Triangular ditetragoltriate

Triangular ditetragoltriate
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Triddet
Coxeter diagram	ab3oo ba3oo&#zc
Elements
Cells	6 triangular prisms, 9 rectangular trapezoprisms
Faces	6 triangles, 18 isosceles trapezoids, 18 rectangles
Edges	9+18+18
Vertices	18
Vertex figure	Notch
Measures (based on variant with trapezoids with 3 unit edges)
Edge lengths	Edges of smaller triangle (18): 1
	Lacing edges (9): 1
	Edges of larger triangle (18): ${\frac {2+{\sqrt {6}}}{2}}\approx 2.22475$
Circumradius	${\frac {6+{\sqrt {6}}}{6}}\approx 1.40825$
Central density	1
Related polytopes
Army	Triddet
Regiment	Triddet
Dual	Triangular tetrambitriate
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A2≀S2, order 72
Convex	Yes
Nature	Tame

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 $(1-b){\frac {\sqrt {6}}{3}}$.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {2+{\sqrt {6}}}{2}}$ ≈ 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:

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

External links

• Klitzing, Richard. "triddet".