Decagonal-cuboctahedral duoprism

Decagonal-cuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymDaco
Coxeter diagramx10o o4x3o ()
Elements
Tera10 cuboctahedral prisms, 8 triangular-decagonal duoprisms, 6 square-decagonal duoprisms
Cells80 triangular prisms, 60 cubes, 10 cuboctahedra, 24 decagonal prisms
Faces80 triangles, 60+240 squares, 12 decagons
Edges120+240
Vertices120
Vertex figureRectangular scalene, edge lengths 1, 2, 1, 2 (base rectangle), (5+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\approx 1.90211}$
Hypervolume${\displaystyle {\frac {25{\sqrt {10+4{\sqrt {5}}}}}{6}}\approx 18.13542}$
Diteral anglesCope–co–cope: 144°
Tradedip–dip–squadedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Central density1
Number of external pieces24
Level of complexity20
Related polytopes
ArmyDaco
RegimentDaco
DualDecagonal-rhombic dodecahedral duotegum
ConjugateDecagrammic-cuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(10), order 960
ConvexYes
NatureTame

The decagonal-cuboctahedral duoprism or daco is a convex uniform duoprism that consists of 10 cuboctahedral prisms, 6 square-decagonal duoprisms, and 8 triangular-decagonal duoprisms. Each vertex joins 2 cuboctahedral prisms, 2 triangular-decagonal duoprisms, and 2 square-decagonal duoprisms.

Vertex coordinates

The vertices of a decagonal-cuboctahedral duoprism of edge length 1 are given by all permutations of the last three coordinates of:

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

Representations

A decagonal-cuboctahedral duoprism has the following Coxeter diagrams:

• x10o o4x3o () (full symmetry)
• x5o o4x3o () (decagons as dipentagons)
• x10o x3o3x ()
• x5x x3o3x ()