Square-hexagonal duoantiprism

Square-hexagonal duoantiprism
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Shiddap
Coxeter diagram	s8o2s12o ()
Elements
Cells	48 digonal disphenoids, 12 square antiprisms, 8 hexagonal antiprisms
Faces	96+96 isosceles triangles, 12 squares, 8 hexagons
Edges	48+48+96
Vertices	48
Vertex figure	Gyrobifastigium
Measures (based on polygons of edge length 1)
Edge lengths	Lacing (96): ${\sqrt {\frac {6-{\sqrt {2}}-2{\sqrt {3}}}{2}}}\approx 0.74889$
	Edges of squares (48): 1
	Edges of hexagons (48): 1
Circumradius	${\frac {\sqrt {6}}{2}}\approx 1.22475$
Central density	1
Related polytopes
Army	Shiddap
Regiment	Shiddap
Dual	Square-hexagonal duoantitegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(I2(8)×I2(12))/2, order 192
Convex	Yes
Nature	Tame

The square-hexagonal duoantiprism or shiddap, also known as the 4-6 duoantiprism, is a convex isogonal polychoron that consists of 8 hexagonal antiprisms, 12 square antiprisms, and 48 digonal disphenoids. 2 hexagonal antiprisms, 2 square antiprisms, and 4 digonal disphenoids join at each vertex. It can be obtained through the process of alternating the octagonal-dodecagonal duoprism. However, it cannot be made uniform, as it generally has 3 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:${\sqrt {\frac {66+26{\sqrt {3}}+{\sqrt {1922+1104{\sqrt {3}}}}}{97}}}$ ≈ 1:1.33530.

Vertex coordinates

The vertices of a square-hexagonal duoantiprism based on squares and hexagons of edge length 1, centered at the origin, are given by:

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