# Hexagonal duoprism

Hexagonal duoprism
Rank4
TypeUniform
Notation
Bowers style acronymHiddip
Coxeter diagramx6o x6o ()
Elements
Cells12 hexagonal prisms
Faces36 squares, 12 hexagons
Edges72
Vertices36
Vertex figureTetragonal disphenoid, edge lengths 3 (base) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {2}}\approx 1.41421}$
Inradius${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Hypervolume${\displaystyle {\frac {27}{4}}=6.75}$
Dichoral anglesHip–6–hip: 120°
Hip–4–hip: 90°
Central density1
Number of external pieces12
Level of complexity3
Related polytopes
ArmyHiddip
RegimentHiddip
DualHexagonal duotegum
ConjugateNone
Abstract & topological properties
Flag count864
Euler characteristic0
OrientableYes
Properties
SymmetryG2≀S2, order 288
Flag orbits3
ConvexYes
NatureTame

The hexagonal duoprism or hiddip, also known as the hexagonal-hexagonal duoprism, the 6 duoprism or the 6-6 duoprism, is a noble uniform duoprism that consists of 12 hexagonal prisms, with 4 joining at each vertex. It is also the 12-5 gyrochoron. It is the first in an infinite family of isogonal hexagonal dihedral swirlchora and also the first in an infinite family of isochoric hexagonal hosohedral swirlchora.

This polychoron can be alternated into a triangular duoantiprism, although it cannot be made uniform.

A unit hexagonal duoprism can be vertex-inscribed into the antifrustary distetracontoctachoron and ditetrahedronary dishecatonicosachoron.

## Vertex coordinates

Coordinates for the vertices of a hexagonal duoprism of edge length 1, centered at the origin, are given by:

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

Simpler coordinates can be given in 6-dimensional space as all permutations of

• ${\displaystyle \pm {\dfrac {\sqrt {6}}{6}}\left(2,\,-1,\,-1,\,2,\,-1,\,-1\right)}$,

where the sum of the first three coordinates is 0. Multiplying these coordinates by ${\displaystyle {\sqrt {6}}}$ gives a set of integral coordinates.

## Representations

A hexagonal duoprism has the following Coxeter diagrams:

• x6o x6o () (full symmetry)
• x3x x6o () (G2×A2 symmetry, one hexagon seen as ditrigon)
• x3x x3x () (A2≀S2 symmetry, both hexagons seen as ditrigons)
• xux xxx6ooo&#xt (G2×A1 axial)
• xux xxx3xxx&#xt (A2×A1 symmetry, ditrigonal axial)