# Triangular-hexagonal prismantiprismoid

Triangular-hexagonal prismantiprismoid
Rank4
TypeIsogonal
SpaceSpherical
Bowers style acronymThipap
Coxeter diagramx6s2s6o ()
Elements
Vertex figureMonoaugmented rectangular pyramid
Cells18 wedges, 6 triangular prisms, 6 triangular antiprisms, 6 ditrigonal trapezoprisms
Faces36 isosceles triangles, 12 triangles, 36 isosceles trapezoids, 18 rectangles, 6 ditrigons
Edges18+18+36+36
Vertices36
Measures (as derived from unit-edge hexagonal-dodecagonal duoprism)
Edge lengthsShort edges of ditrigons (18): 1
Side edges (36): ${\displaystyle \sqrt2 ≈ 1.41421}$
Edges of triangles (36): ${\displaystyle \sqrt3 ≈ 1.73205}$
Long edges of ditrigons (12): ${\displaystyle 1+\sqrt3 ≈ 2.73205}$
Circumradius${\displaystyle \sqrt{3+\sqrt3} ≈ 2.17533}$
Central density1
Euler characteristic0
Related polytopes
ArmyThipap
RegimentThipap
DualTriangular-hexagonal tegmantitegmoid
Topological properties
OrientableYes
Properties
Symmetry(G2×G2)/2, order 72
ConvexYes
NatureTame

The triangular-hexagonal prismantiprismoid or thipap, also known as the edge-snub triangular-hexagonal duoprism or 3-6 prismantiprismoid, is a convex isogonal polychoron that consists of 6 ditrigonal trapezoprisms, 6 triangular antiprisms, 6 triangular prisms, and 18 wedges. 1 triangular prism, 1 triangular antiprism, 2 ditrigonal trapezoprisms, and 3 wedges join at each vertex. It can be obtained through the process of alternating one class of edges of the hexagonal-dodecagonal duoprism so that the dodecagons become ditrigons. However, it cannot be made uniform, as it generally has 4 edge lengths, which can be minimized to no fewer than 2 different sizes.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\displaystyle \frac{1+\sqrt5}{2}}$ ≈ 1:1.61803.

## Vertex coordinates

The vertices of a triangular-hexagonal prismantiprismoid, assuming that the triangular antiprisms and triangular prisms are uniform of edge length 1, centered at the origin, are given by:

• ${\displaystyle ±\left(0,\,\frac{\sqrt3}{3},\,±\frac12,\,\frac{3\sqrt3+2\sqrt6}{6}\right),}$
• ${\displaystyle ±\left(0,\,\frac{\sqrt3}{3},\,±\frac{2+\sqrt2}{2},\,-\frac{\sqrt6}{6}\right),}$
• ${\displaystyle ±\left(0,\,\frac{\sqrt3}{3},\,±\frac{1+\sqrt2}{2},\,-\frac{3\sqrt3+\sqrt6}{6}\right),}$
• ${\displaystyle ±\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12,\,\frac{3\sqrt3+2\sqrt6}{6}\right),}$
• ${\displaystyle ±\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac{2+\sqrt2}{2},\,-\frac{\sqrt6}{6}\right),}$
• ${\displaystyle ±\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac{1+\sqrt2}{2},\,-\frac{3\sqrt3+\sqrt6}{6}\right).}$

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by:

• ${\displaystyle ±\left(±\frac{\sqrt5-1}{4},\,\frac{3\sqrt3+\sqrt{15}}{12},\,0,\,\frac{\sqrt3}{3}\right),}$
• ${\displaystyle ±\left(±\frac{\sqrt5-1}{4},\,\frac{3\sqrt3+\sqrt{15}}{12},\,±\frac12,\,-\frac{\sqrt3}{6}\right),}$
• ${\displaystyle ±\left(±\frac{1+\sqrt5}{4},\,-\frac{3\sqrt3-\sqrt{15}}{12},\,0,\,\frac{\sqrt3}{3}\right),}$
• ${\displaystyle ±\left(±\frac{1+\sqrt5}{4},\,-\frac{3\sqrt3-\sqrt{15}}{12},\,±\frac12,\,-\frac{\sqrt3}{6}\right),}$
• ${\displaystyle ±\left(±\frac12,\,-\frac{\sqrt{15}}{6},\,0,\,\frac{\sqrt3}{3}\right),}$
• ${\displaystyle ±\left(±\frac12,\,-\frac{\sqrt{15}}{6},\,±\frac12,\,-\frac{\sqrt3}{6}\right).}$

A variant based on a uniform hexagonal-dodecagonal duoprism has vertices given by:

• ${\displaystyle ±\left(0,\,1,\,±\frac12,\,\frac{2+\sqrt3}{2}\right),}$
• ${\displaystyle ±\left(0,\,1,\,±\frac{2+\sqrt3}{2},\,-\frac12\right),}$
• ${\displaystyle ±\left(0,\,1,\,±\frac{1+\sqrt3}{2},\,-\frac{1+\sqrt3}{2}\right),}$
• ${\displaystyle ±\left(±\frac{\sqrt3}{2},\,-\frac12,\,±\frac12,\,\frac{2+\sqrt3}{2}\right),}$
• ${\displaystyle ±\left(±\frac{\sqrt3}{2},\,-\frac12,\,±\frac{2+\sqrt3}{2},\,-\frac12\right),}$
• ${\displaystyle ±\left(±\frac{\sqrt3}{2},\,-\frac12,\,±\frac{1+\sqrt3}{2},\,-\frac{1+\sqrt3}{2}\right).}$