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The rhombicosidodecahedron shares its vertex arrangement with three nonconvex uniform polyhedra: the small stellated truncated dodecahedron, the small dodecicosidodecahedron (having the triangular and pentagonal faces in common), and the small rhombidodecahedron (having the square faces in common).
Coxeter, Longuet-Higgins & Miller (1954) define uniform polyhedra to be vertex-transitive polyhedra with regular faces. They define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property.
Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters: [C] Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92.
In geometry, the great truncated icosidodecahedron (or great quasitruncated icosidodecahedron or stellatruncated icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U 68. It has 62 faces (30 squares, 20 hexagons, and 12 decagrams), 180 edges, and 120 vertices. [1]
Greg Egan's applet to display uniform polyhedra using Wythoff's construction method; A Shadertoy renderization of Wythoff's construction method; KaleidoTile 3 Free educational software for Windows by Jeffrey Weeks that generated many of the images on the page. Hatch, Don. "Hyperbolic Planar Tessellations"
Of all vertex-transitive polyhedra, it occupies the largest percentage (89.80%) of the volume of a sphere in which it is inscribed, very narrowly beating the snub dodecahedron (89.63%) and small rhombicosidodecahedron (89.23%), and less narrowly beating the truncated icosahedron (86.74%); it also has by far the greatest volume (206.8 cubic ...
In geometry, the rhombicosahedron is a nonconvex uniform polyhedron, indexed as U 56. It has 50 faces (30 squares and 20 hexagons), 120 edges and 60 vertices. [1] Its vertex figure is an antiparallelogram.
In geometry, the rhombidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U 38. It has 54 faces (30 squares, 12 pentagons and 12 pentagrams), 120 edges and 60 vertices. [1] It is given a Schläfli symbol t 0,2 {5 ⁄ 2,5}, and by the Wythoff construction this polyhedron can also be named a cantellated great dodecahedron.