These puzzles appeared in Issue 04 of the magazine.
In the brief tradition of Chalkdust cover articles there is a developing discussion of how mathematics and art are related.
Art is simply the making of representations. Art happens when a person has an idea or a vision that exists in their imagination (the mind’s eye) and is impelled to communicate said idea by making a visible manifestation (representation) of it in the material world. The idea or vision on its own is not art. Art occurs amid the struggle to make a representation of the idea that the artist can show to other people. Art may be relatively `fine’ or popular, conceptual or objective, highbrow or applied, yet still fall within this definition. Judgements about the quality of art are made largely by consensus among the cognoscenti in a given art milieu. These judgements are subject to change over time as the perception of works of art are always modified by the current `cultural environment’ and fashion.
So artists and mathematicians share the `having of ideas’. But what then? Mathematicians communicate their ideas—yes. But ideas in maths take the form of theorems or conjectures about numbers, space or other abstract entities. The quality of these ideas is first assessed by proof. Can the idea be shown to be true? And second, if the idea is true, is it interesting? That is, does it usefully contribute to the mass of existing mathematics? Communication of mathematical ideas may require the invention of new symbols or diagrammatic forms, etc, but these are in the nature of being a new language, not art.
In my view then, art and mathematics share the magical process of `idea getting’ but essentially differ in where they go with those ideas. If maths is to be considered an art, it would have to be a sort of `super-art’ or art `to a higher power’. Easier, I think, to class mathematics as the science of number, space, shape and structure, etc—the abstract entities that exist in our minds.
Imagine an intelligent alien’s perception of our arts and our mathematics. Our art would be more or less incomprehensible, depending on how alien the being was; but our maths would be as true for the alien as it is for us. Furthermore, good mathematics will not diminish with time or go out of fashion.
There is an affinity between some mathematicians and some artists. Certainly, it is a most pernicious error that scientific and artistic talent exclude each other—an idea unfortunately common among school counsellors. The common ground between art and science/maths that leads us to the `getting of ideas’ is the activity we call play. The thing of it—the thrilling thing, the magical thing—is the moment when one discovers a new idea, or pattern, or conceptual framework, or whatever: the eureka moment! And are these moments not usually approached through playing in the mind with new combinations and orderings of existing mental constructs?
I had one of my most memorable eureka moments sometime in 1971 while sitting on a dead tree in Epping forest. At that time, I had been collaborating on an automatic projective line drawing program with `hidden line removal’, going where Autocad later arrived. I was considering algorithmic approaches to colouring surfaces in projective drawings. I realised that if I thought of objects in the scene as being represented mathematically as arrays of vertices and planes in some coordinate space, then I could solve for the equation of the line going from an eyepoint through a particular pixel in the image plane and into the scene (as in the diagram). From the equation of the line, I could find the closest surface along the path and then compute the colour and illumination value for that pixel based on the defined colour on the surface, along with its relationship to any light source or other light-emitting surfaces. And so I had invented ray tracing—the foundation of all computer generated synthetic imaging for special effects in cinema, television and gaming. Of course, I neither invented it first nor alone—and I certainly had neither the persistence nor vision to pursue ray tracing to practical or rewarding development. But its discovery was a thrill, as were the few simple pictures I made using the technique in a primitive manner on the pen plotter available.
As spheres have become my most persistent motif, I will end with two more related works that play on the division and articulation of spherical surfaces: Sphere Architecture and Star Sphere.
[ Pictures: Central Quadratic: Used with permission from UCL Art Museum, University College London; Spheres, Sphere Architecture and Star Sphere: Used with permission from John Crabtree ]
This issue, Top Ten features the top ten colours of chalk! Then vote here on the top ten parts of a circle for Issue 05!
At 10, it’s perfect for colouring in sideburns: brown chalk.
A re-entry at number 9, it’s yellow chalk by UCL’s second most successful band, Coldplay.
You might think that maths and psychedelic hallucinations tend not to mix very well. But you would be mistaken! There are a series of visual hallucinations known as form constants that are highly geometric, and a mathematical model of them has provided us with some fascinating insight into how our visual cortex (the part of the brain that processes the information we receive from our eyes) works.
These hallucinations were first observed in patients who had taken mescaline, a psychedelic drug produced from a cactus found in South America. Form constants have subsequently been reported in a number of other altered states such as sensory deprivation, waking/falling asleep states, near death experiences and by individuals with synaesthesia. Some people even report seeing these patterns after closing their eyes and applying firm pressure to both eyelids for a few seconds!
The mathematical model we referred to was described in a paper by Bressloff et al., and is based on anatomical features of our brain. It seems that the visual cortex has certain symmetry properties, such as reflective, translational and even a novel shift-twist symmetry. Its electrical activity can be represented mathematically and—a bit of group theory, some eigenvectors and a couple of transformations later—has steady state solutions to the resulting equations that are remarkably similar to the observed hallucinogenic experiences. Groovy!
Disclaimer: Chalkdust does not advocate pressing hard on your eyelids.
[Written in collaboration with Samuel Mills. Pikachu adapted from picture by Matt Levya, CC BY 2.0; Hallucination pictures taken with kind permission from PC Bressloff, JD Cowan, M Golubitsky, PJ Thomas and MC Wiener, What geometric visual hallucinations tell us about the visual cortex, Neural Computation 14(3) (2002), 473–91.]
Our original prize crossnumber is featured on pages 48 and 49 of Issue 04.
- Although many of the clues have multiple answers, there is only one solution to the completed crossnumber. As usual, no numbers begin with 0. Use of Python, OEIS, Wikipedia, etc. is advised for some of the clues.
- One randomly selected correct answer will win a £100 Maths Gear goody bag. Three randomly selected runners up will win a Chalkdust t-shirt. The prizes have been provided by Maths Gear, a website that sells nerdy things worldwide, with free UK shipping. Find out more at mathsgear.co.uk
- To enter, submit the sum of the across clues via this form by 7 January 2017. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 21 January 2017.
Moonlighting agony uncle Professor Dirichlet answers your personal problems. Want the Prof’s help? Send your problems to email@example.com.
This year has been a bit tough for us financially, and the only summer holiday we were able to afford was a week camping in the not-so-sunny East Midlands. My work colleagues, however, won’t stop talking about their blissful trips to the beaches of southern Europe. I don’t want to seem jealous but I wish they’d stop chatting about their suntans. How can I move the conversation on?
— Beyond the pale, Selly Oak