Digonal-square prismantiprismoid

Digonal-square prismantiprismoid
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Dispap
Coxeter diagram	x4s2s4o ()
Elements
Cells	4 tetragonal disphenoids, 8 wedges, 4 rectangular trapezoprisms
Faces	16 isosceles triangles, 16 isosceles trapezoids, 4+4 rectangles
Edges	8+8+8+16
Vertices	16
Vertex figure	Mirror-symmetric notch
Measures (as derived from unit-edge square-octagonal duoprism)
Edge lengths	Short edges of rectangles (8): 1
	Side edges (8+16): ${\sqrt {2}}\approx 1.41421$
	Long edges of rectangles (8): $1+{\sqrt {2}}\approx 2.41421$
Circumradius	${\sqrt {\frac {3+{\sqrt {2}}}{2}}}\approx 1.48563$
Central density	1
Related polytopes
Army	Dispap
Regiment	Dispap
Dual	Digonal-square tegmantitegmoid
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(B2×B2)/2, order 32
Convex	Yes
Nature	Tame

The digonal-square prismantiprismoid or dispap, also known as the edge-snub digonal-square duoprism, 2-4 prismantiprismoid, digonal duoexpandoprism, or digonal duotruncatoalterprism, is a convex isogonal polychoron and the first member of the duoexpandoprism, duotruncatoprism, and duotruncatoalterprism families. It consists of 4 rectangular trapezoprisms, 4 tetragonal disphenoids, and 8 wedges. 1 tetragonal disphenoid, 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 square-octagonal 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.

A variant with regular tetrahedra and squares can be vertex-inscribed into a rectified tesseract.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {1+{\sqrt {3}}}{2}}$ ≈ 1:1.36603.

Vertex coordinates

The vertices of a digonal-square prismantiprismoid, assuming that the tetragonal disphenoids are regular and are connected by squares of edge length 1, centered at the origin, are given by:

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

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by:

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

An additional variant based on a unit square-octagonal duoprism has vertices given by:

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