# Triangular ditetragoltriate

Triangular ditetragoltriate
Rank4
TypeIsogonal
Notation
Bowers style acronymTriddet
Coxeter diagramab3oo ba3oo&#zc
Elements
Cells6 triangular prisms, 9 rectangular trapezoprisms
Faces6 triangles, 18 isosceles trapezoids, 18 rectangles
Edges9+18+18
Vertices18
Vertex figureNotch
Measures (based on variant with trapezoids with 3 unit edges)
Edge lengthsEdges of smaller triangle (18): 1
Lacing edges (9): 1
Edges of larger triangle (18): ${\displaystyle {\frac {2+{\sqrt {6}}}{2}}\approx 2.22475}$
Circumradius${\displaystyle {\frac {6+{\sqrt {6}}}{6}}\approx 1.40825}$
Central density1
Related polytopes
ArmyTriddet
RegimentTriddet
DualTriangular tetrambitriate
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryA2≀S2, order 72
ConvexYes
NatureTame

The triangular ditetragoltriate or triddet is a convex isogonal polychoron and the first member of the ditetragoltriate family. It consists of 6 triangular prisms and 9 rectangular trapezoprisms. 2 triangular prisms and 4 rectangular trapezoprisms join at each vertex. However, it cannot be made uniform. It is the first in an infinite family of isogonal triangular prismatic swirlchora.

It can be obtained as the convex hull of 2 similarly oriented semi-uniform triangular duoprisms, one with a larger xy triangle and the other with a larger zw triangle. If the two triangles have edge lengths a and b, the lacing edges have length ${\displaystyle (1-b){\frac {\sqrt {6}}{3}}}$.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\displaystyle {\frac {2+{\sqrt {6}}}{2}}}$ ≈ 1:2.22474. This value is also the ratio between the two sides of the two semi-uniform duoprisms.

## Vertex coordinates

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

• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,0,\,{\frac {3{\sqrt {2}}+2{\sqrt {3}}}{6}}\right),}$
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,\pm {\frac {2+{\sqrt {6}}}{4}},\,-{\frac {3{\sqrt {2}}+2{\sqrt {3}}}{12}}\right),}$
• ${\displaystyle \left(0,\,{\frac {3{\sqrt {2}}+2{\sqrt {3}}}{6}},\,0,\,{\frac {\sqrt {3}}{3}}\right),}$
• ${\displaystyle \left(0,\,{\frac {3{\sqrt {2}}+2{\sqrt {3}}}{6}},\,\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,0,\,{\frac {3{\sqrt {2}}+2{\sqrt {3}}}{6}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}}\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {2+{\sqrt {6}}}{4}},\,-{\frac {3{\sqrt {2}}+2{\sqrt {3}}}{12}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {6}}}{4}},\,-{\frac {3{\sqrt {2}}+2{\sqrt {3}}}{12}},\,0,\,{\frac {\sqrt {3}}{3}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {6}}}{4}},\,-{\frac {3{\sqrt {2}}+2{\sqrt {3}}}{12}},\,\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}}\right).}$