# Jeremy Smith

Art is in the family - my parents met in art school, dad was a cartoon animator, and my brother is an artist - but so is science - brother 2 teaches mathematics, my cousin runs a wheat physiology lab, and brother in law is actually in the Physics department at Arizona State.

I'm a long-time programmer, and keen on complexity because it's on the cutting edge of science. But lately I've been making 'toys' - various printable kits that you construct to make polyhedra, puzzles, and games. I have workshops at science and arts related festivals, and teachers are often ecstatic to discover these (free) resources.

Although some stuff is educational, I am driven by the fun factor and the wow factor, which I love to share. In my own explorations I am continually blown away by some new discovery that is inherent in the numbers, in the mathematics. I tell folks when I hit a rich vein, and wonder where everyone else is. That's why I'm coming to this conference - I'm sure they're all here.

This family of four 'banders' come from the platonic solids.

Draw a line on a tetrahedron from the midpoint of an edge to the midpoint of an adjacent edge and keep going. The line soon finds itself to form a loop, as do two other lines if you exhaust all possibilities. If you expand these three bands onto the surface of the mid-sphere you arrive at the three-bander. Because the Platonic solids are symmetrical, so are the bands, being geodesic circles.

A similar exercise will form the four-bander from the octahedron and the six-bander from the icosahedron. The cube will form the three- and four-bander, as a geodesic path can be found by chasing the opposite edge or the adjacent edge (but not both).

Triangles provide only one bander model since triangles only have one possible track, but like the square the pentagon also has two tracks - adjacent mid-points and opposite, and these (on the dodecahedron) produce the six-bander and the beautiful ten-bander.

Cube of Chess

A 4x4x4 cube of wood is dissected into 32 pieces, sixteen identical pieces of size 1.5 cubes and sixteen of size 2.5 cubes. The latter is five kinds differing only in the half-cube on top which has been bisected mostly using common symmetrical cuts.

The thirty two pieces are of course a chess set. By giving the pieces a waist turning the half-cube on the base cube, the set looks like a chess set (rather than a cubist sculpture, though don't sell short your Staunton stock).

I saw the Graham Lanier chess set 1966 in the NY MoMA, which fit impressively into a 2x4x6 box. At some point I thought 'but why not a cube'. I just had to do it.

The box, by the way, turns into the board.

These hexagonal pieces are a rendition of Socolar-Taylor aperiodic tiles. I made them to explore aperiodic tiling in general, and these in particular. I've explored rule one but not yet all of rule two, or mirror image possibilities. In the meantime I've created a number of designs for friends and family.

I rendered them spherically here just for fun. (Aperiodic tiling on the surface of a sphere?)

The tile is hexagonal and a path crosses each edge on the left or right hand side. These paths must match up (this is rule #1. There is a second rule, using the little pennant flags.). There are six ways to internally arrange the paths in each hexagon but otherwise all the tiles are functionally identical.

Socolar and Taylor discovered this tile in March 2010. It is the first single aperiodic tile. Penrose found a 2-tile aperiodic set in 1974 and prior to 1966 they were unknown.

Is this an 'einstien' (one stone)? - a single tile that can aperiodically tile an infinite plane?