# Hexagonal-hexagonal prismantiprismoid

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Hexagonal-hexagonal prismantiprismoid
Rank4
TypeIsogonal
SpaceSpherical
Notation
Bowers style acronymHihipap
Coxeter diagramx6s2s12o ()
Elements
Cells36 wedges, 12 ditrigonal trapezoprisms, 6 hexagonal prisms, 6 hexagonal antiprisms
Faces72 isosceles triangles, 72 isosceles trapezoids, 36 rectangles, 12 hexagons, 12 ditrigons
Edges36+36+72+72
Vertices72
Vertex figureMonoaugmented isosceles trapezoidal pyramid
Measures (as derived from unit-edge dodecagonal duoprism)
Edge lengthsShort edges of ditrigons (36): 1
Side edges (72): ${\displaystyle \sqrt2 ≈ 1.41421}$
Edges of hexagons (72): ${\displaystyle \frac{\sqrt2+\sqrt6}{2} ≈ 1.93185}$
Long edges of ditrigons (36): ${\displaystyle 1+\sqrt3 ≈ 2.73205}$
Circumradius${\displaystyle 1+\sqrt3 ≈ 2.73205}$
Central density1
Related polytopes
ArmyHihipap
RegimentHihipap
DualHexagonal-hexagonal tegmantitegmoid
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
Symmetry(G2×I2(12))/2, order 144
ConvexYes
NatureTame

The hexagonal-hexagonal prismantiprismoid or hihipap, also known as the edge-snub hexagonal-hexagonal duoprism or 6-6 prismantiprismoid, is a convex isogonal polychoron that consists of 6 hexagonal antiprisms, 6 hexagonal prisms, 12 ditrigonal trapezoprisms, and 36 wedges. 1 hexagonal antiprism, 1 hexagonal prism, 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 dodecagonal duoprism so that one ring of 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:${\displaystyle \frac{7+3\sqrt3+\sqrt{384+174\sqrt3}}{22}}$ ≈ 1:1.74436.

## Vertex coordinates

The vertices of a hexagonal-hexagonal prismantiprismoid based on a dodecagonal duoprism of edge length 1, centered at the origin, are given by:

• ${\displaystyle \left(0,\,±\frac{\sqrt2+\sqrt6}{2},\,±\frac12,\,\frac{2+\sqrt3}{2}\right),}$
• ${\displaystyle \left(0,\,±\frac{\sqrt2+\sqrt6}{2},\,±\frac{2+\sqrt3}{2},\,-\frac12\right),}$
• ${\displaystyle \left(0,\,±\frac{\sqrt2+\sqrt6}{2},\,±\frac{1+\sqrt3}{2},\,-\frac{1+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\frac{\sqrt2+\sqrt6}{2},\,0,\,±\frac12,\,-\frac{2+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\frac{\sqrt2+\sqrt6}{2},\,0,\,±\frac{2+\sqrt3}{2},\,\frac12\right),}$
• ${\displaystyle \left(±\frac{\sqrt2+\sqrt6}{2},\,0,\,±\frac{1+\sqrt3}{2},\,\frac{1+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{\sqrt2+\sqrt6}{4},\,±\frac12,\,\frac{2+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{\sqrt2+\sqrt6}{4},\,±\frac{2+\sqrt3}{2},\,-\frac12\right),}$
• ${\displaystyle \left(±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{\sqrt2+\sqrt6}{4},\,±\frac{1+\sqrt3}{2},\,-\frac{1+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\frac{\sqrt2+\sqrt6}{4},\,±\frac{3\sqrt2+\sqrt6}{4},\,±\frac12,\,-\frac{2+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\frac{\sqrt2+\sqrt6}{4},\,±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{2+\sqrt3}{2},\,\frac12\right),}$
• ${\displaystyle \left(±\frac{\sqrt2+\sqrt6}{4},\,±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{1+\sqrt3}{2},\,\frac{1+\sqrt3}{2}\right).}$