Stellated truncated hexahedronIn geometry, the stellated truncated hexahedron (or quasitruncated hexahedron, and stellatruncated cube) is a uniform star polyhedron, indexed as U19. It has 14 faces (8 triangles and 6 octagrams), 36 edges, and 24 vertices. It is represented by Schläfli symbol t'{4,3} or t{4/3,3}, and Coxeter-Dynkin diagram, . It is sometimes called quasitruncated hexahedron because it is related to the truncated cube, , except that the square faces become inverted into {8/3} octagrams.
Great cubicuboctahedronIn geometry, the great cubicuboctahedron is a nonconvex uniform polyhedron, indexed as U14. It has 20 faces (8 triangles, 6 squares and 6 octagrams), 48 edges, and 24 vertices. Its square faces and its octagrammic faces are parallel to those of a cube, while its triangular faces are parallel to those of an octahedron: hence the name cubicuboctahedron. The prefix great serves to distinguish it from the small cubicuboctahedron, which also has faces in the aforementioned directions.
DodecadodecahedronIn geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by , and . The edges of this model form 10 central hexagons, and these, projected onto a sphere, become 10 great circles. These 10, along with the great circles from projections of two other polyhedra, form the 31 great circles of the spherical icosahedron used in construction of geodesic domes.
Great icosidodecahedronIn geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r{3,}. It is the rectification of the great stellated dodecahedron and the great icosahedron. It was discovered independently by , and . The figure is a rectification of the great icosahedron or the great stellated dodecahedron, much as the (small) icosidodecahedron is related to the (small) icosahedron and (small) dodecahedron, and the cuboctahedron to the cube and octahedron.
Truncated great dodecahedronIn geometry, the truncated great dodecahedron is a nonconvex uniform polyhedron, indexed as U37. It has 24 faces (12 pentagrams and 12 decagons), 90 edges, and 60 vertices. It is given a Schläfli symbol t{5,5/2}. It shares its vertex arrangement with three other uniform polyhedra: the nonconvex great rhombicosidodecahedron, the great dodecicosidodecahedron, and the great rhombidodecahedron; and with the uniform compounds of 6 or 12 pentagonal prisms.
Small stellated truncated dodecahedronIn geometry, the small stellated truncated dodecahedron (or quasitruncated small stellated dodecahedron or small stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U58. It has 24 faces (12 pentagons and 12 decagrams), 90 edges, and 60 vertices. It is given a Schläfli symbol t{,5}, and Coxeter diagram . It shares its vertex arrangement with three other uniform polyhedra: the convex rhombicosidodecahedron, the small dodecicosidodecahedron and the small rhombidodecahedron.
Great stellated truncated dodecahedronIn geometry, the great stellated truncated dodecahedron (or quasitruncated great stellated dodecahedron or great stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U66. It has 32 faces (20 triangles and 12 decagrams), 90 edges, and 60 vertices. It is given a Schläfli symbol t0,1{5/3,3}.
Truncated great icosahedronIn geometry, the truncated great icosahedron (or great truncated icosahedron) is a nonconvex uniform polyhedron, indexed as U55. It has 32 faces (12 pentagrams and 20 hexagons), 90 edges, and 60 vertices. It is given a Schläfli symbol t{3,} or t0,1{3,} as a truncated great icosahedron. Cartesian coordinates for the vertices of a truncated great icosahedron centered at the origin are all the even permutations of (±1, 0, ±3/τ) (±2, ±1/τ, ±1/τ3) (±(1+1/τ2), ±1, ±2/τ) where τ = (1+√5)/2 is the golden ratio (sometimes written φ).
RhombidodecadodecahedronIn geometry, the rhombidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U38. It has 54 faces (30 squares, 12 pentagons and 12 pentagrams), 120 edges and 60 vertices. It is given a Schläfli symbol t0,2, and by the Wythoff construction this polyhedron can also be named a cantellated great dodecahedron. Cartesian coordinates for the vertices of a uniform great rhombicosidodecahedron are all the even permutations of (±1/τ2, 0, ±τ2) (±1, ±1, ±) (±2, ±1/τ, ±τ) where τ = (1+)/2 is the golden ratio (sometimes written φ).
Snub dodecadodecahedronIn geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U_40. It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol sr{,5}, as a snub great dodecahedron. Cartesian coordinates for the vertices of a snub dodecadodecahedron are all the even permutations of (±2α, ±2, ±2β), (±(α+β/τ+τ), ±(-ατ+β+1/τ), ±(α/τ+βτ-1)), (±(-α/τ+βτ+1), ±(-α+β/τ-τ), ±(ατ+β-1/τ)), (±(-α/τ+βτ-1), ±(α-β/τ-τ), ±(ατ+β+1/τ)) and (±(α+β/τ-τ), ±(ατ-β+1/τ), ±(α/τ+βτ+1)), with an even number of plus signs, where β = (α2/τ+τ)/(ατ−1/τ), where τ = (1+)/2 is the golden mean and α is the positive real root of τα4−α3+2α2−α−1/τ, or approximately 0.
