Hexagonal-hexagonal prismantiprismoid

Hexagonal-hexagonal prismantiprismoid
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Notation
Bowers style acronym	Hihipap
Coxeter diagram	x6s2s12o ()
Elements
Cells	36 wedges, 12 ditrigonal trapezoprisms, 6 hexagonal prisms, 6 hexagonal antiprisms
Faces	72 isosceles triangles, 72 isosceles trapezoids, 36 rectangles, 12 hexagons, 12 ditrigons
Edges	36+36+72+72
Vertices	72
Vertex figure	Monoaugmented isosceles trapezoidal pyramid
Measures (as derived from unit-edge dodecagonal duoprism)
Edge lengths	Short edges of ditrigons (36): 1
	Side edges (72): $\sqrt2 ≈ 1.41421$
	Edges of hexagons (72): $\frac{\sqrt2+\sqrt6}{2} ≈ 1.93185$
	Long edges of ditrigons (36): $1+\sqrt3 ≈ 2.73205$
Circumradius	$1+\sqrt3 ≈ 2.73205$
Central density	1
Related polytopes
Army	Hihipap
Regiment	Hihipap
Dual	Hexagonal-hexagonal tegmantitegmoid
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(G2×I2(12))/2, order 144
Convex	Yes
Nature	Tame

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:$\frac{7+3\sqrt3+\sqrt{384+174\sqrt3}}{22}$ ≈ 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:

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