Hexagonal duoexpandoprism

Hexagonal duoexpandoprism
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Hiddep
Coxeter diagram	xo6xx ox6xx&#zq
Elements
Cells	36 tetragonal disphenoids, 72 wedges, 36 rectangular trapezoprisms, 12+12 hexagonal prisms
Faces	144 isosceles triangles, 144 isosceles trapezoids, 72+72 rectangles, 24 hexagons
Edges	72+72+144+144
Vertices	144
Vertex figure	Mirror-symmetric triangular antiprism
Measures (based on two hexagonal-dodecagonal duoprisms of edge length 1)
Edge lengths	Edges of duoprisms (72+72+144): 1
	Lacing edges (144): ${\sqrt {2}}\approx 1.41421$
Circumradius	${\sqrt {3+{\sqrt {3}}}}\approx 2.17533$
Central density	1
Related polytopes
Army	Hiddep
Regiment	Hiddep
Dual	Hexagonal duoexpandotegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	G2≀S2, order 288
Convex	Yes
Nature	Tame

The hexagonal duoexpandoprism or hiddep is a convex isogonal polychoron and the fifth member of the duoexpandoprism family. It consists of 24 hexagonal prisms of two kinds, 36 rectangular trapezoprisms, 72 wedges, and 36 tetragonal disphenoids. Each vertex joins 2 hexagonal prisms, 1 tetragonal disphenoid, 3 wedges, and 2 rectangular trapezoprisms. It can be obtained as the convex hull of two orthogonal hexagonal-dodecagonal duoprisms, or more generally hexagonal-dihexagonal duoprisms, and a subset of its variations can be constructed by expanding the cells of the hexagonal duoprism outward. However, it cannot be made uniform.

This is one of a total of five polychora that can be obtained as the convex hull of two orthogonal hexagonal-dihexagonal duoprisms. To produce variants of this polychoron, if the polychoron is written as ao3bc oa3cb&#zy, c must be in the range $c<b+{\frac {a{\sqrt {3}}}{2}}$. It generally has circumradius ${\sqrt {a^{2}+b^{2}+ab{\sqrt {3}}+c^{2}}}$.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$2+{\sqrt {3}}-{\sqrt {3+2{\sqrt {3}}}}$ ≈ 1:1.18959.

Vertex coordinates

The vertices of a hexagonal duoexpandoprism, constructed as the convex hull of two orthogonal hexagonal-dodecagonal duoprisms of edge length 1, centered at the origin, are given by:

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