# Digonal-square prismantiprismoid

Digonal-square prismantiprismoid Rank4
TypeIsogonal
SpaceSpherical
Notation
Bowers style acronymDispap
Coxeter diagramx4s2s4o (       )
Elements
Cells4 tetragonal disphenoids, 8 wedges, 4 rectangular trapezoprisms
Faces16 isosceles triangles, 16 isosceles trapezoids, 4+4 rectangles
Edges8+8+8+16
Vertices16
Vertex figureMirror-symmetric notch
Measures (as derived from unit-edge square-octagonal duoprism)
Edge lengthsShort edges of rectangles (8): 1
Side edges (8+16): $\sqrt2 ≈ 1.41421$ Long edges of rectangles (8): $1+\sqrt2 ≈ 2.41421$ Circumradius$\sqrt{\frac{3+\sqrt2}{2}} ≈ 1.48563$ Central density1
Related polytopes
ArmyDispap
RegimentDispap
DualDigonal-square tegmantitegmoid
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
Symmetry(B2×B2)/2, order 32
ConvexYes
NatureTame

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 tetrgonal 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:$\frac{1+\sqrt3}{2}$ ≈ 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:

• $\left(0,\,±\frac12,\,±\frac12,\,±1\right),$ • $\left(±\frac12,\,0,\,±1,\,±\frac12\right).$ An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by:

• $\left(0,\,±\frac12,\,±\frac{\sqrt3-1}{2},\,±\frac12\right),$ • $\left(±\frac12,\,0,\,±\frac12,\,±\frac{\sqrt3-1}{2}\right).$ An additional variant based on a unit square-octagonal duoprism has vertices given by:

• $\left(0,\,±\frac{\sqrt2}{2},\,±\frac12,\,±\frac{1+\sqrt2}{2}\right),$ • $\left(±\frac{\sqrt2}{2},\,0,\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$ 