Hexagonal ditetragoltriate

Hexagonal ditetragoltriate
File:Hexagonal ditetragoltriate.png (OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Hiddet
Elements
Cells	36 rectangular trapezoprisms, 12 hexagonal prisms
Faces	72 isosceles trapezoids, 72 rectangless, 12 hexagons
Edges	36+72+72
Vertices	72
Vertex figure	Notch
Measures (based on variant with trapezoids with 3 unit edges)
Edge lengths	Edges of smaller hexagon (72): 1
	Lacing edges (36): 1
	Edges of larger hexagon (72): ${\frac {2+{\sqrt {2}}}{2}}\approx 1.70711$
Circumradius	${\sqrt {\frac {5+2{\sqrt {2}}}{2}}}\approx 1.97844$
Central density	1
Related polytopes
Army	Hiddet
Regiment	Hiddet
Dual	Hexagonal tetrambitriate
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	G2≀S2, order 288
Convex	Yes
Nature	Tame

The hexagonal ditetragoltriate or hiddet is a convex isogonal polychoron and the fourth member of the ditetragoltriate family. It consists of 12 hexagonal prisms and 36 rectangular trapezoprisms. 2 hexagonal prisms and 4 rectangular trapeozpirsms join at each vertex. However, it cannot be made uniform. It is the first in an infinite family of isogonal hexagonal prismatic swirlchora.

This polychoron can be alternated into a triangular double antiprismoid, which is also nonuniform.

It can be obtained as the convex hull of 2 similarly oriented semi-uniform hexagonal duoprisms, one with a larger xy hexagon and the other with a larger zw hexagon.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {2+{\sqrt {2}}}{2}}$ ≈ 1:1.70711. This value is also the ratio between the two sides of the two semi-uniform duoprisms.

Vertex coordinates

The vertices of a hexagonal ditetragoltriate, assuming that the trapezoids have three equal edges of length 1, centered at the origin, are given by:

• $\left(0,\,\pm 1,\,0,\,\pm {\frac {2+{\sqrt {2}}}{2}}\right),$
• $\left(0,\,\pm 1,\,\pm {\frac {2{\sqrt {3}}+{\sqrt {6}}}{4}},\,\pm {\frac {2+{\sqrt {2}}}{4}}\right),$
• $\left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {2+{\sqrt {2}}}{2}}\right),$
• $\left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2{\sqrt {3}}+{\sqrt {6}}}{4}},\,\pm {\frac {2+{\sqrt {2}}}{4}}\right),$
• $\left(0,\,\pm {\frac {2+{\sqrt {2}}}{2}},\,0,\,\pm 1\right),$
• $\left(0,\,\pm {\frac {2+{\sqrt {2}}}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {2{\sqrt {3}}+{\sqrt {6}}}{4}},\,\pm {\frac {2+{\sqrt {2}}}{4}},\,0,\,\pm 1\right),$
• $\left(\pm {\frac {2{\sqrt {3}}+{\sqrt {6}}}{4}},\,\pm {\frac {2+{\sqrt {2}}}{4}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right).$