Square-square prismantiprismoid

Square-square prismantiprismoid
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Sispap
Coxeter diagram	x4s2s8o ()
Elements
Cells	16 wedges, 8 rectangular trapezoprisms, 4 square prisms, 4 square antiprisms
Faces	32 isosceles triangles, 32 isosceles trapezoids, 8+16 rectangles, 8 squares
Edges	16+16+32+32
Vertices	32
Vertex figure	Monoaugmented isosceles trapezoidal pyramid
Measures (as derived from unit-edge octagonal duoprism)
Edge lengths	Short edges of rectangles (16): 1
	Side edges (32): ${\sqrt {2}}\approx 1.41421$
	Edges of squares (32): ${\sqrt {2+{\sqrt {2}}}}\approx 1.84776$
	Long edges of rectangles (16): $1+{\sqrt {2}}\approx 2.41421$
Circumradius	${\sqrt {2+{\sqrt {2}}}}\approx 1.84776$
Central density	1
Related polytopes
Army	Sispap
Regiment	Sispap
Dual	Square-square tegmantitegmoid
Abstract & topological properties
Flag count	1408
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(B2×I2(8))/2, order 64
Convex	Yes
Nature	Tame

The square-square prismantiprismoid or sispap, also known as the edge-snub square-square duoprism or 4-4 prismantiprismoid, is a convex isogonal polychoron that consists of 4 square antiprisms, 4 square prisms, 8 rectangular trapezoprisms, and 16 wedges. 1 square prism, 1 square antiprism, 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 duoprism so that one ring of 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:${\frac {3+{\sqrt {2}}+{\sqrt {32+13{\sqrt {2}}}}}{7}}$ ≈ 1:1.64463.

Vertex coordinates

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

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