KaleidoHedron. The symmetries of an icosahedron are used to replicate a pattern across the surface of a sphere.

What are the symmetries of an icosahedron? Pick any pair of adjoining vertices, and call them v1 and v2. You can rotate the icosahedron so that any of its 12 vertices ends up at the original position of v1, and any of the 5 adjoining vertices ends up at the original position of v2. That gives 60 distinct rotations. An optional reflection, in the plane passing through the icosahedron’s centre and the original positions of v1 and v2, yields a total of 120 symmetries.

Each of these symmetries carries a different triangular piece of the icosahedron’s surface (one sixth of a triangular face) into the triangle bounded by (the original positions of) v1, the point halfway between v1 and v2, and the centre of one of the faces that has v1 and v2 as corners. All the symmetries can be generated by repeated reflections in the three planes that pass through the centre of the icosahedron and any of the three edges of this triangle.

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