Hosohedron

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An n-gonal hosohedron is a way to tile a sphere with n lunes (digonal faces) that all meet at the same two opposite points, like the North and South poles. In a regular hosohedron, written as {2,n}, each lune has an interior angle of 2π/n (360/n degrees). The tiling has two shared vertices, and the n lunes fit together around the sphere.

The dual shape is the n-gonal dihedron {n,2}, which has two opposite faces. The {2,2} case is special: it is self-dual and is both a hosohedron and a dihedron. If you cut each lune in half, you get an n-gonal bipyramid, showing another way the pieces can be arranged.

You can truncate a hosohedron to obtain an n-gonal prism. If you take the idea to the limit, you get an apeirogonal hosohedron, a two-dimensional tiling by infinite lunes. The tetragonal hosohedron (n = 4) is related, topologically, to the Steinmetz solid—the intersection of two perpendicular cylinders.

In higher dimensions this idea becomes a hosotope. A regular hosotope with symbol {2,p,...,q} still has just two vertices. The 2D case {2} is a digon. The name hosohedron comes from the Greek for “as many,” reflecting that there can be as many faces as you want. It was introduced by Vito Caravelli in the 18th century.