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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 ...
Adding six squares or eight equilateral triangles results in the cubohemicotahedron or octahemioctahedron, respectively. [ 17 ] 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 shape with the same exterior appearance as the dodecadodecahedron can be constructed by folding up these nets: 12 pentagrams and 20 rhombic clusters are necessary. . However, this construction replaces the crossing pentagonal faces of the dodecadodecahedron with non-crossing sets of rhombi, so it does not produce the same internal st
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.
In geometry, the excavated dodecahedron is a star polyhedron that looks like a dodecahedron with concave pentagonal pyramids in place of its faces. Its exterior surface represents the Ef 1 g 1 stellation of the icosahedron. It appears in Magnus Wenninger's book Polyhedron Models as model 28, the third stellation of icosahedron.
The icosahedron and dodecahedron are dual to each other. The small stellated dodecahedron and great dodecahedron are dual to each other. The great stellated dodecahedron and great icosahedron are dual to each other. The Schläfli symbol of the dual is just the original written backwards, for example the dual of {5, 3} is {3, 5}.
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 .
It comes from a topological construction from the snub dodecahedron with the kis operator applied to the pentagonal faces. In this construction, all the faces are computed to be the same distance from the center. 80 of the triangles are equilateral, and 60 triangles from the pentagons are isosceles.