I was fortunate enough to visit downtown Los Angeles just in time to see this beautiful geometry installation by Hybycozo at the City National Plaza. The purple object on the left is an icosahedron, and the green one on the right — a truncated octahedron. The icosahedron is one of the 5 Platonic solids discovered several thousand years ago. Here is a picture of the Platonic solids from Johannes Kepler’s work, Harmonices Mundi, in 1619:
The icosahedron is constructed of 20 equilateral triangles, with 5 triangles meeting at each vertex. The truncated octahedron is an Archimedian solid, also discovered in antiquity. It is constructed of six squares and 8 regular hexagons. Here is Johannes Kepler’s drawings of the truncated octahedron, also from Harmonices Mundi:
With Geometiles, you can easily create your own mini-versions of these solids. You can glue small battery-operated candles for the lighting effect:
It may not be obvious how the truncated octahedron is obtained from the octahedron by truncation. This is more clearly seen when you construct both solids using different color Geometiles.
In the picture below, on the left is an octahedron about to be truncated; in fact, the truncation process has begun by chopping off one of the orange corners. On the right is what you get after you’ve finished truncating corners, and “sealing” the openings with squares. Note that the truncation has to be performed so that what you are left with are regular hexagons.