Hexagonal-octagonal duoprismatic prism

Hexagonal-octagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymHaop
Coxeter diagramx x6o x8o
Elements
Tera8 square-hexagonal duoprisms, 6 square-octagonal duoprisms, 2 hexagonal-octagonal duoprisms
Cells48 cubes, 6+12 octagonal prisms, 8+16 hexagonal prisms
Faces48+48+96 squares, 16 hexagons, 12 octagons
Edges48+96+96
Vertices96
Vertex figureDigonal disphenoidal pyramid, edge lengths 3 (disphenoid base 1), 2+2 (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {9+2{\sqrt {2}}}}{2}}\approx 1.71962}$
Hypervolume${\displaystyle 3({\sqrt {3}}+{\sqrt {6}})\approx 12.54462}$
Diteral anglesShiddip–hip–shiddip: 135°
Sodip–op–sodip: 120°
Sodip–cube–shiddip: 90°
Hodip–hip–shiddip: 90°
Sodip–op–hodip: 90°
Height1
Central density1
Number of external pieces16
Level of complexity30
Related polytopes
ArmyHaop
RegimentHaop
DualHexagonal-octagonal duotegmatic tegum
ConjugateHexagonal-octagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryG2×I2(8)×A1, order 384
ConvexYes
NatureTame

The hexagonal-octagonal duoprismatic prism or haop, also known as the hexagonal-octagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 hexagonal-octagonal duoprisms, 6 square-octagonal duoprisms, and 8 square-hexagonal duoprisms. Each vertex joins 2 square-hexagonal duoprisms, 2 square-octagonal duoprisms, and 1 hexagonal-octagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

This polyteron can be alternated into a triangular-square duoantiprismatic antiprism, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a triangular-square prismatic prismantiprismoid, which is also nonuniform.

Vertex coordinates

The vertices of a hexagonal-octagonal duoprismatic prism of edge length 1 are given by all permutations of the third and fourth coordinates of:

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

Representations

A hexagonal-octagonal duoprismatic prism has the following Coxeter diagrams:

• x x6o x8o (full symmetry)
• x x3x x8o (hexagons as ditrigons)
• x x6o x4x (octagons as ditetragons)
• x x3x x4x
• xx6oo xx8oo&#x (hexagonal-octagonal duoprism atop hexagonal-octagonal duoprism)
• xx3xx xx8oo&#x
• xx6oo xx4xx&#x
• xx3xx xx4xx&#x