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In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces. The snub dodecahedron has 92 faces (the most of the 13 Archimedean solids): 12 are pentagons and the other 80 are equilateral triangles .
In 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. [ 1 ] It is given a Schläfli symbol sr{ 5 ⁄ 2 ,5}, as a snub great dodecahedron .
Let φ be the golden ratio.The 12 points given by (0, ±1, ±φ) and cyclic permutations of these coordinates are the vertices of a regular icosahedron.Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the 8 points (±1, ±1, ±1) together with the 12 points (0, ±φ, ± 1 / φ ) and cyclic permutations of these coordinates.
It is explicitly called a pentatruncated pentagonal hexecontahedron since only the valence-5 vertices of the pentagonal hexecontahedron are truncated. [2]Its topology can be constructed in Conway polyhedron notation as t5gD and more simply wD as a whirled dodecahedron, reducing original pentagonal faces and adding 5 distorted hexagons around each, in clockwise or counter-clockwise forms.
Truncated dodecahedron: 3.10.10: 20 triangles 12 decagons: 90 60 I h: Truncated icosahedron: 5.6.6: 12 pentagons 20 hexagons 90 60 I h: Rhombicosidodecahedron: 3.4.5.4: 20 triangles 30 squares 12 pentagons 120 60 I h: Truncated icosidodecahedron: 4.6.10: 30 squares 20 hexagons 12 decagons 180 120 I h: Snub dodecahedron: 3.3.3.3.5: 80 triangles ...
Pentagonal pyramids are added to the 12 pentagonal faces of the snub dodecahedron, and triangular pyramids are added to the 20 triangular faces that do not share an edge with a pentagon. The pyramid heights are adjusted to make them coplanar with the other 60 triangular faces of the snub dodecahedron. The result is the pentagonal ...
The snub disphenoid name comes from Johnson (1966) classification of the Johnson solid. [12] However, this solid was first studied by Rausenberger (1915). [13] [14] It was studied again in the paper by Freudenthal & van d. Waerden (1947), which first described the set of eight convex deltahedra, and named it the Siamese dodecahedron. [15] [14]
Net (click to enlarge) The pentakis snub dodecahedron is a convex polyhedron with 140 triangular faces, 210 edges, and 72 vertices. It has chiral icosahedral symmetry ...