# Square-dodecagrammic duoprism

Square-dodecagrammic duoprism
Rank4
TypeUniform
Notation
Coxeter diagramx4o2x12/5o ()
Elements
Cells12 cubes, 4 dodecagrammic prisms
Faces12+48 squares, 4 dodecagrams
Edges48+48
Vertices48
Vertex figureDigonal disphenoid, edge lengths (62)/2 (base 1) and 2 (base 2 and sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5-2{\sqrt {3}}}{2}}}\approx 0.87633}$
Hypervolume${\displaystyle 3(2{\sqrt {3}})\approx 0.80385}$
Dichoral anglesStwip–12/5–stwip: 90°
Cube–4–stwip: 90°
Cube–4–cube: 30°
Height1
Central density5
Number of external pieces28
Level of complexity12
Related polytopes
DualSquare-dodecagrammic duotegum
ConjugateSquare-dodecagonal duoprism
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryB2×I2(12), order 192
ConvexNo
NatureTame

The square-dodecagrammic duoprism, also known as the 4-12/5 duoprism, is a uniform duoprism that consists of 12 cubes and 4 dodecagrammic prisms, with 2 of each at each vertex.

## Vertex coordinates

The vertex coordinates of a square-dodecagrammic duoprism, centered at the origin and with unit edge length, are given by:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {3}}-1}{2}},\,\pm {\frac {{\sqrt {3}}-1}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2-{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2-{\sqrt {3}}}{2}}\right)}$.

## Representations

A square-dodecagrammic duoprism has the following Coxeter diagrams:

• x4o x12/5o () (full symmetry)
• x x x12/5o () (I2(12)×A1×A1 symmetry, dodecagrammic prismatic prism)
• x4o x6/5x () (B2×G2 symmetry)
• x x x6/5x () (G2×A1×A1 symmetry)