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In geometry, a dodecahedron (from Ancient Greek δωδεκάεδρον (dōdekáedron); from δώδεκα (dṓdeka) 'twelve' and ἕδρα (hédra) 'base, seat, face') or duodecahedron [1] is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid.
A regular dodecahedron or pentagonal dodecahedron [notes 1] is a dodecahedron composed of regular pentagonal faces, three meeting at each vertex. It is an example of Platonic solids , described as cosmic stellation by Plato in his dialogues, and it was used as part of Solar System proposed by Johannes Kepler .
For example, with 4 square faces, and 60-degree rhombic faces, and D 4h dihedral symmetry, order 16. It can be seen as a cuboctahedron with square pyramids attached on the top and bottom. In 1960, Stanko Bilinski discovered a second rhombic dodecahedron with 12 congruent rhombus faces, the Bilinski dodecahedron. It has the same topology but ...
It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry. Uniform polyhedra can be divided between convex forms with convex regular polygon faces and star forms. Star forms have either regular star polygon faces or vertex figures or both. This list includes these:
3D model of a elongated dodecahedron. In geometry, the elongated dodecahedron, [1] extended rhombic dodecahedron, rhombo-hexagonal dodecahedron [2] or hexarhombic dodecahedron [3] is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular depending on the shape of the rhombi.
It has 24 faces (12 pentagons and 12 pentagrams), 60 edges, and 20 vertices. [1] It has extended Schläfli symbol b{5, 5 ⁄ 2 }, as a blended great dodecahedron , and Coxeter diagram . It has 4 Schwarz triangle equivalent constructions, for example Wythoff symbol 3 | 5 ⁄ 3 5 , and Coxeter diagram .
In geometry, the trapezo-rhombic dodecahedron or rhombo-trapezoidal dodecahedron is a convex dodecahedron with 6 rhombic and 6 trapezoidal faces. It has D 3h symmetry. A concave form can be constructed with an identical net, seen as excavating trigonal trapezohedra from the top and bottom. It is also called the trapezoidal dodecahedron. [1]
A variant with pyritohedral symmetry is called a dyakis dodecahedron [5] [6] or diploid. [7] It is common in crystallography. A dyakis dodecahedron can be created by enlarging 24 of the 48 faces of a disdyakis dodecahedron. A tetartoid can be created by enlarging 12 of the 24 faces of a dyakis dodecahedron. 3D model of a dyakis dodecahedron [8]