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.
