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 .  Now choose values for x, y and z, as the vector (x,y,z), for the vector (0,0,0) we have , so this is a solution of the polynomial.  Similarly (-1,0,1) gives .  On the other hand (1,1,1) gives , so this is not a solution.  The set of solutions with real numbers (if any exist) defines an algebraic surface.  For example for this polynomial we have the following:

Kolibri , Herwig Hauser

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 and the double cone itself is the solutions of the equation .  Many mathematicians, including Euler and Monge started studying surfaces in the same way, and also started to make models.  By the middle of the nineteenth century model making had become and industry and there were catalogues of published models.  In fact the Norwegian  mathematician Sophus Lie received funding for many of his trips to France and Germany by the need to find models for the university, not for scientific collaboration.  This proved a golden age for models however and by the 1930s they were dying out.  The craftsmanship and accuracy of these models is mind-blowing.  The act of creating a complex object simply from ideas, without having seen it before is hard enough when one is sketching graphs in two dimensions.  Yet these craftsmen were able to achieve it in three dimensions.

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

This gives a new angle to games from Chess to noughts and crosses (tic-tac-toe in the US).  These games provide some interesting variants (bid hex is already on Facebook), it also opens up possibilities for mathematical analysis.  This excellent paper, does that job very well coming up with delightfully weird results for example in bidding noughts and crosses the centre is an optimal move if and only if the number of starting chips is not 5! (Theorem 6.7 P 26)

It also has an appendix on methods of tie breaking, including chips and  and claiming:

Coin Flips, and taking turns nicely. Some tie-breaking methods are not worth discussing.

P30

Coin flips, however are not ruled out entirely.  The deepest result of the paper shows that, if bids are allowed to very continuously (allowing any number rather than just whole numbers) then bidding games become equivalent to random turn games (where the next player to move is determined by coin toss). 

Random turn games led me to an Ivars Petersons’ blog post, describing random turn Hex.  It turns out that this is actually easier to analyse than the same game with alternating moves.  Hex is a famous example as it can be shown that there is a winning strategy for the player who moves first (as every game must end and no game ends in a draw), however finding the strategies for large boards is a very difficult problem.  Peterson describes a paper that shows that the chance of winning random turn hex (with both players playing optimally) is the same as the chance of winning where moves alternate but are made at random.  

Both these papers describe toy examples, however both bidding games and random turn games describe real world conflicts where the right to move can depend on chance, or on how much one is willing to invest.  From this point of view bidding games where bids are lost (whether or not you win) might also be interesting.  

It was hard for me to believe that I had not already come across Ivars Peterson’s blog, and I look forward to reading the archives.  However the latest post really caught my eye.  He has dug through the photo archives of LIFE magazine, and unearthed some mathematical gems.  These pictures give some glimpses into the history of our mathematical culture.  In particular, a couple show geometric models and an incredible collection of tools to teach plane and solid geometry.  What has happened to these models and toys?  I am reminded of the fact that when Sophus Lie was travelling from Norway his travels where usually funded by the university to collect mathematical models rather than do research.  Today, do any universities actively maintain libraries of mathematical models?  As I have commented before and will almost certainly do again, these are an important way both to draw people into mathematics and to generate the intuition to do research in many areas.