Particle Navigator: Heesch's Problem

"Heesch's Problem asks the following question: How many times can a tile be surrounded by congruent copies of itself? That is, how many layers made of copies of the tile can you place around the tile. The layers are called coronas, and the maximum number of coronas that can surround a tile is called the Heesch number of the tile."
(http://math.uttyler.edu/cmann/math/heesch/heesch.htm)

On my trip to Orlando I happen to run into Klaus Heesch who runs Juicy Temples design studio. Klaus was the first person to commission art from me, when I was going to school at the University of Central Florida. We ran into each other briefly, then arranged to have some coffee and morning drawing time later in the week. After catching up and drawing a little bit, I showed Klaus some of the pattern work I was doing. He then asked me if I have heard of Heesch numbers. Only recognizing the word as his last name, I said no. It turns out, his relative, Heinrich Heesch, was a mathematician who came up with the notion. (see top of post) I thought this was very cool and coincidental, seeing as I have been experimenting with tile patterns in my drawings. Knowing these shapes exist and that there are many more to discover is exciting to me. I have started thinking about them as I draw, and have already attempted to find shapes that would work. It is very difficult to do this without knowing any theory or equations related to the idea, so I am not expecting to come up with anything new any time soon. Just trial and error. I am, however, memorizing some of the existing Heesh tiles, and starting to apply them to some of my work. I plan on developing this idea further. For now, here are some examples of Heesh shapes, as well as new posts from my pattern journal.

Klaus Heesch of Juicy Temples, with a sweet drawing he did of me.

The first shape discovered to have a heesch number of 1. Only once and the tiles be set around the same shape completely covering it without having any gaps

This example has a Heesch number of 2. Twice the same shape can be wrapped around, but not a third.

And here, the Heesh number is 3. So far 5 is the top number. These shapes are aperiodic and finite. Shape like squares can exist on an infinite plane and therefor do now count.

Here are my studies:

And last, this mornings drawing.

2 reactions:

lil blogger
August 12, 2009 at 5:16 PM

SO SIC SOTO!!!! Let's let our minds enter the replicated mosaic repeat pattern of our good nature'd souls!!

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