# A sphere

A tiling of the sphere with 120 spherical triangles with angles 36°, 60°, 90°. The great circles in this tiling are the intersections of the sphere with all the (15) symmetry planes of a dodecahedron 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 dodecahedron. 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 (4,6,10), in the three vertices we get a dodecahedron, an icosahedron and a (3,5,3,5), on the three sides we get a (5,6,6), a (3,10,10) and a (3,4,5,4).

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 dodecahedron (*532) (Symmetry)