Truncated great dodecahedron

In geometry, the truncated great dodecahedron is a nonconvex uniform polyhedron, indexed as U37. It has 24 faces (12 pentagrams and 12 decagons), 90 edges, and 60 vertices.[1] It is given a Schläfli symbol t{5,​5⁄2}.

Truncated great dodecahedron
TypeUniform star polyhedron
ElementsF = 24, E = 90
V = 60 (χ = −6)
Faces by sides12{5/2}+12{10}
Wythoff symbol5
2 5/3 | 5
Symmetry groupIh, [5,3], *532
Index referencesU37, C47, W75
Dual polyhedronSmall stellapentakis dodecahedron
Vertex figure
10.10.5/2
Bowers acronymTigid
3D model of a truncated great dodecahedron

It shares its vertex arrangement with three other uniform polyhedra: the nonconvex great rhombicosidodecahedron, the great dodecicosidodecahedron, and the great rhombidodecahedron; and with the uniform compounds of 6 or 12 pentagonal prisms.


Nonconvex great rhombicosidodecahedron
Great dodecicosidodecahedron
Great rhombidodecahedron

Truncated great dodecahedron
Compound of six pentagonal prisms
Compound of twelve pentagonal prisms

This polyhedron is the truncation of the great dodecahedron:

The truncated small stellated dodecahedron looks like a dodecahedron on the surface, but it has 24 faces, 12 pentagons from the truncated vertices and 12 overlapping as (truncated pentagrams).

Name Small stellated dodecahedron Truncated small stellated dodecahedron Dodecadodecahedron Truncated
great
dodecahedron		 Great
dodecahedron
Coxeter-Dynkin
diagram
Picture

Small stellapentakis dodecahedron
Small stellapentakis dodecahedron
TypeStar polyhedron
Face
ElementsF = 60, E = 90
V = 24 (χ = −6)
Symmetry groupIh, [5,3], *532
Index referencesDU37
dual polyhedronTruncated great dodecahedron
3D model of a small stellapentakis dodecahedron

The small stellapentakis dodecahedron (or small astropentakis dodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the truncated great dodecahedron. It has 60 intersecting triangular faces.

See also

References
  1. Maeder, Roman. "37: truncated great dodecahedron". MathConsult.

Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208

Animated truncation sequence from {​5⁄2, 5} to {5, ​5⁄2}


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