Martin Hunt's Website
An Essay about Tiling

The Rutherford Bohr Atom
I'm interested in physics. Years ago, I realized that one of the ways that I could get around my lack of math was to try and use my experience in art to visualize what was going on in the subatomic world. I wasn't interested in representations of things like electrons whizzing around protons.

This kind of representation wouldn't tell me anything that I didn't already know.

I have found several ways of representing the physical processes. The most obvious is to write computer programs that simulate the behaviour of the process I wish to consider. This is the basis of the Linofo images which are explorations of the properties of the electromagnetic field. I've written programs that simulate the action of gravity on objects floating in space. I don't think that my simulations are very accurate, in the sense that they are very simple and surely don't encompass all of the subtleties of the thing simulated. But even so, I've been amazed at how even simple simulations share recognizable characteristics of the real world.

I've also done some work with Lindenmayer-systems, which provide a fascinating way of exploring biological structures. I first encountered L-Systems in an amazing book called "The Algorithmic Beauty of Plants " . With L-systems you write programs that then write programs that are too complicated for a human to understand, but that can be used to create images. While the Linofo and gravity programs deal with trajectories of particles, the L-system simulations deal with patterns generated by growth and has strong ties to genetics and embryology.

Gravity

This image was made by simulating the movement of 100 masses floating in a plane in space. The masses start arranged in a grid, and as time progresses fall toward the centre and then oscillate about the centre. Any one trajectory, as shown by the foreground path seems pretty arbitrary, but the symettry of the whole group shows how deterministic the system really is. The image is from a serigraph print made from an image generated on an Atari 1040 computer.

A Tree

This image of a tree was generated from a few simple rules that act as a "seed".
Click on the image for an enlarged view.

The physicists I was reading were saying that they didn't know what subatomic particles were, but they knew what some of their properties were. (David Bohm, in his work " The Undivided Universe " has other ideas, but that's another story.) They were content to not attempt visualization, and just work with what they knew about the properties. So I started to see subatomic particles in terms of their properties, instead of in visual terms like, say, billiard balls. This lead to the thought that subatomic particles were things with "intrinsic properties". This lead me to wonder if I could find forms, or shapes, that had intrinsic properties, and whether I could use those intrinsic properties to help me visualize the intrinsic properties of the objects of subatomic reality. So I started looking for forms that would behave like an electron, instead of look like an electron, because after all, an electron doesn't look like anything.

Click on the picture to see
what a bunch of spinners together look like.
The first step was to find a shape that would behave like anything. At first I wasn't sure what it meant for a shape to behave. One night I made a little shape I call a "spinner". This shape had visual properties. Groups of spinners had visual properties that individual spinners didn't.

A year or so after I started exploring this, I picked up Grunbaum and Shephard's wonderful " Tilings and Patterns " in the library . This book was a revelation. I saw at once that my "forms with intrinsic properties" were "tiles". The book was a book of mathematics, and it provided a systematic survey of all of the possible ways of tiling a plane. Fortunately for me, the language used was largely diagramatic, and I could understand a lot of it. This book also made me realize how my interests were linked to those of MC Escher, whose work I had admired for years.

Among the many interesting types of tiles listed were tiles that produced "aperiodic" tilings. An aperiodic tiling is patterned, and there are elements that repeat throughtout the pattern, but the repetition is more like flowers in a field {link here to image of penrose tiling] than like squares in a grid. There are many kinds of aperiodic tilings, but my favourite are those invented by Roger Penrose. He invented two related types, the "Kites and Darts" and the "Penrose Rhombs". (See Darlene's Floor for an example of a tiling done with the Penrose Rhombs.) Penrose tiles are bihedral (two differnent tile shapes are used to make the tiling) and generate a pattern that has a 5 sided symettry. One can make tilings with stars and rhombs or with pentagons and rhombs that are very closely related to the Penrose Tilings. Also, anything with a 5 sided symmetry has the proportion of the Golden Mean [link here to description of Golden Mean] occuring and reoccuring throughout the piece. This proportion was considered by the Greeks to be the most perfect possible, and is also a proportion that is found again and again in nature.

Any tile generates a characteristic pattern. The thing that strikes me about a pattern is that it is made of physical things, but is not itself a physical thing. The tiles are the "thing", even if only colour on a computer screen. That the pattern is not physical can be demonstrated by just mussing up the tiling. Physically, it's the same, you have the same tiles. But the pattern has disappeared. If the tiles are "things" then the pattern is a "meta-thing" (after Gregory Bateson in " Steps Toward an Ecology of Mind "). This is the "the whole is greater than the sum of the parts" phenomenon, and represents a very general kind of relationship.

I take pleasure in contemplating a whole chain of thing/meta-thing relationships.

  • A proton is a meta_thing upon quarks.
  • Atoms are meta-things upon protons, neutrons and electrons.
  • A molecule is a meta-thing upon atoms.
  • Matter is a meta_thing upon molecules.
  • Life is a meta_thing upon matter.
  • The mind is a meta_thing upon life.

The key point about the relationship of things and meta-things is that the meta-things have properties that are not found in the things, even though those meta-properties are completely dependant on the properties of the things. In general, you can't predict the properties of the meta-thing from knowledge of the properties of the thing. This is seen very clearly in tilings, because the pattern of the whole is not at all apparent when you just look at the unit tiles.

Click on the image to see what the tiling looks like.

These shapes are Roger Penrose's "kite" and "dart", which form an aperiodic tiling. I think the pattern these tiles make is surprising.
The tiling is transformed further if you decorate the surface of each tile. This is an area that was explored by MC Escher. Escher used the surface decoration as a device to allow him to incorporate recognizable objects from the physical world into his pictures. He felt that his work was not "art" unless there were recognizable forms, I think because he thought that pure abstract forms were too sterile to excite emotions in an audience. He was old fashioned, and I have no problem with that for I greatly admire his work. For my work however, I feel that recognizable objects would obscure the "intrinsic properties" that I'm interested in.

Click on the image to see what the tiling looks like.
Assembling a Penrose tiling is very much like a game and is very challenging. I once made Penrose Tile fridge magnets. These were sets of 50 tiles, with magnets on the back, so that you could make tesselations on your fridge. They were great fun. You assemble the tiles according to a set of rules, but even if you follow the rules you often find yourself with spaces into which you can't fit a tile. Then you have to undo a bunch of your work and find an alternative tiling that doesn't lead to that problem.
If you aren't careful when you assemble a tiling, you often find that you can't get one part of the pattern to line up with another part. This can happen if you start building along a base, and then build first up on the ends of the base, and then try to fill in the space in the middle. You generally find that the cumulative error in the placing of the tiles has caused the two areas to each be displaced from the ideal. Within each of these domains, the error between neigbouring tiles is small, and is not significant. But between two domains, the error may be quite large, and this leads to a crack between the two domains that can't be filled by tiles.

I have found myself fascinated by these cracks. Unless you are very careful to always work along a front that grows evenly, the cracks are inevitable. There is no way that you can be precise enough that the pattern won't shift from the ideal as you work. This inevitability of imperfection reminds me a lot of the Heisenberg Uncertainty Principle. And the fact that you can't avoid cracks unless you are careful to work along a front reminds me of the principle in physics that all actions must be local.

Click on the image to see an enlarged view.

Considerations of things and metathings has caused me to view tilings as metaphors for other aspects of our world that share these kinds of relations. An obvious metaphor is to see the tiles as standing for individuals, and the tiling as representing society. With this metaphor in mind, various properties of the tilings start to speak of things that we observe. For instance, a set of tiles may be used as the basis for different types of tiling. This is perhaps similar to the fact that humans are individually much alike, but that the patterns of society that they create are often neither similar nor compatible. In this context, the cracks make a statement about the nature of the barrier between cultures.
One of the implications of the fun that one has in assembling a Penrose tiling is that it's a job that couldn't be done by Mother Nature. Nature is blind and has no foresight. Nature has no way of detecting a mistake and backing up and trying again. So, it was predicted that there would be no crystals that had a five sided symmetry, because 5 sided symmetries were aperiodic. It was thought that if nature tried to make a crystal with units that had a 5 sided symmetry that they'd end up with a material with so many cracks that it wouldn't be a crystal. Yet, in the 1980's, crystals were discovered that had a clear 5 fold symmetry. The quantum mechanical explanation is intricate, but it turns on the idea that quantum effects, under certain circumstances, manifest themselves over much larger groups of atoms than had been previously realized. Generally, quantum effects were considered to be restricted to the "micro" realm, and the 5 fold "quasicrystals" demonstrated that sometimes quantum effects are significant in the "macro" realm of our everyday experience.

Roger Penrose , the man who invented the Penrose tiles, is a physicist and mathematician at Oxford. He's one of my heroes. One time I rode my bicycle across Canada and his book " The Emperor's New Mind " was my companion. On the last morning of my ride, I decided to go straight on to Saint John, my destination, rather than hang out in Fredericton for the day. Penrose was in Fredericton on that very day! If only I'd known, I'd have stayed to listen to his talk.

Penrose has some very interesting ideas about the way that quantum mechanical properties manifest themselves in a structure like a brain. As I understand it, his idea is that quantum effects that are normally only significant in the subatomic realm are needed to account for the manifest properties of the brain. The brain is a huge "macro quantum object".

Personally, I find this linkage among tiles, physics, and the mind to be very interesting and satisfying.

Click here to see other tilings that I've done.

The Algorithmic Beauty of Plants,
by Przemyslaw Prusinkiewicz, J.S. Hanan, Aristid Lindenmayer,
ISBN 0387946764,
1st edition - Published in 1996 by Springer - Verlag

Tilings & Patterns,
Branko Grunbaum, Geoffrey C. Shephard,
ISBN 0716711931,
Published by W. H. Freeman & Company, 1986

The Emperor's New Mind
Roger Penrose
ISBN 0-19-286198-0,
Published by Oxford University Press

The Undivided Universe
David Bohm and Basil J. Hiley, 1995
ISBN:0 415 12185 X

Steps Toward an Ecology of Mind
Bateson, G. (1972)
New York: Chandler Publishing

Copyright 2000, Martin Hunt
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Revised: 01/23/00