cs4388

Hyperbolic Planar Tesselations

Don Hatch's is favor in designing computer graphics and computational geometry. He graduated at U.C. Berkeley, studied PhD in Mathematics. He is favors in Logic and Combinatorial Set Theory.

In 2003 April, He published "A $20 Holyhedron".

In 98 and 99, he demo a application called "From Outer Space to In-Your-Face" in SGI's booth at IITSEC and SIGGraph, which was the predecessor to Keyhole's EarthViewer3D.

Artist Statement

“A software engineering or R&D position that allows me to use my skill in designing simple robust solutions to well-defined challenging problems in computer graphics and computational geometry.”

Artwork Highlight:

Hyerbolic tessellations

{p,q}:{3,7} Truncation:1

Computing Process:

Each tessellation is represented by a Schlafli symbol of the form {p,q}, which means that q regular p-gons surround each vertex. There exists a hyperbolic tessellation {p,q} for every p,q such that (p-2)*(q-2) > 4.

Enumeration of Uniform Tessellation:

A tessellation (a.k.a. tiling) is uniform if its faces are regular and its symmetry group (including reflections) is transitive on the vertices. A uniform tessellation whose dual is also uniform is called regular. A uniform tessellation that is not regular is called semi-regular. We'll consider spherical, planar, and hyperbolic tilings all at once, using "Schwarz polygons" (a generalization of Schwarz triangles) to generate the symmetry groups, and using a generalized Coxeter-Dynkin symbol to name the resulting tessellation. (The Wythoff symbol could probably be used instead, except that I find the Wythoff symbol to be horribly unintuitive and I don't see how to extend it to Schwarz polygons of more than 3 sides). We will only consider tilings composed of convex (non-star) polygons.

Analysis:

The tessellation is very attractive. The line can lead your eye into the symbol. You can feel the tessellation give you a dignity feel. The artist not only shows front view of the tessellation, he also shows the symbol in other view, the tessellation then become another beautiful artwork. Furthermore, this kind of tessellation can be reused in various patterns not only the “Schwarz polygons”. It can be replaced by other type of symbol. The artist can generate other kind of beautiful tessellation.

Link:

Tessellation http://www.hadron.org/~hatch/HyperbolicTessellation/

Schwars-Christoffel http://en.wikipedia.org/wiki/Schwarz-Christoffel_mapping

Wythoff http://en.wikipedia.org/wiki/Wythoff

Coxeter-Dynkin http://en.wikipedia.org/wiki/Coxeter-Dynkin_diagram