Rank4
TypeSegmentotope
Notation
Coxeter diagramxx5/2xo5ox&#x
Elements
Cells1 doe
12 pap
12 stiscu
30 trip
Faces60+60 triangles
30+60 squares
12+12 pentagons
12 pentagrams
Edges60+60+120 3-fold
30 6-fold
Vertices20+60
Measures (edge length 1)
Circumradius${\displaystyle \approx 1.43217272634}$
Height${\displaystyle {\sqrt {\frac {3{\sqrt {5}}-1}{8}}}\approx 0.84470438118}$
Related polytopes
Convex hullDodecahedron atop truncated icosahedron
Convex coreDodecahedron atop chamfered dodecahedron
Abstract & topological properties
OrientableNo
Properties
SymmetryH3×I, order 120
ConvexNo
NatureWild

The dodecahedron atop rhombidodecadodecahedron is a segmentochoron. It consists of 1 dodecahedron, 1 rhombidodecadodecahedron, 12 pentagonal antiprisms, 12 pentagrammic cuploids, and 30 triangular prisms.

It appears as a facet of the medial hecatonicosafaceted prismatodishecatonicosachoric alterprism.

## Vertex coordinates

The vertices of a dodecahedron atop rhombidodecadodecahedron of edge length 1 are given by all permutations of the first three coordinates of:

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\sqrt {\frac {3{\sqrt {5}}-1}{8}}}\right),}$

along with all even permutations of the first three coordinates of:

• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0,\,0\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,{\sqrt {\frac {3{\sqrt {5}}-1}{8}}}\right),}$
• ${\displaystyle \left(\pm 1,\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,{\sqrt {\frac {3{\sqrt {5}}-1}{8}}}\right).}$