An interesting additional example can be found in spherical geometry. On a sphere, a properly sized regular pentagon can be divided into five equilateral triangles. Simply inscribe a regular icosahedron in the sphere and use the contact points to define the vertices of the pentagon and its component triangles.

Some interesting additional examples can be found in spherical geometry. The simplest involves dividing a pentagon into five equilateral triangles. Simply inscribe a regular icosahedron in the sphere and use the contact points to define the vertices of the pentagon and its component triangles.

With Archimedean solids we find a couple less obvious examples:

• A small rhombicuboctahedron inscribed in a sphere contains regular-octagonal loops that encompass four equilarlteral triangles and five squares, allowing that size regular octagon to be divided accordingly.

• A small rhomicoisdodecahedron similarly contains a regular decagonal loop, which can be divided to give five equilateral triangles, five squares and a regular pentagon. The pentagon in this case is smaller than that in the regular icosahedral loop described above, so a radial division would not yield equilateral triangles.

We may consider these spherical-geometry divisions as relatives of the planar ones in which the curvature of the sphere forces a reduction in the number of sides in the large polygon and any centrally located component polygon. Thus on the plane a regular hexagon is divisible into six equilateral triangles, but on the sphere the number of sides is reduced to five in the inscribed-icosahedron division. Similarly, on the plane the regular dodecagon is divisible into equilateral triangles, squares and a central regular hexagon; the two Archimedean-solid based divisions above reduce the number of outer sides from twelve to eight or ten with the central piece following suit.