A sphere
A tiling of the sphere with 48 spherical triangles with angles 45°, 60°, 90°. The great circles in this tiling are the intersections of the sphere with all the (9) symmetry planes of a cube 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 cube. 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 (4,6,8), in the three vertices we get a cube, a octahedron and a cuboctahedron, on the three sides we get a (4,6,6), a (3,8,8) and a (3,4,4,4).
We can also see the other analogue tilings of the sphere.
© matematita
immagine di Francesca Lazzaroni
The image belongs to the sections...:
Coxeter dossier (From "XlaTangente")
Spherical geometry (Other geometries)
The symmetry group of the cube (*432) (Symmetry)
