A tiling of the sphere with 24 spherical triangles with angles 60°, 60°, 90°. The great circles in this tiling are the intersections of the sphere with all the (6) symmetry planes of a regular tetrahedron with center in the center of the sphere.
One of these triangles corresponds to a kaleidoscope where we can reconstruct objects with the same symmetry as a regular tetrahedron. The seven points we see in the figure show where we have to place a point-object in the kaleidoscope in order that the points we see are the vertices of a polyhedron such that all its faces are regular polygons: in the center of the spherical triangle we get a (4,6,6), in the three vertices we get a regular tetrahedron (in two cases) or a regular octahedron, on the three sides we get a (3,6,6) (in two cases) or a cuboctahedron and a (3,4,4,4).
We can also see the other analogue tilings of the sphere.
immagine di Francesca Lazzaroni