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The concave equilateral dodecahedron, called an endo-dodecahedron. [clarification needed] A cube can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions. A regular dodecahedron is an intermediate case with equal edge lengths. A rhombic dodecahedron is a degenerate case with the 6 crossedges reduced to ...
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 28 faces: twelve regular pentagons arranged in four panels of three pentagons each, four equilateral triangles (shown in blue), and six pairs of isosceles triangles (shown in yellow). All edges of the tetrated dodecahedron have the same length, except for the shared bases of these isosceles triangles, which are approximately 1.07 times ...
Adding six squares or eight equilateral triangles results in the cubohemicotahedron or octahemioctahedron, respectively. [ 20 ] The cuboctahedron 2-covers the tetrahemihexahedron , which accordingly has the same abstract vertex figure (two triangles and two squares: 3 ⋅ 4 ⋅ 3 ⋅ 4 {\displaystyle 3\cdot 4\cdot 3\cdot 4} ) and half the ...
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 .
A dodecahedron can be considered a regular tiling of 12 pentagons on the surface of a sphere, with Schläfli symbol {5,3}, having three pentagons around each vertex. One may also consider a degenerate tiling by two hemispheres , with the great circle between them subdivided into five equal arcs, as a pentagonal tiling with Schläfli symbol {5,2}.
The shape of the "Crystal Dome" used in the popular TV game show The Crystal Maze was based on a pentakis dodecahedron. In Doctor Atomic, the shape of the first atomic bomb detonated in New Mexico was a pentakis dodecahedron. In De Blob 2 in the Prison Zoo, domes are made up of parts of a Pentakis Dodecahedron. These Domes also appear whenever ...
The stellated rhombic dodecahedron, a compound of three flattened octahedra used in Escher's Study for Stars and Waterfall. The compound of three octahedra has the same symmetry group as a single octahedron. It is an isohedral deltahedron, meaning that its faces are equilateral triangles and that it has a symmetry taking every face to every ...