Sunday, January 23, 2011

Magnetics Part 1: Hyperbolic Planes

After being inspired by Vi Hart, I started playing with my magnetics set. It's a building toy that works really well for making polyhedra and other mathematical stuff.

I started by making a hyperbolic plane. There are several ways to create one. You can have four pentagons meet at each vertex (432°), or three heptagons (386°), or three triangles, one square, and one pentagon (378°). Having more than 360° at each vertex prevents the plane from being flat, and instead warps it into a hyperbolic plane.

Wikipedia has a complete list; it uses compressed plane tilings (Poincaré disk model) rather than 3-D hyperbolic planes but they're mathematically the same.

I built a small section of an Order-7 triangular tiling, which has seven triangles (420°) at each vertex. Its Schläfli symbol is {3,7}, indicating 7 triangles around each vertex. ({3,3} is a tetrahedron, {3,4} an octahedron, {3,5} an icosahedron, and {3,6} a flat plane; {3,8} and above are hyperbolic surfaces that don't tile well.)

The green is the center vertex and seven triangles, with blue outer edge. The red and yellow are the next outward layer of triangles from the arbitrary center point; the silver is a nice aesthetic edge.

Vi Hart has a lot of neat stuff with hyperbolic planes; she makes them with apple slices and balloons.

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