Triangular duoexpandoprism

Triangular duoexpandoprism
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Triddep
Coxeter diagram	xo3xx ox3xx&#zy
Elements
Cells	9 tetragonal disphenoids, 18 wedges, 6+6 triangular prisms, 9 rectangular trapezoprisms
Faces	36 isosceles triangles, 12 triangles, 36 isosceles trapezoids, 18+18 rectangles
Edges	18+18+36+36
Vertices	36
Vertex figure	Mirror-symmetric triangular antiprism
Measures (based on two triangular-hexagonal duoprisms of edge length 1)
Edge lengths	Lacing edges (36): ${\frac {\sqrt {6}}{3}}\approx 0.81650$
	Edges of duoprisms (18+18+36): 1
Circumradius	${\frac {2{\sqrt {3}}}{3}}\approx 1.15470$
Central density	1
Related polytopes
Army	Triddep
Regiment	Triddep
Dual	Triangular duoexpandotegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A2≀S2, order 72
Convex	Yes
Nature	Tame

Different view

The triangular duoexpandoprism or triddep is a convex isogonal polychoron and the second member of the duoexpandoprism family. It consists of 12 triangular prisms of two kinds, 9 rectangular trapezoprisms, 18 wedges, and 9 tetragonal disphenoids. Each vertex joins 2 triangular prisms, 1 tetragonal disphenoid, 3 wedges, and 2 rectangular trapezoprisms. It can be obtained as the convex hull of two orthogonal triangular-hexagonal duoprisms, or more generally triangular-ditrigonal duoprisms, and a subset of its variations can be constructed by expanding the cells of the triangular 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 triangular-ditrigonal duoprisms. To produce variants of this polychoron, if the polychoron is written as ao3bc oa3cb&#zy, c must be in the range $c<b+{\frac {a}{2}}$. It generally has circumradius ${\sqrt {\frac {a^{2}+b^{2}+ab+c^{2}}{3}}}$.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {3+{\sqrt {15}}}{6}}$ ≈ 1:1.14550.

Vertex coordinates

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

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

External links

• Klitzing, Richard. "triddep".