# Hexagonal ditetragoltriate

Hexagonal ditetragoltriate
File:Hexagonal ditetragoltriate.png
Rank4
TypeIsogonal
Notation
Bowers style acronymHiddet
Elements
Cells36 rectangular trapezoprisms, 12 hexagonal prisms
Faces72 isosceles trapezoids, 72 rectangless, 12 hexagons
Edges36+72+72
Vertices72
Vertex figureNotch
Measures (based on variant with trapezoids with 3 unit edges)
Edge lengthsEdges 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 density1
Related polytopes
ArmyHiddet
RegimentHiddet
DualHexagonal tetrambitriate
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryG2≀S2, order 288
ConvexYes
NatureTame

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:

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