One of the concepts we talked about today were the outcomes of arranging different numbers of equilateral triangles at a vertex, both in Euclidean and non-Euclidean terms.

- 3 triangles that meet at a vertex make a pyramid (a
**tetrahedron**). - 4 triangles that meet at a vertex create an
**octahedron**, which looks like two tetrahedrons connected at a face. - 5 triangles that meet at a vertex make an
**icosahedron**(20-faced polyhedron). - 6 triangles that meet at a vertex make a plane, creating the regular {3,6} tessellation.
- 7 triangles that meet at a vertex make a saddle-like representation of a hyperbolic plane.

One of the ways we can visualize what this would look like is with **hyperbolic paper**. Go to http://homepages.wmich.edu/~drichter/hyperbolicimbed.htm to see a colorful representation of hyperbolic paper.

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Here's an article about using paper to model hyperbolic geometry.

And another article about the collaboration between Escher and Coxeter.

Although the philosophy of math isn't explicitly part of the course, ideas like non-Euclidean geometry pushed the boundaries of mathematical thought into those rarefied realms. We briefly asked the idea of whether math is "invented" or "discovered" and Prof. Bolker said it was his experience, and that of most other mathematicians, that there was the sense that it was discovered.

The act of discovery of an object implies the independent existence of that object. It was already there, waiting to be discovered. Max Tegmark, a physicist at MIT, has a theory that the universe itself is ultimately mathematical! Here is a short interview with him. On his website he postulates the existence of four levels of parallel universes, Level 4 being "Mathematical Multiverse". He admits it's a crazy idea, but if you want to pursue this heady concept, check it out.

===Matt===

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Matt and our class discussions really pushed me to investigate Escher's work, including his work with Coxeter. I was amazed to learn in class that Escher had created a number of beautiful works that demonstrated tiling yet had no idea about the mathematical concepts we had been working on in class until meeting Coxeter.

I found an interesting overview of Coxeter and his works here. It has an amazing drawing that exemplifies some of the concepts we have been discussing with tiling. The aforementioned piece does not really seem to focus on Escher, but does make an interesting reference to some work he did on a geometry book that was published by Lewis Carroll (who also wrote *Alice in Wonderland*). I thought that was a very interesting tie to a figure we had talked about in class, and I honestly did not know much of Carroll's work in geometry. It might be interesting to investigate.

As I continue to look through Google, I stumbled upon this interesting video of Coxeter talking about Poincare's works with the hyperbolic plane. It was interesting to hear it coming from Coxeter himself!

Finally, I looked a little bit more into tiling, which we had covered this lecture. One of the first results that came up (as is typically the case for many searches it seems), was the Wikipedia article, available here. It is pretty technical, and goes into *much* more detail than we did, but has graphic representations of some of the tilings above and *much* more. The aforementioned article discusses uniform tilings in the hyperbolic plane. For what Wikipedia calls "Tiling by regular polygons" (Note: I am not sure if this is the mathematically accepted term, or if and how such tiling is distinguished in mathematical terms. This would be interesting to look up as well), which also goes into detail but has many helpful illustrations, click here.

I then did a quick Google search for "tiling hyperbolic plane," and came across this pretty easily digestible article, which covers some of the tiling concepts we talked about. See if you can pick out Lobachevski and Poincare in the pictures toward the beginning. One awesome part of this article is that it takes you step-by-step through Coexter tiles in the hyperbolic plane. As you go through the pages, the drawings start from a Poincare disk model of the hyperbolic plane with one shape to the completely tiled disk, with steps in between so the reader can better understand the process.

I hope people find this information helpful!

-Rob Moray

A slightly off topic question, is it possible for someone to post the complete solution for homework 1? I mean I understand the concept and all that, but it would be helpful to see the solution clearly, since my is such a mess. Thank you.

Lin

I was really interesting in Escher's artwork, it's a amazing work of Art and Math…I really want to know how Escher worked? Was he consider math structure before he started painting? And did he have strong math skill? Why he painted like that? what's the information in the work? I think it's a really tough, even for someone with strong mathematical skill. I mean every detail he calculated and painted in his work is perfect !!!

Jun