Truncated dodecahedron Truncated dodecahedron In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges. This polyhedron can be formed from a dodecahedron by truncating (cutting off) the corners so the pentagon faces become decagons and the corners become triangles. It is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated icosahedral honeycomb. The area "A" and the volume "V" of a truncated dodecahedron of edge length "a" are: Cartesian coordinates for the vertices of a truncated dodecahedron with edge length 2"φ" − 2, centered at the origin, are
First stellation of rhombic dodecahedron First stellation of rhombic dodecahedron In geometry, the first stellation of the rhombic dodecahedron is a stellation of the rhombic dodecahedron. This polyhedron is sometimes called "Escher's solid"; it appears in M. C. Escher's works "Waterfall" and in a study for "Stars" (although "Stars" itself features a different shape, the compound of three octahedra). It has 48 faces, all triangles. It has 26 vertices in total: 6 with degree 8, 8 with degree 6, and 12 with degree 4. It has 72 edges in total, giving an Euler characteristic of 26 + 48 − 72 = +2. Its vertices are