Rectified triangular duoprism

Rectified triangular duoprism
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Retdip
Coxeter diagram	xo3ou uo3ox&#zq
Elements
Cells	9 tetragonal disphenoids, 6 rectified triangular prisms
Faces	36 isosceles triangles, 6 triangles, 9 squares
Edges	18+36
Vertices	18
Vertex figure	Wedge
Measures (based on triangles of edge length 1)
Edge lengths	Edges of base triangles (18): 1
	Lacing edges (36): ${\sqrt {2}}\approx 1.41421$
Circumradius	${\frac {\sqrt {15}}{3}}\approx 1.29099$
Central density	1
Related polytopes
Army	Retdip
Regiment	Retedip
Dual	Joined triangular duotegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A2≀S2, order 72
Convex	Yes
Nature	Tame

The rectified triangular duoprism or retdip is a convex isogonal polychoron that consists of 6 rectified triangular prisms and 9 tetragonal disphenoids. 3 rectified triangular prisms and 2 tetragonal disphenoids join at each vertex. It can be formed by rectifying the triangular duoprism.

It can also be formed as the convex hull of 2 oppositely oriented semi-uniform triangular duoprisms, where the edges of one triangle are exactly twice as long as the edges of the other.

The ratio between the longest and shortest edges is 1:${\sqrt {2}}$ ≈ 1:1.41421.

Vertex coordinates

Coordinates for the vertices of a rectified triangular duoprism based on equilateral triangles of edge length 1, centered at the origin, are given by:

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

Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora:

• Rectified triangular prism (6): Triangular duotegum
• Tetragonal disphenoid (9): Triangular duoprism
• Triangle (6): Triangular duotegum
• Square (9): Triangular duoprism
• Isosceles triangle (36): Triangular duoexpandoprism
• Edge (18): Rectified triangular duoprism
• Edge (36): Semi-uniform hexagonal duoprism

External links

• Klitzing, Richard. "retdip".