Izzy Triangles

Exploring the 64 possible colorings of an equilateral triangle cut into 6 segments by its medians. Each segment can be black or white, but the figure is unchanged by a third of a turn, so colorings come in orbits of three — and only 24 of the 64 are distinct.

Press Count to walk through all 64. The grid beside it holds one slot for each of the 24 distinct colorings, and fills as each new one appears — so you can see which patterns have turned up, not just how many.

When the colouring on the left is one you have already seen, it is a third or two-thirds of a turn away from one in the grid, and that one is ringed. The speed control runs from a quarter of the original pace up to twice it; the slowest setting is the default, because at speed the repeats go by faster than you can match them.

Where this came from

I met Mark Saul through Ethan Berman’s i2camp. Among the things he put in front of children was IZZI, an edge-matching puzzle of 64 square cardboard tiles designed by Frank Nichols and published by Binary Arts in 1992. Each square is cut into eight triangles by its diagonals and its two mid-lines, every triangle black or white, and the game is to lay all 64 in an 8×8 square with each touching edge matching colour to colour. Watching that, I wanted to know what the same idea looked like on a triangle.

The connection is closer than inspiration. IZZI uses 64 of 70 possible tiles, and 70 is exactly the number of eight-triangle colourings counted up to rotation — Burnside over the four turns of a square:

which is the calculation on this page, one figure over. Six wedges with three-fold symmetry give

Both figures happen to have 64 in them — IZZI’s tile count, and this page’s colourings — and that is a coincidence rather than a correspondence.

Sol LeWitt would have called it a wall drawing

Stating a rule and then exhibiting every variation it permits is a conceptual-art method as much as a mathematical one. Sol LeWitt built all 122 Variations of Incomplete Open Cubes in 1974 and showed them as a grid, because the complete set is the work — no single cube is the point. The 4×6 gallery above is the same gesture: the rule is a third of a turn, and what you get for obeying it is 24 objects and no more.