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Endododecahedron


Endododecahedron

The endodocehedron, also called the concave pyrohedral dodecahedron, is the concave solid corresponding to the interior void formed when each face of a regular dodecahedron is folded along a diagonal and resulting faces are unfolded to form a cube. It is equilateral and has 12 faces, 30 edges, and 20 vertices.

EndododecahedronByWedges

The endodocehedron with unit edge lengths corresponds to removing six square-base oblique wedges of edge length phi (where phi is the golden ratio), height 1/2 and ridge length 1/2 from a cube of edge length phi.

EndododecahedronNet

The endododecahedron can be constructed from 6 pieces as shown above, where each edge has unit length, assembled so that the central edges of adjacent pieces are oppositely oriented.

The endododecahedron of edge length a has surface area

 S=3sqrt(5+2sqrt(5))a^2,

and volume

 V=1/2phia^3,

where phi is again the golden ratio.

The skeleton of the endodecahedron is the dodecahedral graph.


See also

Cube, Equilateral Polyhedron, Wedge

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Cite this as:

Weisstein, Eric W. "Endododecahedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Endododecahedron.html

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