The strange quest: Mathematics as Concrete Art

October 10, 2009

I have to confess that this post has not been an easy one to write. I wanted to express some ideas that are difficult to put into words. The central, rather playful, thesis is that pure mathematics itself is a branch of concrete art.

Let me begin with some easy facts. This month, I have had the great fortune to be able to take part in a studio exhibition with a group of constructive/concrete artists, including members of the systems group from the 1960′s.  The exhibition was curated by Trevor Clarke in Peter Lowe’s studio.  As a result I have had a chance to have some fascinating conversations with several artists, including Peter Lowe, Trevor Clarke and Jeffrey Steele.

IMG_0131

Spirograph by Richard Grimes

One goal of the exhibition is to start dialogues between artists and technicians, in the spirit of the studio exhibitions that started the systems group in the 1950′s around Adrian Heath and Kenneth and Mary Martin.  With that in mind I would like to give some of the ideas that emerged for me from the conversations.

Constructive and Concrete art arose from a natural conclusion of the process of abstraction. In the case of concrete art this is explicit and stated in Van Doesberg’s “Manifesto of Concrete Art”. Abstraction began by cutting away the figurative and symbolic content of artworks. As this program progresses more and more is cut away until, in a natural conclusion, one is left with nothing. Nothing is a fascinating concept. It is certainly not a trivial one, as we see with relatively late arrival of zero as a number. It does not, however, give a large space in which ideas can work. An empty canvas is an empty canvas and one ends up unable to tell the profound from the lazy. Concrete art emerges from this vacuum as the attempt to produce artworks that are not empty but have no figurative or symbolic meaning. It seems that this goal can be achieved in two distinct ways. One can either take the subconscious or irrational approach that leads to mysticism or the hyper-rational approach to create small works with their own logic.  For obvious reasons I want to consider the second here.

This would seem to argue for a very subjective art, as we must not only consider different personal opinions about a piece, but the individual world that each piece inhabits. Constructivism is more ambitious than this. The idea of removing figure and symbol is not nihilism, but a desire to address raw or objective beauty. It is of course fully accepted that no such beauty exists. This leads to a strange quest, where the goal is known to be unobtainable.

Being interviewed by Peter Lowe about hyperbolic geometry.

Being interviewed by Peter Lowe about hyperbolic geometry.

I come into this from a different point of view. My art does not contain mathematics in order to have no content, but to communicate mathematics. The mathematics is precisely the symbolic meaning. Yet what is mathematics? My personal definition is that mathematics is any concept that can be considered without reference to the real world. I know that this is an intellectual land grab, but I favour overlapping disciplines anyway. Putting this definition together with the constructivist quest for beauty led to some interesting similarities. Let us consider a parallel history of the two topics.

In the late nineteenth century, while painting was starting the move to abstraction with the work of impressionists and others, mathematics was starting a re-examination of its axiomatic roots. Just as art became more abstract the concepts and fields of mathematics were being cut back to rest on top of the set theory of Cantor and Dedekind.  By the 1930′s the impossibilities inherent in both quests were becoming apparent. A year after Van Doesberg published the “Manifesto of Concrete Art”, Göodel published “On formally undecidable propositions of Principia Mathematica and related systems”.  This work showed that whatever axioms one considered (that allow arithmetic) there would always be holes, statements that the axioms did not say were true or false, and one could never be sure that there was not a contradiction a statement both true and false. This was the end of the dream of a perfect mathematical machine. Pure mathematics thus joined in the strange quest, seeking patterns and structure without the possibility of obtaining a final goal.

Work by Gary Woodley

Work by Gary Woodley

In fact by the 1940′s the two subjects were recognising their similarities. Hardy published “A mathematician’s Apology” in 1940 that claimed that mathematics was an art form. With the humility that only a Cambridge academic can feel for his own place in the world he declared:

“A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”

The quest of a mathematician, to Hardy, was to find beauty and truth, yet without defining exactly what he meant by either. This bears a striking similarity to the vision of constructivism that I described above.  It is no surprise therefore that, perhaps unaware that mathematics had been declared an art, in 1949 Max Bill considered “The Mathematical Approach in Contemporary Art”.

I want to reverse Bill and consider that perhaps the mathematical structure itself, from gauge theory to groups, from motives to matrices from the games of Conway to the technical depth of Grothendieck, stopping on the way to take in the Hopf fibration and bifurcation, the Penrose tiling, and the 57-cell, is simply one giant work of concrete art put together by a cast of thousands.  An edifice built with some logical consistency on the Zermelo-Frankael axioms and the fudge factor axiom of choice.

So here’s to everyone pursuing the strange quest in the belief that the universe has an inexhaustible supply of secrets, and there will always be new beauty to be found even in some of its simplest corners.

Works by Trevor Clarke and John Bremner

Works by Trevor Clarke and John Bremner

The show

A studio presentation linking a selection of historical and contemporary autonomous works with a focus on modular investigations including:

Alexander Rodchenko*
Anthony Hill
Dirk Verhaegen
Edmund Harriss
Freddy Van Parys
Gary Woodley
Getulio Alviani
Jean Spencer
John Bremner
Kenneth Martin
Mary Martin
Peter Lowe
Richard Grimes
Trevor Clarke

Curated by Trevor Clarke in response to an invitation from Peter Lowe to stage a studio exhibition.

*reconstructions

10 Comments | Art, General, Main Article, Mathematics | Tagged: , , , , , , , , , , , , , , | Permalink
Posted by gelada

Surfaces 1: The ooze of the past

March 21, 2009

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

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

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

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 x^3+x^2z^2-y^2 = 0.  Now choose values for x, y and z, as the vector (x,y,z), for the vector (0,0,0) we have 0^3+0^20^2-0^2 = 0, so this is a solution of the polynomial.  Similarly (-1,0,1) gives (-1)^3+(-1)^21^2-0^2 = 0.  On the other hand (1,1,1) gives 1^3+1^21^2-1^2 = 1, 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 x^3+x^2z^2-y^2 = 0, Herwig Hauser

Kolibri x^3+x^2z^2-y^2 = 0, 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 x^2-y = 0 and the double cone itself is the solutions of the equation x^2 + y^2 - z = 0.  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).

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

manraysurface

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

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

Tripartite Unity, Max Bill

6 Comments | Art, Main Article, Mathematics | Tagged: , , , , , , , , , , , , , , , , , , | Permalink
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Book Review: Mathematics a Very Long Introduction

November 21, 2008

I recently received my copy of the wonderful The Princeton Companion to Mathematics.  In the title of this piece, I could not help making the obvious joke, as the editor Tim Gowers also wrote the brilliant Mathematics: A Very Short Introduction
.  I first started looking for a book like this as an undergraduate.  I wanted to get some overview of mathematics so I could have an idea how the different courses on offer might fit together.  Over the years I have cobbled together some vague ideas of what the different areas of maths are, and more importantly to me, what questions they try to answer.  Reading this book is like turning on the light (to steal and corrupt Andrew Wiles’ metaphor).  There is a very serious attempt to explain what mathematics is and what the areas studied are.  Also to give an impression of how they relate.

The book starts by identifying three methods of doing mathematics: Algebra, manipulating symbols; Geometry, using visualization; and Analysis, using limiting processes.  Obviously these are very short description of longer passages of text aiming to sum up very subtle differences.  Having explained these three methods the book starts to describe areas of mathematics, identifying the problem that the three methods are also areas of knowledge.  The introduction then gets more specific defining some of the basic objects of mathematical study, from numbers through to manifolds, via groups and geometries.  It concludes with a discussion of the goals of mathematics research.  This final section is particularly valuable as it motivates mathematics as a way to answer questions, and talks about the general questions that we consider.

The second part of the book is historical, charting how modern mathematics came into being.  This is followed by a more detailed discussion of 99 mathematical concepts.  It is not every day that \pi lies nestled between Phase Transitions and Probablility Distributions.  The fourth part returns to the branches of mathematics and discusses them in more detail, showing how they consider, translate and transform the basic concepts.  My apologies if this is starting to sound like a list!  The book has many parts, each of which is worthy of at least a brief description as the structure has clearly been well thought through so that mathematics can be viewed from many different angles.  This is especially true of the fifth part that considers 35 important theorems and open problems.  This allows important single results to stand out, rather than considering the area in which they lie.  Part six is similar in relation to the history chapter.  It considers specific mathematicians (the cut off date was important work by the 1950s, so the list can be kept below 100!) rather the grand sweep of mathematical progress.  The book concludes with a section on the influence of mathematics and some final thoughts.  The section on the influence of mathematics is rather short, however this is in accordance with the books stated goals of describing mathematics, rather than its applications.  At over 1000 pages there is hardly space to add a topic which could take up a possibly larger volume on its own.

Along with my excitement on opening the book, however, I felt a certain sense of trepidation.  How would my own area be treated?  Unfortunately this was justified, as it did not do so well.  Tesselation is only mentioned in the briefest of ways and the Penrose tiling turns up only as a model for quasicrystals.  Of course I accept that the book already covers a vast area, and mathematics is diverse enough that any book would have to miss something out.  However I feel that what is missing is not just the particular topics on which I am working, but the whole area in which I work.  I think that this is a more general problem than just this book, so it is a little unfair.  However the book gives a clear target to attack, rather than relying on anecdotal evidence and insinuation.  If asked what general area of mathematics I work in I would say geometry.  However, although the book does describe geometry as the study of mathematics through visualisation, when it comes to describe geometry as a mathematical area the only topic considered is the study of manifolds (P 4-5).  More tellingly even after describing geometry as visualisation the book states that:

If you look at a typical research paper in geometry, will it be full of pictures? Almost certainly not. (P 1)

Similarly the figures in the book, though reasonably plentiful are not considered worthy of their own contents list.

So what is left out by this?  Here lies part of the problem.  With the most natural term “Geometry” taken away to mean something else there are not even clearly defined terms for it.  Discrete Geometry is certainly a large part of it as is the study of tilings, I would like to include sphere packings, space filling curves, fractals and nurbs.  Maybe “visual mathematics” is a good name.  This is meant to be a broad area, not a specific research topic, and manifolds especially in low dimensions would be both an object of study and an important tool, however I believe that the topic is a lot broader and contains many other concepts.  Some books in the area might include bibles like Tilings and Patterns, The Symmetries of Things and Regular Polytopes, and many others worthy of mention such as Polyominoes and Indra’s Pearls.

Why is this area important? Firstly it shares with number theory the distinction of being able to ask difficult questions in very simple language.  It is therefore an ideal way to explain mathematical concepts.  In fact the book does exactly this, using the classification of the regular polytopes to explain classification (P 52-53) and sphere packings to talk about the discovery of mathematical patterns (P 58-59).  As it also produces images, that are often strikingly beautiful, it might even go beyond number theory in some respects.  People can be more drawn to images than to even the simple equations of number theory.  It is true, however, that there is nothing to compete in grandeur to the quest and proof of Fermat’s last theorem.  Though a natural extension of Hilbert’s 18th problem is a big open question.  This asks if there is a single tile that will tile the plane but not periodically.

It is also of importance educationally, working with this sort of geometry and visualisation in 3 and 4 dimensions generates intuition.  This intuition is a valuable skill for any area of work that creates objects whether in the real world (architecture and engineering) or in a computer (3d models for films and prototypes), as well as in mathematics.

Finally the that has been a consistent source of mathematics for thousands of years.  The existence of irrational numbers for example was found by considering the diagonal of a square. Valuable contributions have also come from the study of perspective, leading to the projective plane and the work of Coxeter in group theory.  The list could go on.  Nor is it an area that will stop giving as it lies so close to undecidable problems.  For example the domino problem, which asks if a set of shapes will tile the plane is undecidable.  Though, as with all mathematics, there is a danger of simply wandering aimlessly down abstract paths, there are almost certainly beautiful and useful ideas waiting to be found.  This is especially true as computers allow investigation of areas that are too complex to explore by hand.  Klienian groups (as shown in Indra’s Pearls) and a wonderful example of this (though the wikipedia page contains no images!)

It is true that the influences section of the book is stronger on visual mathematics.  It is here that you find the Penrose tiling for example.  In particular I read with delight the wonderful section on Mathematics Art.  It is great to see this acknowledged by mathematicians, especially citing Max Bill (P 950) and Constuctivism (P 948-9) along with the more traditional M C Escher (P 950-1) and Albrecht Dürer (P 945).

As I said before, though I see a lack of visual mathematics (as defined above) in the companion, this is perhaps more as a problem in the current priorities of mathematicians than the companion itself.  In fact one of the great strengths of the book is that it helps clarify what mathematics currently is, so we can have a debate about how it might change and what it should become.  Another example of this is Doron Zeilberger’s Opinion 92 pointing out that the importance of learning to program a computer to mathematicians.

To conclude I want to return to what the book does cover.  It sets out with the admirable goal of explaining all of modern mathematics at a level comprehensible by people with a good level of high school mathematics.  This goal, was dropped according to the preface (P xiii), as it was accepted that different topics needed different backgrounds.  However much of the book is still accessible at the original level and even the harder parts should be accessible to maths graduates.  The level of success achieved is remarkable and it is a great thing to see a stellar cast of mathematicians describing their topics in the simplest possible terms, letting motivation and ideas rule over technical detail.  I certainly look forward to the many hours I will spend reading.  As mathematics is getting larger having even a shallow understanding of the different areas being studied is hard.  This book will become an essential tool.  Hopefully it will inspire and challenge all mathematicians.  It is possible to explain our research beyond our close colleagues!

2 Comments | Book Review, Mathematics | Tagged: , , , , , , , , , , , , | Permalink
Posted by gelada

Working with constraints

November 2, 2008

There is a theory that one can have too much freedom, at least in art.  With constraints the imagination is forced to work harder, and might achieve an elegance and beauty unobtainable when a simple answer can be used.  Rules and constraints can also be broken, at the right time, giving an aesthetic that is unobtainable for a completely free setting.  In particular a set of rules gives a possibility of some idea of perfection, and thus adding a deliberate flaw can, for many, increase the aesthetic appeal of a work of art.  It is not possible to have a flaw in a work which is completely free.  

 These ideas have been developed many times, especially in the modern era as freedom became more fashionable, and thus setting rules became more rebellious than breaking them (as an aside I would be interested if anyone has examples of such ideas from before the middle of the 19th century).  I would like to mention two of my heroes who also wrote theory on this subject.  The first is Max Bill, who gave a vision of concrete art, stating in his essay The Mathematical Approach in Contemporary Art (no prizes for guessing my interest):

…despite the fact the basis of this Mathematical Approach to Art is in reason, its dynamic content is able to launch us on astral flights which soar into unknown and still uncharted regions of the imagination.

The second is the Oulipo, a literary group, described by Raymond Queneau as:

 

Rats qui ont à construire le labyrinthe dont ils se proposent de sortir.

(Rats who construct the labyrinth from which they propose to escape)

 

The ideas of the Oulipo are perhaps more explicitly similar to the points I mention at the beginning, for the simple reason that my thinking has been heavily influenced by them.  An archetypal example of their work and possible my favourite novel is George Perec’s Life: A Users Manual.  This work had an incredibly complex set of rules for its structure, yet wears it lightly.  One can read, and enjoy, the book without considering, or even being aware, of the fiendish construction process.  In addition the story of the book, (grand, complex artistic schemes) describes the book itself.  The schemes in the book eventually fail, and the book itself only has 99 chapters rather than the 100 that the rules declare it should have.  One of its many themes, therefore, is a beautiful study on many levels of the aesthetics of flawed perfection.

One final idea that working with constraints introduces is the element of puzzle.  Can one find the constraints or rules given the finished piece.  This is an idea that I want to develop more fully, so for the moment I will just leave a forward reference.

The ideas of the Oulipo and Max Bill seem so close that it is amazing to find so few links between the two traditions, for example using google.  While I was writing this there were only 93 hits for “Max Bill” and Oulipo, ”concrete art” and oulipo did even worse gaining just 23 and “constructive art” and oulipo only 2!  I think that it is therefore safe to assume that the ideas were developed separately in Literature and visual art.  Can one use this convergent evolution to justify the ideas, though simple are deep, profound and interesting?

Non-philosophic constraints

I did not mention above that one can also take on constraints for purely utilitarian reasons.  Either one is forced to because of the materials one is using, or certain constraints can solve a practical problem.  As a personal example I have recently been playing with images that can be created on squared paper.  For no other reason than I have been using a pad of squared paper as a notebook (and that I am not happy with my drawing skills).  A simple piece made on the computer from some of this thinking is below (guesses on the construction are again welcome, but no prize this time!):

 

Root 5 grids

Root 5 grids

7 Comments | Art, Main Article, Puzzle | Tagged: , , , , , , , , , , | Permalink
Posted by gelada

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