I hope the title is not too confusing, given my previous posts on surfaces. The title of this post is also the title of a work of mine that I have mentioned here a couple of times before. It has just gone up in the common room of the Maths Department of Imperial College. If you are a London mathematician take a look and let me know what you think!
Here is the work in place. The order was chosen by Anne Rowlands and Andy Pope. I love seeing how they chose to interpret my work!
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Art | Tagged: 4d, Art, communication, geometry, Imperial College London, Mathematics, painting, surfaces, Tiling, visual maths | 
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Posted by gelada
A novelist is, like all mortals, more fully at home on the surface of the present than in the ooze of the past.
Vladimir Nabokov, Strong Opinions
Curves and surfaces are a wonderful visual representation of mathematics. They can move from the simple and profound to the complex and intriguing. They have even been accused of being part of a sinister plot. In addition the mathematics behind them is becoming increasingly useful in many areas, algebraic statistics for example. I began this article with the idea that interest in the physical and visual sides of these objects was in a rather sharp decline. The curves and surfaces courses that I studied had rather few images, and beyond the Science Museum I had not seen a decent collection of mathematical models (and they have hidden a lot of theirs). However on scratching the surface of the subject I found a huge wealth of material. In fact so much that I have decided to split up my post (this one ended up at 1600 words anyway!). This first post will introduce mathematical surfaces and give some snapshots from their history and links to art. I plan two subsequent posts (this will be edited when they come online). Firstly an introduction to algebraic surfaces, secondly a discussion of splines and nurbs and how they provide a practical tool to designers (without them having to worry too much about the maths).
So what is a surface from the point of view of mathematics? It is a two-dimensional topological manifold but this is just jargon. Start instead by thinking of a sheet that might be folded and draped. We normally want to consider surfaces that are smooth, which means that the sheet has no creases.
Vowel, Alison Watt
However surfaces can get wilder in several ways. For example they can bend round and connect back to themselves. In fact we consider surfaces like this all the time. The surface (english meaning not mathematical) of any object you might pick up is like this. When the surface connects back onto itself and has no edges, it is compact. For example the surface of a ball is a surface, called, unsurprisingly, a sphere. Surfaces can also have holes. The simplest example being the torus which is the surface of a ring donut.
Loop in Layers, Eva Hild
The number of holes through a surface is called the genus and, for compact surfaces that we can create in three dimensions, the genus gives a complete topological description. Topology considers what happens when the surface is stretched and deformed but not glued or torn. This is the reason you will sometimes hear that a topologist cannot tell the difference between a donut and a coffee cup, as both have one hole.
A second strange behaviour comes from the famous Möbius strip. To make this we take a strip of paper and put one twist in it. The two ends of the strip are then connected. This creates a surface with only one side, as by walking along the surface, without going over the edge we can get from one side to the other.
Minimal Möbius, Benjamin Storch
The Möbius strip is not a compact surface, as it has an edge. However it can be made into a compact surface by attaching its boundary to the boundary of a disc. This gives a compact surface called the Klein bottle, that retains the property that it has only one side. However the property of being able to get from one side to the other is a topological one. We cannot create two sides simply by stretching and bending. How does this correspond to my statement above that the topological information is given by genus alone? The answer is that I cheated. I added the vague terms that we could make the surface in three dimensions. This is impossible for Klein bottle unless we allow the surface to cut through itself.
Klien Bottle, Alan Bennett
Surfaces with this property are called non-orientable there is an analogous counting concept to genus for such surfaces, but it is a little more complicated. However genus and the distinction between orientable and non-orientable completely describe the topology of any compact smooth two dimensional surface.
One concept that I will mention in passing is that of minimal surface. These arose as surfaces that minimised area subject to some constraints. For example containing a certain line in three dimensions. They can be hard to find precisely by analytic methods, yet soap bubbles can find them very quickly. More recently the definition has become surfaces which have zero average curvature. More importantly however they can be made out of lego:
The Catalan Minimal Surface, Andrew Lipson
Topology is an interesting area that helps to understand some of the processes of modern mathematics. There are obvious differences between objects with the same topology (donuts and coffee cups), yet they do share certain features. Such features cannot be changed by a well defined (though large) set of operations. More importantly up to those operations we can understand all possible behaviour. We are therefore able to give a complete classification of surfaces. One of the origins of topology comes from the study of surfaces defined in a concrete way that yields a zoo of examples. These surfaces are called algebraic surfaces.
Algebraic surfaces are the set of points in three dimensions that give the solution to polynomials with three variables. For example, consider the polynomial 
Kolibri
Click the image above for a site with a wonderful zoo of examples. You can really find nearly anything you want for example to express your love. This zoo of examples is one of the reasons that mathematicians moved to topology to get an idea of the limits of what could happen. Even with a computer it is not trivial how to construct a model for any polynomial. However such programs do exist. You can find your own examples and generate models to play with in your 3d programs.
However when mathematicians started considering such equations that did not have these luxuries. Things began in the seventeenth century with Descartes and the use of algebra and co-ordinates to study geometry. By the eighteenth century it had been established that many previously studied curves, such as the conic sections were in fact the solutions of polynomials. For example the parabola is the set of solutions of the equation 
Surface of order three with four real double points (A1). Schilling
Even though today such models are no longer an essential part of any mathematics library there are still many places they can be enjoyed. Many universities still have (dusty) collections, as does the Science Museum in London (including polished wood models). Although it is not that satisfying to see a three dimensional model as an image there are also many places to see large collections online. The collection of the University of Groningen has a large number of Schilling models as well as several other. The University of Arizona and the University of Tokyo also have model collections online. The development of 3d printing allows for a far simpler method of constructing models, this site also has interesting details of the original process including the recipe for the modelling clay. Finally Angela Vierling-Claassen has a large amount of material and research on these models, including a photographic catalogue of the collection at MIT.
As you might have guessed from some of the images these surfaces have provided inspiration for artists, especially the modern movements of constructivism and surrealism. It is debatable however how much these artists engaged with the mathematics or simply regarded used surfaces as objet trouvé (a term which Duchamp himself found in the writings of Poincaré, which used it to describe mathematical theorems). For Man Ray this is almost certainly the case. He photographed the collection of the Poincaré Institute in Paris and went on to produce a series of painting entitled Shakespearean Equations. Using someone else’s words he described these:
At the beginning of my career I once classed myself amongst the photometrographers. My works are purely photometric. Take … the Shakespearean Equations, you will notice that no plastic idea entered these works, it is scientific thought which dominates.
Man Ray, Self-Portrait
Mathematical object, Man Ray
The work of Naum Gabo and his brother Antoine Pevsner certainly involved some of the mathematics, particularly in the case of Gabo’s Linear constructions. However it seems that this was still an endeavour that was independent of the mathematics community beyond the initial motivation. In fact:
Although he always denied it Pevsner based his Developable Surfaces on a concept found in certain mathematical models.
Anthony Hill Constructivism — the European Phenomenon
Developable Surface, Antoine Pevsner
Finally how could I miss Maxwell Demon regular Max Bill. Bill of course considered mathematical ideas to be central to his work, and perhaps fundamental to the future of art. His work included consideration of surfaces, including the potentially independent discovery of the Möbius strip and Tripartite Unity, which also has a beautiful mathematical structure.
Tripartite Unity, Max Bill
6 Comments | Art, Main Article, Mathematics | Tagged: Alison Watt, Antoine Pevsner, Art, communication, Concrete art, Euler, geometry, Hild, Man Ray, Mathematics, Max Bill, models, Monge, Nabokov, painting, Sophus Lie, Storch, surfaces, visual maths | Permalink
Posted by gelada
This piece was originally written for a poster on my art work (shown below). It had to be shortened, partly as a poster can only have so much text and partly as the font was my own design, so I had to typeset by hand. This took a long time!
My art work comes directly from my mathematics research, in fact it is hard for me to see a clear line where one starts and the other finishes. My mathematical work is therefore very visual, playing off the intuition I gain from the aesthetic considerations in pictures. However, aesthetics should not be seen as something foreign to mathematics. In fact you will often hear mathematicians refer to work as `beautiful’ or `ugly’. In a subject where results can see practical use years after they were proved, aesthetics and taste are essential tools. In fact, even when an area of mathematics does have applications, the reason people choose to study it is often its beauty, rather than its practical use.
The problem with mathematical beauty is that it can be well hidden. The language of mathematics requires years of study, almost initiation, to use with any fluency. Perhaps music gives a good analogy. Imagine trying to appreciate the genius of J S Bach purely from his written music. This would require the ability to read music, but also an understanding of the structure of the musical scale, fugues and canons. In listening, these requirements disappear and one can simply be swept up in the music. Visual images can play the same role for mathematics, revealing its beauty.
The importance of revealing the beauty of mathematics (and science in general) is the power of inspiration. Inspiration, far more than practical applications, has the power to bring people to the subject with the willingness to put effort into learning the more difficult technical details. In my work I attempt to reveal this beauty, without equations or formulae, and thus inspire people into learning more about mathematics and science.
2 Comments | Art, Main Article | Tagged: Art, communication, Mathematics, painting | Permalink
Posted by gelada
Having put so much time into making the image and setting up this space, I thought it was time to add some more content. Maybe the weekend will be the weekly time to publish, but probably I will be inconsistent.
This is a piece I wrote in reaction to working with artists at an art science show at Imperial College London in April 2008. My piece in the show is below:
Surfaces in three and four dimensions
Reading the words of Raymond Brownell to describe his mathematical art work, I was struck by an interesting observation. The things that draw me to maths or science art are quite the reverse of many in the field. Instead of seeking the impersonal the perfect the rigourous, I seek to find the personsal, the lucky stroke, the error. When Raymond writes that his style is all about using techniques to make the personal, such as brush strokes, disappear my only reaction is to think of how I want to compose work solely of brushstrokes, the mathematical forms blurred but apparent. Even on a computer I seek ways in which the generating algorithm can introduce subtle changes to what, in the mathematically perfect world should be the same thing.
A related topic occurs in discussions surrounding this show. People have been talking about the nature of the process as apposed to finished work. Science and maths are by their nature an unfinishable process. In general understanding something opens the ways to more questions rather than closing off old ones. Personally I love the feeling of being able to finish a piece of work. It is so much more final for me than the submission or publication of a paper. I understand that art too is a process and in looking at a body of work by a particular artist one can see how ideas developed and were explored. However in my personal work I do get a feeling of finishing when I have produced a work from a particular idea that allows me to move on to the next.
Maybe this is natural as I am coming from maths into art and most are going the opposite way. Therein lies a problem for me, in some ways the work I crave to make is virtuoso, coming from skill, and that skill nearly always takes years to produce. I cringe when I read reviews of Tomma Abts, whose work I adore, and who seems to be a far more skilled painter than I could hope to be in any reasonable time, criticised for her bad or even autistic brushwork. The problem is not only that I am not able to make such strokes, but I am not even sure what they would be.
Maybe I could regret not going to art school, or alternatively take time to enroll in one now. I find two reasons for not doing this. Firstly it would be a disservice to the superb technical and intellectual eductaion that I received instead. I speak in glowing terms about a mathematical education not my own reaction to it. I firmly believe that it is one of the best training for clear thinking available. This education in fact allows me to overcome some of my artistic issues, an inability to draw, by using the power of a computer and then a laser cutter to etch the designs onto canvas, even wet paint. This in fact leads to the second reason. I do not need much of the knowledge an art training provides. That is of course not to say that I do not envy and desire skills such as drawing, merely that I can compensate for my own lack. Time spent learning them would therefore be time not spent learning something else.
I must therefore accept that time, practice combined with informal discussions must do the training for me and I can simply dream of the time when, as I think is the case for Tomma Abts, reviewers of my work have to mention the shoddy brushwork in order to bring balance to an otherwise glowing review.
Apologies for the wordy and slightly rambling style of this short essay. I was inspired to write by listening to a podcast of Stephen Fry and think that I got slightly infected by my poor imitation of his style.
2 Comments | Art, Main Article | Tagged: Abts, Art, Brownell, painting | Permalink
Posted by gelada
