Computer Graphics and Geometric Ornamental Design

Craig S. Kaplan. PhD thesis, 2002

Inspired by the wonderful page Robert Glenn Scharein has dedicated to his PhD thesis Interactive Topological Drawing, I decided to give my own thesis a happy little home on the internet. Like Robert, I wanted to include some of the figures from the thesis with which I was particularly pleased. From this page, you can download my thesis in a couple of different formats suitable for on-screen viewing or printing. You can also sample some of my favourite pages.

If you enjoy looking at pretty thesis pages, I also recommend the finely illustrated work of Sascha Rogmann: Wachstumsfunktionen von Pflasterungen. Sadly, I can't read German.


Throughout history, geometric patterns have formed an important part of art and ornamental design. Today we have unprecedented ability to understand ornamental styles of the past, to recreate traditional designs, and to innovate with new interpretations of old styles and with new styles altogether.

The power to further the study and practice of ornament stems from three sources. We have new mathematical tools: a modern conception of geometry that enables us to describe with precision what designers of the past could only hint at. We have new algorithmic tools: computers and the abstract mathematical processing they enable allow us to perform calculations that were intractable in previous generations. Finally, we have technological tools: manufacturing devices that can turn a synthetic description provided by a computer into a real-world artifact. Taken together, these three sets of tools provide new opportunities for the application of computers to the analysis and creation of ornament.

In this dissertation, I present my research in the area of computer-generated geometric art and ornament. I focus on two projects in particular. First I develop a collection of tools and methods for producing traditional Islamic star patterns. Then I examine the tessellations of M.C. Escher, developing an “Escherization” algorithm that can derive novel Escher-like tessellations of the plane from arbitrary user-supplied shapes. Throughout, I show how modern mathematics, algorithms, and technology can be applied to the study of these ornamental styles.


My thesis contains a lot of graphics, and so resolution does matter. However, higher-resolution images make for much larger PDFs. Therefore, I'm making several versions available.


Whenever possible, one's thesis should be write-only. Nevertheless, I understand if once in a while a visitor to this page may choose to read what I've written. Shockingly, this hypothetical reader may occasionally run across errors of spelling, punctuation, grammar, citation, cross-reference, exposition, fact, content, organization, intent, and overall structure. Some of these errors may even be unintentional. I welcome you to send me any errors you find, along with your suggestions, criticisms, and ideas. If enough significant errors emerge, I'll add an errata page here.

Favourite Figures

I enjoy creating expository diagrams. My thesis already had a great deal of graphics by virtue of being in the field of computer graphics, but I wanted to use graphics as much as possible to explain concepts, or failing that to help clarify the writing. What follows are some of my favourite figures from the thesis. These aren't the results, they're just the pretty diagrams. Click on each one for a full-size version and some occasional commentary.

[page 25] [page 35] [page 54] [page 58]
[page 68] [page 69] [page 74] [page 107]
[page 110] [page 122] [page 128] [page 131]
[page 164] [page 173] [page 194]