Octagonal duoprism

Octagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Odip
Coxeter diagram	x8o x8o ()
Elements
Cells	16 octagonal prisms
Faces	64 squares, 16 octagons
Edges	128
Vertices	64
Vertex figure	Tetragonal disphenoid, edge lengths √2+√2 (bases) and √2 (sides)
Measures (edge length 1)
Circumradius	$\sqrt{2+\sqrt2} ≈ 1.84776$
Inradius	$\frac{1+\sqrt2}{2} ≈ 1.20711$
Hypervolume	$4(3+2\sqrt2) ≈ 23.31371$
Dichoral angles	Op–8–op: 135°
	Op–4–op: 90°
Central density	1
Number of external pieces	16
Level of complexity	3
Related polytopes
Army	Odip
Regiment	Odip
Dual	Octagonal duotegum
Conjugate	Octagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(8)≀S2, order 512
Convex	Yes
Nature	Tame

The octagonal duoprism or odip, also known as the octagonal-octagonal duoprism, the 8 duoprism or the 8-8 duoprism, is a noble uniform duoprism that consists of 16 octagonal prisms, with 4 joining at each vertex. It is also the digonal double gyrotrapezohedroid and the 16-7 gyrochoron. It is the first in an infinite family of isogonal octagonal dihedral swirlchora, the first in an infinite family of isochoric octagonal hosohedral swirlchora and also the first in an infinite family of isochoric digonal tegmatic swirlchora.

The octagonal duoprism can be vertex-inscribed into a small rhombated tesseract or small prismatotetracontoctachoron.

This polychoron can be alternated into a square duoantiprism, although it cannot be made uniform. Eight of the octagons can also be alternated into long rectangles to create a square-square prismantiprismoid, which is also nonuniform.

It can form a non-Wythoffian uniform hyperbolic tiling with 288 octagonal duoprisms at each vertex with a bitetracontoctachoron as the vertex figure, called an octagonal duoprismatic tetracomb.

Gallery

• 

  Wireframe, cell, net

Vertex coordinates

Coordinates for the vertices of an octagonal duoprism of edge length 1, centered at the origin, are given by:

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

Representations

An octagonal duoprism has the following Coxeter diagrams:

• x8o x8o (full symmetry)
• x4x x8o (BC2×I2(8) symmetry)
• x4x x4x (BC2×BC2 symmetry, both octagons as ditetragons)
• xwwx xxxx4xxxx&#xt (BC2×A1 axial, octagonal prism-first)

Related polychora

Non-adjacent cells of the octagonal duoprism can be augmented with square pucofastegiums. If 8 cells are augmented in this way, so that all the cupolas blend with the prisms into small rhombicuboctahedra, the result is the uniform small rhombated tesseract.

An octagonal duoprism of edge length 1 contains the vertices of a hexadecachoron of edge length $\sqrt{4+2\sqrt2}$, since the hexadecachoron is the 8-3 step prism.

External links

• Bowers, Jonathan. "Category A: Duoprisms".

• Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".

• Klitzing, Richard. "odip".

• Wikipedia Contributors. "8-8 duoprism".