CS798 Lecture notes

These are notes and additional references that complete and augment the material presented in class. Note that the material is organized by "lecture topic" and not "class period".

Lecture 1: Introduction to ornament

I began the course with some general administrivia. All that information is spread around these web pages.

The main lecture consisted of a general introduction to the world of ornamental design. I discussed common characteristics of ornament and some of the history. I then made some observations about why we seem predisposed to cover surfaces with ornament, and why we're in a good position to study the subject today.

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Lecture 2: Introduction to symmetry theory

A general introduction to the informal and formal ideas of symmetry. Naturally, I built up to a discussion of discrete symmetry groups of isometries in the Euclidean plane.

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Lecture 3: Other forms of symmetry

Building on the basic ideas of discrete symmetry groups, I developed some more sophisticated symmetry classifications based on extensions to the usual idea of "isometry". I covered counterchange symmetry, n-coloured symmetry (just an introduction), and two-sided symmetry. I closed by briefly mentioning some other random ideas, such as symmetries in other numbers of dimensions, using other kinds of automorphisms (such as similarities), and using other surfaces (such as the sphere). The topic of non-Euclidean symmetry will reappear later in the term.

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Lecture 4: Celtic knotwork

This lecture discussed Celtic art, with a focus on knotwork and the paper by Cromwell connecting Celtic friezes with two-sided symmetry groups.

Lecture 5: Introduction to tiling theory

I plan to spend a large part of the course talking about tiling theory or topics motivated by tiling theory. So I had a lot of basic definitions to get through in this lecture.

Lecture 6: Polygonal tilings

This lecture did a small amount of useful math in proving the existence of regular and semi-regular (Archimedean tilings), then went on to an informal presentation of interesting tilings by simple polygons. Most of the material is from Chapter 2 of Tilings and Patterns, specifically Sections 2.1, 2.4, and 2.5.

Lecture 7: Theory of isohedral tilings

I developed enough machinery to show that the isohedral tilings can in fact be completely classified via a small symbolic description of the interaction of a single tile's edges with those of its immediate neighbours. There's really only one place you need to look for this material: Chapter 6 of Tilings and Patterns. In my copy of the book, the spine is completely broken at the beginning of Section 6.2. Mind you, you also need to be familiar with Sections 2.7 and 4.3 as background. My thesis provides an alternate introduction to the subject.

Lecture 7: Implementation of isohedral tilings

Now we take a big jump from the mathematics of isohedral tilings to details of a software implementation. If we want to draw tilings in practice, we need to parameterize the space of possible shapes of tiles in each isohedral type. Though this problem has been studied by several people in related forms, I present my version as it appears in the Escherization paper and my thesis.

Lecture 8: Escher and his periodic tessellations

Finally, we get to talk about the great modern master of geometric art, M.C. Escher. I spent some time talking about Escher's life and work, including the important influence of Islamic art on his quest for "regular division of the plane". Most of the background information in this lecture comes from Schattscheider's book Visions of Symmetry, though some details came from other books about Escher.

Lecture 9: Metamorphoses and Deformations

I covered two popular, related ornamental styles based on continuous transformation: Escher's metamorphosis drawings and Huff's Parquet Deformations. The lecture was constructed by beginning with Escher's Metamorphosis II and showing how all the important transition devices appear there. As far as I know, there are very few published references about any of this (except for brief mentions in two of my papers, as mentioned below).

Lecture 10: Nonperiodic tilings

Before I jump with both feet into the mathematics of aperiodic tilings, I stop to appreciate some of the more attractive systems of tilings that are merely nonperiodic without necessarily being aperiodic. This lecture is simply a survey of some ways to tile the plane nonperiodically, though it concludes with a discussion of the existence of aperiodic tile sets.

Many lovely examples of these and other tiling-related topics can be found in the tiling section of the Geometry Junkyard.

Lecture 11: Aperiodic tilings

Now we get really get into the mathematics behind aperiodic tiles. At the outset, it's not clear that we care about all this math from an ornamental design point of view. It doesn't seem likely that our aesthetic appreciation of nonperiodic and aperiodic tilings is any different. When presented with a tiling, we see just that tiling and not implicitly all other tilings that can be formed by the same shapes, in which case aperiodicity might not matter that much. On the other hand, Penrose tilings are quasiperiodic, which we could argue matters to human perception (though that's far from clear). Moreover, one of the difficult aspects of aperiodic tiles is proving that they tile the plane. Since we're never going to assemble more than a finite patch, does this matter aesthetically?

However, this material is absolutely beautiful and most deserving of study. Everybody with an interest in math and art should be introduced to the Penrose tiles and get an understand of how they work.

This material is mostly taken from Chapter 10 of Tilings and Patterns. You might also want to take a look at the Penrose tiling section of the Geometry Junkyard. Be sure to read the article on toilet paper plagiarism! And make sure to carefully study the cartoon Bob the Angry Flower Goes Through Customs!

I'm pretty happy with the quality of the slides I created to demonstrate the aperiodic tile sets I discussed. Here's a PDF of the slides to refresh your memory and in case anyone out on the web can use them for teaching. Of course, when I make slides I just put the pictures on them, and speak from notes. No bulleted lists here!

Lecture 12: Further properties of Penrose tiles

Penrose's aperioric sets have so many interesting mathematical properties that I can't resist demonstrating some of them. We can either view these as useful properties that might yield decorative applications, or maybe as attractive mathematical facts in and of themselves.

Lecture 13: Euclidean and non-Euclidean geometry

Because I retain my love for the pure math I did as an undergrad, I spare no expense in presenting a formal treatment of geometry on the way to a practical system for Euclidean and non-Euclidean geometry. Almost all of the first part of this lecture is taken from the opening chapters of Marvin J. Greenberg's book Euclidean and Non-Euclidean Geometries: Development and History (W.H. Freeman, 1993). This is a wonderful book that I heartily recommend to anyone with an interest in geometry or the history of mathematics. The book intersperses the math with chapters on history and philosophy, and includes biographical information about the sometimes curious personalities of those involved in the discovery of non-Euclidean geometry. As always, a very brief summary can be found at the very start of Chapter 2 of my thesis.

Lecture 14: Islamic star patterns

The Islamic decorative tradition spread quickly with the rise of Islamic rule in the ninth and tenth centuries. Today, remnants of that tradition can be found across southern Europe, northern Africa, the Middle East, India, and western Asia. Small pockets of artisans continue to practice Islamic art on a smaller scale.

One of the best known varieties of Islamic art are Islamic star patterns, abstract geometric arrangements of stars and other shapes. In this lecture, I discuss some techniques for producing and decorating star patterns. Much of this discussion comes from my own work. I also highly recommend the following references.

Many other references can be found through obvious web searches or by looking through the citations in Chapter 3 of my thesis.

Here are the specific topics I covered.

Lecture 15: Geometry and Islamic art wrap-up

The previous lectures wove together many threads having to do with non-Euclidean geometry and Islamic art. In this lecture I draw those threads together and try to provide some closing remarks for each. The lecture can be broken down roughly into three components: "further results on Islamic star patterns", "other forms of Islamic art", and "other artistic uses of non-Euclidean geometry".

Lecture 16: Polyhedra

Polyhedra have long been an important part of mathematical art. We could argue that they don't belong in the same group as other kinds of ornament because they are 3D objects in and of themselves. But polyhedra have many properties in common with planar ornament because of their connection to spherical geometry. And they appear repeatedly in some of the same contexts. So we'll study them in this course. The first and ultimate reference for all things polyhedral is George Hart's website. It also includes links to other useful sites. Other consistently useful online sources of information are Wikipedia and Mathworld.

There are many other systems of polyhedra that yield attractive shapes. Some feel more "classification based": define a set of properties that your polyhedra should have, and find a way to determine all polyhedra that satisfy those properties. Other techniques are more "notation based": define an open-ended notation where strings produce polyhedra, and search for interesting examples.

Lecture 17: Projections of polytopes, polyhedral wrap-up

Polytopes are the generalizations of polyhedra to any number of dimensions, though the term is sometimes used to refer specifically to four dimensions. It's worth taking a look at some properties of 4D polytopes and how we might construct representations of them in 3D.

Most of the images and slides used in this lecture come directly from the "Four-dimensional forms" lecture in George Hart's computer science and sculpture course. Also worth checking out are Bathsheba Grossman's website (click on "math models", for example) and David Richter's page of Zome projects. Web searching can locate many other amazing examples of Zometool sculptures.

Lecture 18: Remapping the plane

Once you have created an attractive 2D pattern in the plane, there are still some things you can do to it to increase its appeal. One possibility is to define a transformation of the plane and pass the design through that transformation. For more information on this topic, see Robert Dixon's book Mathographics or his paper "Two Conformal Mappings" from the journal Leonardo.
Craig S. Kaplan Last updated: