumeblue, volumeyellow}]
thglist = Permutations[Complement[Range[0,9], {n, i, l, o, z, e, r}]]
volumetotallist = Map[FromDigits[{n, o, #[[1]], #[[2]], i, n, #[[3]]}]&, thglist]
numberlist=Append[Map[FromDigits, {{n, o}, {n, o, n, e}}], x]
{1,2,5,6,4,0,9}
{125,64,4096}
{{125,64,4096}, {125,4096,64}, {64,125,4096}, {64,4096,125}, {4096,125,64}, {4096,64,125}}
{{3,7,8}, {3,8,7}, {7,3,8}, {7,8,3}, {8,3,7}, {8,7,3}}
{1637218, 1638217, 1673218, 1678213, 1683217, 1687213}
{16,1610,x}

For each member of the list of volumes (see step 3 above) and each possible total volume (see step 5 above) generate an equation for the number of cubes in the unknown pile, and label each equation with the colour of the unknown pile.
eqns = Flatten[Outer[{Switch[#1[[3]], 125, "r", 64, "b", 4096, "y"], #1.numberlist==#2}&, volumelist, volumetotallist, 1], 1]
{{y,105040+4096 x==1637218}, {y,105040+4096 x==1638217}, {y,105040+4096 x==1673218}, {y,105040+4096 x==1678213}, {y,105040+4096 x==1683217}, {y,105040+4096 x==1687213}, {b,6596560+64 x==1637218}, {b,6596560+64 x==1638217}, {b,6596560+64 x==1673218}, {b,6596560+64 x==1678213}, {b,6596560+64 x==1683217}, {b,6596560+64 x==1687213}, {y,202274+4096 x==1637218}, {y,202274+4096 x==1638217}, {y,202274+4096 x==1673218}, {y,202274+4096 x==1678213}, {y,202274+4096 x==1683217}, {y,202274+4096 x==1687213}, {r,6595584+125 x==1637218}, {r,6595584+125 x==1638217}, {r,6595584+125 x==1673218}, {r,6595584+125 x==1678213}, {r,6595584+125 x==1683217}, {r,6595584+125 x==1687213}, {b,266786+64 x==1637218}, {b,266786+64 x==1638217}, {b,266786+64 x==1673218}, {b,266786+64 x==1678213}, {b,266786+64 x==1683217}, {b,266786+64 x==1687213}, {r,168576+125 x==1637218}, {r,168576+125 x==1638217}, {r,168576+125 x==1673218}, {r,168576+125 x==1678213}, {r,168576+125 x==1683217}, {r,168576+125 x==1687213}}

How many cubes are in the remaining pile and what is their colour? This is the only integer solution that can found to any of the above equations.
Select[Map[{#[[1]], Reduce[#[[2]], x, Integers]}&, eqns], (#[[2]]=!=False&)]
{{b,x==21413}}

Ising model simulation

2007-09-28T11:19:00.000Z

This posting shows how to use Mathematica (version 6) to create an interactive simulation of a 2-dimensional Ising model, which is based on some work on Boltzmann machines that I did back in the mid 1980s. This nicely illustrates the use of the Manipulate function in Mathematica, which is by far the easiest way of creating interactive graphical demonstrations that I have ever used.

Define a function for randomly initialising the state of the 2-dimensional Ising model.

init[narray_] := array = RandomInteger[{0,1}, {narray,narray}];

Define a function for doing a single Monte Carlo update of the 2-dimensional Ising model:

update[{pe_,pne_,pn_,pnw_}, narray_] :=
Module[
{probpatch={{pnw,pn,pne}, {pe,0.5,pe}, {pne,pn,pnw}}, i, j, di, dj, arraypatch, freq1, freq0},
{i,j} = RandomInteger[{1,narray}, {2}];
arraypatch = Table[array[[Mod[i+di,narray,1],Mod[j+dj,narray,1]]], {di,-1,1}, {dj,-1,1}];
freq1 = Apply[Times, probpatch arraypatch+(1-probpatch)(1-arraypatch), {0,1}];
freq0 = Apply[Times, (1-probpatch)arraypatch+probpatch(1-arraypatch), {0,1}];
array[[i,j]] = If[Random[] < freq1/(freq0+freq1), 1, 0];
];

Define a function for implementing the interactive graphical demonstration.

Manipulate[
If[narray!=Length[array], init[narray], ];
If[u, Do[update[{p2en[[1]],p2nenw[[1]],p2en[[2]],p2nenw[[2]]}, narray], {nupdate}], ];
ArrayPlot[array],
{{p2en, {0.5,0.5}, "probability(\[RightArrow],\[UpArrow])"}, {e, e}, {1-e, 1-e}, ControlPlacement->Left},
{{p2nenw, {0.5,0.5}, "probability(\[UpperRightArrow],\[UpperLeftArrow])"}, {e, e}, {1-e, 1-e}, ControlPlacement->Left},
{{u,"False","update"}, {True,False}, ControlPlacement->Top},
{{narray,32,"size"}, 3, 64, 1, Appearance->"Labeled", ControlPlacement->Top},
FrameLabel->"Ising Model",
Initialization->(e=0.001; array={}; nupdate=50)
]

When Manipulate is evaluated the initial output typically looks like this:

Use the check box at the top of the output window to switch the simulation on and off. The number of Monte Carlo updates that is applied between the display of each "frame" of the simulation is hardwired to be 50.

The initial size of the 2-dimensional array of "spins" is 32 by 32, which is controlled by the 1D slider at the top of the output window. This slider can be moved at any time, even whilst a simulation is being run.

Each Monte Carlo update is implemented by selecting a spin at random, and calculating 8 probability factors due to its interaction with each of its 8 surrounding spins. Each such factor specifies a multiplicative contribution to the relative likelihood that the central spin has the same/different value compared to its neighbouring spin (this is not the most general interaction that is possible). There are 4 independent probability factors corresponding to the following directions from the central spin: east, north, north-east, and north-west. These probability factors are controlled by the two 2D sliders on the left of the output window. The top 2D slider controls the east (left/right slider movement) and north (up/down slider movement) factors, and analogously the bottom 2D slider controls the north-east and north-west factors.

Denote the probability factor for a pair of spins being the same as p, and for being different as 1-p. The central position of each 2D slider corresponds to a probability factor p=1/2 (i.e. the same contribution whether the spins are the same or different), the left and bottom edges of each 2D slider correspond to p=0 (i.e. force the spins to be different), and the right and top edges correspond to p=1 (i.e. force the spins to be the same). These sliders can be moved at any time, even whilst a simulation is being run.

So, using a 1-dimensional example, the spin configuration 0?0 gives a product of probability factors (1-p)^2 for 010 and p^2 for 000, whereas 1?0 gives p(1-p) for 110 and (1-p)p for 100 (i.e. the same in each case). A Monte Carlo update of the central spin selects ? as being 1 or 0 with a relative probability given by the ratio of the corresponding products of probability factors. This ratio is ((1-p)/p)^2 and 1 in the two examples above, respectively. The 2-dimensional case is a straightforward generalisation of the 1-dimensional case.

Each snapshot of the output below shows the typical result of running a randomly initialised simulation for a few seconds with the 2D slider settings as shown in the snapshot.

Note that extreme values are used for the probability factors in each of these simulations, and because the simulations each have a short duration they do not actually reach equilibrium. For instance, the first simulation enforces such strong positive correlations between adjacent spins that it should eventually degenerate to all the spins having the same colour.

Enjoy playing with this Ising model simulation. It is quite interesting to move the 2D sliders to vary the probability factors as the simulation is running, because the speed of the simulation is sufficiently fast that you get an almost real-time response as the Ising model dynamically adjusts its equilibrium state.

High risk, high payoff research

2007-09-28T07:30:00.000Z

There is an interesting blog posting at Advanced Nanotechnology on The struggle over high risk, high payoff research, which pours scorn on the current low-risk approach to research funding. I have extracted some of the more interesting snippets below.

The blog posting itself makes the following comments:

...the United States science and technology research community has seen a return to a culture which is less likely to pursue high risk/high payoff technology research.

There is a struggle between those who want more High risk, high payoff scientific and technological research and development and those who want only timid, incremental goals who also ridicule even the description of a high payoff technological possibility.

Farber [who] sits on a computer science advisory board at the NSF [says]:

... he has been urging the agency to "take a much more aggressive role in high-risk research." He explains, "Right now, the mechanisms guarantee that low-risk research gets funded. It's always, 'How do you know you can do that when you haven't done it?' A program manager is going to tell you, 'Look, a year from now, I have to write a report that says what this contributed to the country. I can't take a chance that it's not going to contribute to the country.'"

The old head of ARPA Charles Herzfeld says (see here):

...the people that you have to persuade are too busy, don't know enough about the subject and are highly risk-averse ... If the system does not fund thinking about big problems, you think about small problems.

It's all pretty damning stuff. Only slightly tongue-in-cheek, I blame it all on the rise of the use of the spreadsheet as a management and accountancy tool, which makes it far too easy for unimaginative people to become so engrossed with the numbers in their spreadsheets that they overlook the big picture.

Enigma 1461

2007-09-21T12:38:00.001Z

Here is how Mathematica can be used to solve New Scientist Enigma number 1461 in the 22 September 2007 issue. The solution is optimised for clarity rather than brevity.

Generate a list of all the 24-hour clock squares.
squares = Select[Flatten[Outer[100#1+#2&, Range[0,23], Range[0,59]]], IntegerQ[Sqrt[#]]&]
{0, 1, 4, 9, 16, 25, 36, 49, 100, 121, 144, 225, 256, 324, 400, 441, 529, 625, 729, 841, 900, 1024, 1156, 1225, 1444, 1521, 1600, 1849, 1936, 2025, 2116, 2209, 2304}

Generate all 3-tuples from the list of squares. The convention is that each 3-tuple corresponds to {Tom, Dick, Harry}.
(tuples = Tuples[squares, 3]) // Length
35937

Week 1: filter the list of 3-tuples so that T < D < H and also H = T + D.
(triples1a = Select[tuples, #[[1]]<#[[2]]<#[[3]]&]) // Length
(triples1b = Select[triples1a, #[[3]]==#[[1]]+#[[2]]&]) // Length
5456
5

Use the cyclical relationship between the weeks to obtain the corresponding results for weeks 2 and 3.
triples2b=Map[RotateLeft, triples1b];
triples3b=Map[RotateRight, triples1b];

Extract the case where Harry's square is the same for all weeks, keeping only the value of Harry's square.
harrysquare = Intersection[triples1b, triples2b, triples3b, SameTest->(#1[[3]]==#2[[3]]&)][[1,3]]
400

Extract the 3-tuple of squares {Tom, Dick, Harry} for each week that contain the above value of Harry's square.
tdhsquares = Map[Cases[#, {_,_,harrysquare}][[1]]&, {triples1b, triples2b, triples3b}]
{{144,256,400}, {441,841,400}, {625,225,400}}

Extract Tom's squares.
Transpose[tdhsquares][[1]]
{144, 441, 625}

Drawing on air

2007-09-20T13:17:00.000Z

I learn from KurzweilAI.net about a report at PhysOrg.com (see here) on a new virtual reality system called "Drawing on Air" that allows artists to create 3-dimensional objects. They say

By putting on a virtual reality mask, holding a stylus in one hand and a tracking device in the other, an artist can draw 3D objects in the air with unprecedented precision. This new system is called “Drawing on Air,” and researchers have designed the interface to be intuitive and provide the necessary control for artists to illustrate complicated artistic, scientific, and medical subjects.

and

... artists can stylize their curves while drawing by dynamically adjusting line thickness and color. Haptic effects enable artists to intuitively adjust line thickness by applying pressure against an imaginary 3D surface, making drawing in the air feel similar to pushing a paintbrush against paper ...

and

... once you have this ability to sketch in the air, there are so many different artistic directions you can go with it ...

I'm quite excited by this development because it gets us closer to having an artistic medium that has the richness of the visualisation "studio" that we each have inside our head. The system is still too expensive for everyday use, so I will use a programmatic approach to 3-dimensional art for the time being (e.g. see the sculpture here), but I already have plans in the pipeline to add a simple "Drawing on Air" type interface to enable the artist to have more intuitive control over the software that creates their artwork, e.g. a standard gamepad can be used to manipulate curved surfaces to form 3-dimensional sculptures, and lots more besides.

Enigma 1460

2007-09-14T18:58:00.000Z

Here is how Mathematica can be used to solve New Scientist Enigma number 1460 in the 15 September 2007 issue.

This problem is designed to make brute force attack difficult because the set of candidate solutions starts off with (9!)^2=131681894400 elements, i.e. it is the Cartesian product of 2 sets of 9! elements. However, a methodical approach can be used to prune this set down until there is only one case left, i.e. the solution. The main trick is to prune each of the 9! element sets as much as possible before forming the Cartesian product of the residual sets, and keeping track of how many candidate solutions remain at each stage.

Create a list of digits 1,...,9 and a list of all permutations of this.
digits=Range[9]
(digitperms=Permutations[digits])//Length
{1,2,3,4,5,6,7,8,9}
362880

Create a list of the 9 colours and a list of all permutations of this.
colours={"grey","hazel","indigo","jade","khaki","lemon","mauve","navy","orange"};
(colourperms=Permutations[colours])//Length
362880

Without loss of generality, assume that "grey" is at circular position 1, and select all digit permutations where each circular position 2, ... , 9 (i.e. not including the "grey" position) has an odd and even digit on its neighbours.
(digitperms2=Select[digitperms,Apply[And,Map[OddQ[#[[1]]+#[[3]]]&,Most[Partition[#,3,1,{1,1}]]]]&])//Length
2880

Select the colour permutations where "hazel", "indigo", "jade" and "khaki" are in consecutive clockwise circular positions. There is no need to look for wrapped-round cases because "grey" is locked in circular position 1.
(colourperms2=Cases[colourperms,{"grey",___,"hazel","indigo","jade","khaki",___}])//Length
120

Build a Cartesian product list of all remaining digit permutations and colour permutations.
(digitcolourperms=Flatten[Outer[List,digitperms2,colourperms2,1],1])//Length
345600

Select the cases where the "hazel", "indigo" and "jade" digits add up to give the "khaki" digit.
(digitcolourperms2=Select[digitcolourperms,Total[Transpose[Cases[#,{_,"hazel""indigo""jade"}]][[1]]]==Cases[#,{_,"khaki"}][[1,1]]&[Transpose[#]]&])//Length
8640

Select the colour permutations where "lemon", "mauve" and "navy" are in consecutive clockwise circular positions. There is no need to look for wrapped-round cases because "grey" is locked in circular position 1.
(digitcolourperms3=Cases[digitcolourperms2,{_,{"grey",___,"lemon","mauve","navy",___}}])//Length
432

Select the cases where the "lemon" and "mauve" digits add up to give 2 times the "navy" digit.
(digitcolourperms4=Select[digitcolourperms3,Total[Transpose[Cases[#,{_,"lemon""mauve"}]][[1]]]==2Cases[#,{_,"navy"}][[1,1]]&[Transpose[#]]&])//Length
24

Select the cases where the "hazel" digit is the same as the number of times you can find a digit equal to the sum of the digits on its neighbours. This leaves just one possibility which is the required solution.
Select[digitcolourperms4,(x=Transpose[#];Length[Select[Partition[x,3,1,{1,1}],#[[1,1]]+#[[3,1]]==#[[2,1]]&]]==Cases[x,{_,"hazel"}][[1,1]])&][[1]]//Transpose
{{1,grey}, {3,hazel}, {2,indigo}, {4,jade}, {9,khaki}, {5,orange}, {8,lemon}, {6,mauve}, {7,navy}}

Hunting for the fundamental laws of physics

2007-09-12T16:10:00.000Z

Stephen Wolfram has posted here some interesting news about one of his hobbies - hunting for the fundamental laws of physics, where he outlines how he is both developing and using Mathematica to search for fundamental laws of physics that generate simulated universes which have properties that resemble our own real universe.

The power of his approach lies not only in his use of Mathematica, but more fundamentally in his use of very few axioms to define what the fundamental laws of physics are in the first place. His basic object is a network (i.e. nodes and links-between-nodes), and his basic operation is the mutation of a piece of the network (via the application of a set of rules), which thus allows the network structure to have a dynamical behaviour. In this approach the fundamental laws of physics are determined by the choice of the set of network update rules.

It turns out that various simple consistency criteria cause this approach to give rise to both special relativity and general relativity. That is impressive, starting from a rule-based approach!

One of the challenges is to determine the consequences of a particular choice of network update rules, and to ascertain if they correspond to the behaviour of our known real universe. In general, the network behaviour in response to its update rules can be extremely complicated, and working out what is going on can thus be very difficult and time consuming. The development of Mathematica itself is partly driven by the need to create tools for addressing problems such as this.

Wolfram says that he has not yet found a viable candidate for the fundamental laws of physics using this approach, but that he is hard at work both developing and using Mathematica to achieve this goal. As he says

I certainly think it'll be an interesting - almost metaphysical - moment if we finally have a simple rule which we can tell is our universe. And we'll be able to know that our particular universe is number such-and-such in the enumeration of all possible universes.

I wish him luck in this venture. It would be very impressive if he found that a 3-line Mathematica program was all that was needed to generate the behaviour of our known real universe. Even if he is destined not to discover the fundamental laws of physics using this approach, he will nevertheless have created along the way a very useful toolbox for doing lots of other things, i.e. Mathematica.

SciVee - a YouTube for science

2007-09-11T09:54:00.000Z

I learn from ars technica here about SciVee which will

...provide a form of scientific communication that's intermediate between abstracts (which take a few minutes to read) and a full reading of a paper (which can take hours). The primary type of video presentation that SciVee intends to host could be called a "pubcast," in which a researcher provides a short video description of their work that's synchronized to the display of text from the paper.

and also about the process of creating a SciVee presentation

What's the incentive for researchers to put the effort into creating a pubcast? "There's actually a low barrier to entry," Bourne said. "All you need is a webcam and iMovie or Movie Maker." The SciVee site has tutorials for recording and editing video content on both Mac and Windows platforms. Once the resulting video is uploaded, the site's software walks users through synchronizing it with the text of the paper.

I think SciVee will catch on very quickly indeed. There is a large gulf between reading just a paper's abstract and reading the entire paper which SciVee fills well, and because it is video it is (potentially) a very engaging medium with which to attract people's attention. No doubt I will try out SciVee before long, and I will write about my impressions of it here.

The Spline - a movie

2007-09-09T13:32:00.000Z

Here is a little doodle that I created whilst waiting for Sunday lunch to cook:

Apologies to Channel 4 for my blatent plagiarism of their animated logo idea.

Boltzmann brains

2007-09-08T17:13:00.000Z

In the 18 August 2007 issue of New Scientist there is an article Spooks in space by Mason Inman about so-called Boltzmann brains. Quoting from the introduction to the article

Boltzmann posed the question of whether the universe could have arisen from a thermal fluctuation; his work presaged the idea that a fluctuation could also give rise to a conscious entity that sees the universe. In this regard Boltzmann brains are not necessarily actual brains, but rather are a metaphor for observers of the universe that might appear spontaneously.

Thus a Boltzmann brain is a conscious entity that instantaneously pops into existence as a spontaneous fluctuation of matter into a highly ordered form, rather than gradually coming into existence like us through the slow process of evolution gradually rearranging matter into a highly ordered form. The probability of a whole brain suddenly popping into existence in this way is extremely small, because the internal structure of the Boltzmann brain has to be exactly right so that it works as a brain, so you have to wait a very long time for there to be a significant likelihood of a Boltzmann brain coming into existence. This likelihood problem is very deftly avoided by evolution, because it breaks the overall problem of making a brain into very small steps, each of which is much more likely to occur than the whole big jump from start to finish; of course, evolution doesn't know in advance that this is what is going on.

More generally, you could imagine a whole spectrum of processes ranging from the instantaneous rearrangement of matter at one extreme (e.g. Boltzmann brain) all the way through to the gradual rearrangement of matter at the other extreme (e.g. evolution).

In a sufficiently large and long-lived universe Boltzmann brains will come into existence, because then the extremely small likelihood of one coming into existence at any given place and time is offset by the large number of alternative places and times that are available in the whole universe (i.e. space-time). Our universe looks exactly like the sort of place where these conditions are (or will be in the far future) satisfied, so Boltzmann brains will eventually come into existence here.

The eventual existence of Boltzmann brains worries cosmologists, because the laws of physics that are deduced by a Boltzmann brain (which is a conscious observer) would be different from the laws that are deduced by us. The reason for this difference is that Boltzmann brains would typically come into existence a very long time in our future when the universe is much larger, emptier and colder than now, so the typical observations made by a Boltzmann brain would be very different from the typical observations made by us, so a Boltzmann brain would deduce laws of physics that are very different from ours.

This fact worries cosmologists so much that they would very much like to find a way to "banish" Boltzmann brains from existence, for instance by finding some property of the currently known laws of physics that prevents favourable conditions for Boltmann brains from ever arising.

I am not as worried as cosmologists are about the potential existence of Boltzmann brains who would deduce different laws of physics from us. My reasoning is as follows:

Firstly, we are not special in the grand scheme of things, because we are basically the same type of information processors as Boltzmann brains. Both Boltzmann brains and we are two extreme examples of the outcome of a spectrum of processes that have rearranged matter in the universe. Our brains have come into existence via the gradual process of evolution, which stores its intermediate results in the form of DNA, and then uses this as the starting point for the next step in the rearrangement of matter. Whereas Boltzmann brains pop into existence without going via any of these intermediate steps. There is also a whole spectrum of intermediate cases (e.g. partly DNA and partly random chance) that one could imagine; the only way that DNA can evolve is to live a little way into this intermediate regime where there is a small element of random chance. Our brains are not particularly special in this spectrum of possibilities, other than because evolution using DNA (or something analogous) is the only process that has a significant likelihood of making something as complex as our brains in a universe that is as (relatively) young as ours is. The other possible processes for making brains that lie nearer the Boltzmann brain end of this spectrum will need much longer to have a significant likelihood of happening.
Secondly, the laws of physics that we deduce are (at least partly) environmentally determined, so it doesn't matter too much if observers living in very different universes (or at very different times in the same universe) deduce different laws of physics. We set up our experiments and make observations, then we discover a low-complexity "explanation" of all of these observations and call it the "standard model" (or whatever). The scientific method is driven by whatever experimental observations we make, and we have little choice in this other than the freedom to choose which particular experiments we conduct. We grandly give the name "laws of physics" to our explanation of all of our observations, but the way in which this explanation is constructed leaves a lot of room for doubt about its uniqueness or inevitability. It may yet turn out that the laws of physics are somehow unique/inevitable, but that is far from being obvious right now. It is much safer to keep an open mind, and to assume that the laws of physics are simply what we observe-then-guess them to be, and to be thankful that mathematical beauty and elegance has taken us as far as it has in constraining the precise form of the laws of physics. It may yet turn out that there is a lot more mileage to be had in the mathematical beauty/elegance approach, but we should not assume that this is inevitable.

So we are not special in the grand scheme of things, and the laws of physics that we deduce are (at least partly) environmentally determined. Together, these two points mean that I am not as worried as cosmologists about the potential existence of Boltzmann brains.

Wouldn't it be nice to write a computer simulation of a synthetic universe whose properties gradually changed as the simulation proceeded, in which different types of information processing entity (brains, if you wish) emerged at different times during the simulation, and which interacted with their simulated environment to deduce what they called the "laws of physics" that governed their existence? None of these information processing entities would be "special" in any way (although they might believe themselves to be special!), and the "laws of physics" that they deduced would be environmental.

Enigma 1459

2007-09-08T13:39:00.000Z

Here is how Mathematica can be used to solve New Scientist Enigma number 1459 in the 8 September 2007 issue.

This is a straightforward digit manipulation problem which can easily and quickly be solved by a brute force search. In this problem there is no point in trying to do any clever programming.

Initialise the counter.
n=1;

Step the counter until the condition on its digits is satisfied.
While[Not[MemberQ[Map[FromDigits, Permutations[IntegerDigits[n]]], 12n/11]], n++]

Display the counter value that satisfies the condition.
n
1683

West Malvern sunset

2007-09-06T19:35:00.000Z

This is my favourite photo of sunset as seen from my house high up on the west side of the Malvern Hills. I have never seen so many different types of cloud layer in the same small area of the sky.

Unfortunately, this year's summer weather has been useless so there were no opportunities to take good sunset photos. The photo above is one from my archive which I took on 26 July 2001.

Caleb Clarke's favourite painting

2007-09-06T19:17:00.000Z

This one is for Caleb Clarke. He painted this remarkable picture, and then later sold it to me. I think he regrets having sold it, so I am posting a photo of it here for him and others to enjoy.

It was difficult to take this photo because I had mounted the painting in a glass-fronted frame, so I took the photo from slightly to one side in order to avoid glare from reflection of the camera flash in the glass, and then after that I had to do a bit of image processing to massage the photo back to the correct rectangular shape.

Enjoy, Caleb!

The Spline - a musical theme

2007-09-05T20:52:00.000Z

To play this music click here.

Singularity Summit 2007

2007-09-05T14:58:00.000Z

The Singularity Summit 2007 is about to start. I wish I could be there, but I will have to content myself with being an observer on the sidelines.

They say in the introduction on the website:

The Singularity Institute for Artificial Intelligence presents the Singularity Summit 2007, a major two-day event bringing together 18 leading thinkers to address and debate a historical moment in humanity's history – a window of opportunity to shape how we develop advanced artificial intelligence.

The introduction goes on to explain what "The Singularity" is:

For over a hundred thousand years, the evolved human brain has held a privileged place in the expanse of cognition. Within this century, science may move humanity beyond its boundary of intelligence. This possibility, the singularity, may be a critical event in history, and deserves thoughtful consideration.

That sounds like hype, but it is not.

Artificial intelligence is a very active area of academic research, and useful software continuously spins-off from this research. All such software applications are very specific in their scope, but their very specificity means that each such application can excel in its own chosen area. Currently, there is no general-purpose artificial intelligence software that can, like a human can, excel at a wide range of activities. The idea of The Singularity is that sometime during the 21st century we will have progressed artificial intelligence software (and hardware) technology to the point where human-level performance (and beyond) will not only become possible but will be highly likely.

Each generation of artificial intelligence will assist in the development of the next generation (this is what happens even now), and this development process will accelerate as the need for external human assistance in the design process becomes less with each advancing generation of AI, until AI can do the development all by itself. Once AI can develop its own next generation without human assistance the pace of progress can become very rapid indeed, and if external resources such as materials and energy were unlimited (which they are not, of course!) then the pace of progress would become so large that it would seem to be infinite (although it is not, of course!).

Hence the use of the phrase "The Singularity", because it marks a fairly sharp transition between the human intelligence that dominates now and the advanced artificial intelligence that will exist (I hesitate to say "dominate"!) in the future. The AI technology that emerges from The Singularity will be thinking thoughts that we will not be able to accommodate within our limited-ability biological brains, so making predictions about the post-singularity era is fraught with difficulties. There is a more detailed discussion of the term "The Singularity" given here.

What sort of things would have to happen in order to make The Singularity (or a smoothed-out version thereof) possible?

We would use an advanced form of nanotechnology to evolve & grow massively parallel fine-grain computer architectures, rather than use the current approach where we design & build every small detail of the computer architecture ourselves. This type of evolution would use an artificial form of DNA to record the long-term state of the evolutionary process, and this type of growth would make use of molecular self-organisation to assemble the computer. This is essentially a fine-grain form of artificial life.
We would use external training to teach the computer what its observed behaviour should be, rather than internal programming to dictate to the computer what its internal workings should be. This training process would involve interaction of the AI with its environment via sensors (e.g. inputs such as eyes and ears) and effectors (e.g. outputs such as touch and speech), and one possible training environment might be Second Life (or something similar).
We would largely remove the artificial distinction that exists between software and hardware, so that each particular behaviour of electrons/molecules/etc in the computer has a unified existence rather than being split up as hardware+software. Currently, the use of programmable architectures has some of the spirit of this unified approach. A useful behaviour that is learnt by one generation could be optionally hard-wired into the next generation (i.e. Lamarkism), but it is not clear that this would be easy to do in a fine-grain architecture that arises through evolution & growth.

An advantage of using our own technology (rather than the outcome of biological evolution) to implement an artificial intelligence is that we can optionally hard-wire some of its behaviour. This could be achieved by "steering" (e.g. selective breeding) the process of evolution & growth that gives rise to the computer architecture in the first place, which could be used to influence the behaviour of the AI in many ways. This offers the possibility of using our human influence to create "nice" advanced AI, but unfortunately the same technique could also be used to create "nasty" advanced AI.

One thing that always comes up in conversation about this sort of advanced artificial intelligence is "do we need it?", or "do we want it?", and so forth. I'll cut straight to the bottom line. The applications of this type of technology are so wide-ranging, the potential for enhancing the quality of our lives is so great, the potential for defending ourselves against an aggressor who might wish to deploy this technology against our better interests is automatically inbuilt (yes, the "arms race" argument!), and so on, that I see no way of suppressing this technology. We have to learn to live with it. If this sort of advanced artificial intelligence is technically possible (and there is no obvious reason why it is not) then it will eventually come into existence no matter how much we try to stall the process.

So I will be watching what happens at the Singularity Summit 2007 with great interest, and I think you should too.

Update: There is now a report and discussion on the presentations at the Singularity Summit at Reason magazine here entitled "Will Super Smart Artificial Intelligences Keep Humans Around As Pets?". I could find only one mention of "nanotechnology", and then only in the context of optimising resource usage (i.e. making things smaller to get more computing done). There was no mention of the clever use nanotechnology, such as using synthetic DNA to evolve & grow massively parallel fine-grain computer architectures, as I discussed above. I find this omission very odd indeed.

Update: Here are links to some live-blogging on the Singularity Summit 2007 to be found on David Orban's blog, which I heard about here on Tommaso Dorigo's blog:

Singularity on the front page (of The San Francisco Chronicle)
Liveblogging the Singularity Summit 2007?
Liveblogging the Singularity Summit 2007 - Day One - morning
Liveblogging the Singularity Summit 2007 - Day One - afternoon
Liveblogging the Singularity Summit 2007 - Day Two - morning
Liveblogging the Singularity Summit 2007 - Day Two - afternoon

Well, I said "So I will be watching what happens at the Singularity Summit 2007 with great interest, and I think you should too.". If what I read in the above live-blogs is even a vaguely accurate report of the sort of discussion that went on at the Singularity Summit, then I am very disappointed by the airing of so many apparently superficial opinions there. Maybe the purpose of the event was to strut their stuff before the adoring eyes of the press, and the meat of the arguments was hidden away behind the scenes. What a pity.

Update: The presentations given at The Singularity Summit 2007 are now online at http://www.singinst.org/media/singularitysummit2007.

Second Life grid

2007-09-05T10:35:00.000Z

Linden Lab has announced (see here) the launch of the Second Life Grid. This is an exciting development that moves the popular Second Life virtual world towards being

a resource for businesses, organizations and educators for creating a successful virtual presence on the Second Life Grid platform

Linden Lab also says

the Second Life Grid will enable these organizations to understand and create meaningful 3D immersive experiences ... The platform, tools and programs available on SecondLifeGrid.net will provide the foundation needed to create a successful virtual world experience.

From the users' point of view the Second Life Grid offers all the advantages of virtual worlds in general, with none of the disadvantages of the somewhat anarchistic ongoings in Second Life itself. This is a very exciting development because finally we have the promise of a programmable computing environment which we actually inhabit, rather than being limited to using software that we interact with whilst remaining firmly outside of it (as it were).

Ultimately, environments such as this will be so realistic that it will be easy to forget where the real you is, and then the meaning of "real you" becomes rather ambiguous. Have a look at Counterfeit World (a.k.a. Simulacron-3) which is a book-length essay on this problem; I read this when I was a teenager and I was hooked from that point onwards.

One of my interests in programmable virtual worlds such as this is that I can use them to create simulations that mimic what goes on inside my head. I am a highly visual thinker so I understand a concept by expressing it visually as a 3-dimensional simulation in my mind's eye. If I can formulate a concept visually then I can make fluid use of it (think "bird's eye view"), and if I can't then I have to use it in a plodding one-step-at-a-time sort of way (think "worm's eye view").

Prior to the advent of virtual world technology my visualisations have spilt over into the real world in the form of diagrams that I draw to illustrate selected freeze frames of the visualisation, but to other people these diagrams can appear out of context so their full meaning is elusive. However, the promise of virtual world technology is that I can now recreate a more faithful representation of what goes on inside my head when I am visualising.

It is hard work to create a rich virtual world using the current primitive programming tools in Second Life, i.e. the Second Life scripting language. I find that currently the best approach is to do as much as possible of the programming work outside Second Life (using Mathematica, in my case), and to then upload the results to Second Life. Even when I use the best programming tools that are available to me, the overall process of generating virtual world simulations is hard work, but at least it is now feasible whereas before the existence of Second Life it was impossible.

It will be interesting to see how virtual world programming tools develop over the next few years. In the meantime, to be as productive as possible using virtual world technology, it is best to tailor your ambitions to the sorts of things that are relatively easy to do using current programming tools, which means that you have to experiment continuously with the tools to see what they can do for you. For me that means building up my "flying time" in Second Life, and concentrating on doing work there rather than mucking about ... yes, there are distractions!

Enigma 1458

2007-09-04T00:45:00.000Z

Here is how Mathematica can be used to solve New Scientist Enigma number 1458 in the 1 September 2007 issue.

Define the digits used in the 7-digit number.
u = {n6, n5, n4, n3, n2, n1, n0};

Define the reordered digits used in the 6-digit number.
v = {n0, n1, n2, n3, n5, n4};

Define the 7-digit number.
a = FromDigits[u] // Expand
n0 + 10 n1 + 100 n2 + 1000 n3 + 10000 n4 + 100000 n5 + 1000000 n6

Define the 6-digit number.
b = FromDigits[v] // Expand
100000 n0 + 10000 n1 + 1000 n2 + 100 n3 + n4 + 10 n5

Define some inequality constraints on the digits.
c = Apply[And, Map[0 <= # <= 9 &, u]]
0 <= n0 <= 9 && 0 <= n1 <= 9 && 0 <= n2 <= 9 && 0 <= n3 <= 9 && 0 <= n4 <= 9 && 0 <= n5 <= 9 && 0 <= n6 <= 9

Find solutions for the digits in the domain of integers. Searching for 3 solutions and finding only 2 demonstrates that there are only 2 solutions, one of which is the trivial solution with all digits zero.
s = FindInstance[a == 3 b && c, u, Integers, 3]
{{n6->0, n5->0, n4->0, n3->0, n2->0, n1->0, n0->0}, {n6->2, n5->9, n4->3, n3->9, n2->9, n1->7, n0->9}}

Verify the 2 solutions.
a == 3 b /. s
{True, True}

Computer assisted sculpture

2007-09-02T19:39:00.000Z

I am going to show you a way of creating sculptures using Mathematica. For the time being, I will show you just one finished example that I prepared earlier, but I will return to this theme in the future to show more generally how you can do computer-assisted sculpture, and lots more besides.

Here is the finished object (i.e. the letter "X") displayed using Mathematica's tasteful 3D rendering:

The basic trick to creating this sort of sculpture is to start with a sheet of elastic "paper", and to then stretch and fold it to the required shape. The allowed moves are basically the same as in origami, except for the fact that the "paper" used here is elastic. Also, in the example shown here the sheet starts off curled round into a cylinder.

The simplest way to see how the cylinder is deformed into the final letter "X" is to see a video of the whole process:

The various steps shown in the above video are:

Start with a cylinder.

Pinch the top and bottom of the cylinder to bring the front and back sheets of its surface together. The aim of this is to create two separate tubes that will eventually become the left and right halves of the "X". At this point in the video there is an artefact where the front and back sheets of the surface pass through each other; this is a side effect of the interpolation method that I used to create intermediate frames in the video.

Fill out the waist of the above surface to compensate for the fact that the pinching operation (2) has made the front and back sheets of the surface touch all of the way from top to bottom. The aim of this is to recreate a 3D volume contained between the front and back sheets of the surface.

Vertically stretch the left and right tubes of the surface. The aim of this is to begin to make these tubes look a bit more like what they need to be to make an "X".

Bend the top and bottom ends of the left and right right tubes outwards. The aim of this is to make these tubes look even more like what they need to be to make an "X".

Vertically constrict the middle of the surface, and stretch the tubes vertically. The aim of this is to accentuate the left and right tubes of the surface, which makes them look like the required "X".

Sharpen the edges of the surface. The aim of this is to make the final shape of the "X" cleaner and crisper.

Each of the steps above is a simple deformation of the surface, and the sequence of steps is carefully arranged so that the surface gets gradually deformed towards the required shape, i.e. the letter "X". More generally, in addition to the basic deformation operations used above, a different starting shape for the surface could be used (e.g. a plain 2D sheet), or more complex operations could be used such as cutting/gluing the surface to create any sculpture that you want.

In the future I will return to this theme to describe it in more detail. I will also post a link here to a more detailed desciption of the steps, including complete Mathematica code.

Technorati activation

2007-08-31T19:07:00.001Z

Technorati Profile

We are open for business again

2007-08-25T16:08:00.000Z

It has been a long while since I last blogged - luttrellica.blogspot.com was on 25 February 2007 and acenetica.blogspot.com was on 26th January 2007. I have been away on a long "sabbatical" to fight for my right to work and think independently; this fight is ongoing but I hope it will be resolved before too long.

I found that previously running two blogs (i.e. acenetica and luttrellica) in parallel was rather inconvenient because there was no clear separation between their subject matter, so I will now maintain a single blog (with comments) here at www.blogger.com and occasionally link to detailed blogs (without comments) that I maintain in my own separate webspace.

This allows me to post much more detailed and useful material that I have generated using Mathematica, which can then be used by anybody who has previously downloaded the free Mathematica Player.

Usually I will not censor comments, but there are limits to what I will allow commenters to get away with, so I will censor rants, ad hominem attacks, etc. Just use your common sense when you post comments here.