Octagonal-dodecahedral duoprism

Octagonal-dodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymOdoe
Coxeter diagramx8o x5o3o
Elements
Tera12 pentagonal-octagonal duoprisms, 8 dodecahedral prisms
Cells96 pentagonal prisms, 30 octagonal prisms, 8 dodecahedra
Faces240 squares, 96 pentagons, 20 octagons
Edges160+240
Vertices160
Vertex figureTriangular scalene, edge lengths (1+5)/2 (base triangle), 2+2 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {17+4{\sqrt {2}}+3{\sqrt {5}}}{8}}}\approx 1.91589}$
Hypervolume${\displaystyle {\frac {15+15{\sqrt {2}}+7{\sqrt {5}}+7{\sqrt {10}}}{2}}\approx 37.00081}$
Diteral anglesDope–doe–dope: 135°
Podip–op–podip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Podip–pip–dope: 90°
Central density1
Number of external pieces20
Level of complexity10
Related polytopes
ArmyOdoe
RegimentOdoe
DualOctagonal-icosahedral duotegum
ConjugatesOctagrammic-dodecahedral duoprism, Octagonal-great stellated dodecahedral duoprism, Octagrammic-great stellated dodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(8), order 1920
ConvexYes
NatureTame

The octagonal-dodecahedral duoprism or odoe is a convex uniform duoprism that consists of 8 dodecahedral prisms and 12 pentagonal-octagonal duoprisms. Each vertex joins 2 dodecahedral prisms and 3 pentagonal-octagonal duoprisms.

Vertex coordinates

The vertices of an octagonal-dodecahedral duoprism of edge length 1 are given by:

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

as well as all even permutations of the last three coordinates of:

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

Representations

An octagonal-dodecahedral duoprism has the following Coxeter diagrams:

• x8o x5o3o (full symmetry)
• x4x x5o3o (octagons as ditetragons)