Final stellation of the icosahedron 180 triangular faces, which are the outermost triangular regions in the stellation diagram. These join along 270 edges, which in turn meet at 92 vertices, with an Euler characteristic of 2. The 92 vertices lie on the surfaces of three concentric spheres. The innermost group of 20 vertices form the vertices of a regular dodecahedron; the next layer of 12 form the vertices of a regular icosahedron; and the outer layer of 60 form the vertices of a nonuniform truncated icosahedron. The radii of these spheres are in the ratio When regarded as a three-dimensional solid object with edge lengths
Regular dodecahedron Shown here including the inner 12 vertices, which are not connected by the outer hull edges of 6D norm length , form a regular icosahedron. <br>The 3D projection basis vectors ["u","v","w"] used are: Regular dodecahedral objects have found some practical applications, and have also played a role in the visual arts and in philosophy. Iamblichus states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons." In "Theaetetus", a dialogue of Plato, Plato was able to prove that there are just five uniform regular solids; these later became known