Hexagonal-square prismantiprismoid

Hexagonal-square prismantiprismoid
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Hispap
Coxeter diagram	x4s2s12o ()
Elements
Cells	24 wedges, 12 rectangular trapezoprisms, 4 hexagonal prisms, 4 hexagonal antiprisms
Faces	48 isosceles triangles, 48 isosceles trapezoids, 12+24 rectangles, 8 hexagons
Edges	24+24+48+48
Vertices	48
Vertex figure	Monoaugmented isosceles trapezoidal pyramid
Measures (as derived from unit-edge octagonal-dodecagonal duoprism)
Edge lengths	Short edges of rectangles (24): 1
	Side edges (48): ${\sqrt {2}}\approx 1.41421$
	Edges of hexagons (48): ${\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\approx 1.93185$
	Long edges of rectangles (24): $1+{\sqrt {2}}\approx 2.41421$
Circumradius	${\sqrt {\frac {6+{\sqrt {2}}+2{\sqrt {3}}}{2}}}\approx 2.33220$
Central density	1
Related polytopes
Army	Hispap
Regiment	Hispap
Dual	Hexagonal-square tegmantitegmoid
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(B2×I2(12))/2, order 96
Convex	Yes
Nature	Tame

The hexagonal-square prismantiprismoid or hispap, also known as the edge-snub hexagonal-square duoprism or 6-4 prismantiprismoid, is a convex isogonal polychoron that consists of 4 hexagonal antiprisms, 4 hexagonal prisms, 12 rectangular trapezoprisms, and 24 wedges. 1 hexagonal antiprism 1 hexagonal 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 octagonal-dodecagonal 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:${\frac {5+2{\sqrt {3}}+{\sqrt {102+46{\sqrt {3}}}}}{13}}$ ≈ 1:1.68790.

Vertex coordinates

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

• $\left(\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}}\right),$
• $\left(\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{6}}\right),$
• $\left(0,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right).$
• $\left(\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right),$