# Triangular-square prismantiprismoid

Triangular-square prismantiprismoid
Rank4
TypeIsogonal
SpaceSpherical
Bowers style acronymTispap
Coxeter diagramx4s2s6o ()
Elements
Vertex figureMonoaugmented rectangular pyramid
Cells12 wedges, 4 triangular prisms, 4 triangular antiprisms, 6 rectangular trapezoprisms
Faces24 isosceles triangles, 8 triangles, 24 isosceles trapezoids, 6+12 rectangles
Edges12+12+24+24
Vertices24
Measures (as derived from unit-edge hexagonal-octagonal duoprism)
Edge lengthsShort edges of rectangles (12): 1
Side edges (24): ${\displaystyle \sqrt2 ≈ 1.41421}$
Edges of triangles (24): ${\displaystyle \sqrt3 ≈ 1.73205}$
Long edges of rectangles (12): ${\displaystyle 1+\sqrt2 ≈ 2.41421}$
Circumradius${\displaystyle \sqrt{\frac{4+\sqrt2}{2}} ≈ 1.64533}$
Central density1
Euler characteristic0
Related polytopes
ArmyTispap
RegimentTispap
DualTriangular-square tegmantitegmoid
Topological properties
OrientableYes
Properties
Symmetry(B2×G2)/2, order 48
ConvexYes
NatureTame

The triangular-square prismantiprismoid or tispap, also known as the edge-snub triangular-square duoprism or 3-4 prismantiprismoid, is a convex isogonal polychoron that consists of 4 triangular antiprisms, 4 triangular prisms, 6 rectangular trapezoprisms, and 12 wedges. 1 triangular antiprism, 1 triangular prism, 2 rectangular trapezoprisms, and 3 wedges join at each vertex. It can be obtained through the process of alternating one class of edges of the hexagonal-octagonal duoprism so that the octagons become rectangles. 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:${\displaystyle \frac{3+2\sqrt6}{5}}$ ≈ 1:1.57980.

## Vertex coordinates

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

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

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

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

Another variant obtained from the uniform hexagonal-octagonal duoprism has coordinates given by:

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