Enneagonal duoprism

Enneagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Edip
Coxeter diagram	x9o x9o ()
Elements
Cells	18 enneagonal prisms
Faces	81 squares, 18 enneagons
Edges	162
Vertices	81
Vertex figure	Tetragonal disphenoid, edge lengths 2cos(π/9) (bases) and √2 (sides)
Measures (edge length 1)
Circumradius	$\frac{\sqrt2}{2\sin\frac\pi9} ≈ 2.06744$
Inradius	$\frac{1}{2\tan\frac\pi9} ≈ 1.37374$
Hypervolume	$\frac{81}{16\tan^2\frac\pi9} ≈ 38.21495$
Dichoral angles	Ep–9–ep: 140°
	Ep–4–ep: 90°
Central density	1
Number of external pieces	18
Level of complexity	3
Related polytopes
Army	Edip
Regiment	Edip
Dual	Enneagonal duotegum
Conjugates	Enneagrammic duoprism, Great enneagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(9)≀S2, order 648
Convex	Yes
Nature	Tame

The enneagonal duoprism or edip, also known as the enneagonal-enneagonal duoprism, the 9 duoprism or the 9-9 duoprism, is a noble uniform duoprism that consists of 18 enneagonal prisms, with 4 joining at each vertex. It is also the 18-8 gyrochoron. It is the first in an infinite family of isogonal enneagonal dihedral swirlchora and also the first in an infinite family of isochoric enneagonal hosohedral swirlchora.

This polychoron can be subsymmetrically faceted into a triangular triswirlprism, although it cannot be made uniform.

Gallery

• 

  Wireframe, cell, net

Vertex coordinates

The coordinates of an enneagonal duoprism, centered at the origin and with edge length 2sin(π/9), are given by:

• $\left(1,0,1,0\right),$
• $\left(1,0,\cos\left(\frac{j\pi}9\right),±\sin\left(\frac{j\pi}9\right)\right),$
• $\left(1,0,-\frac12,±\frac{\sqrt3}2\right),$
• $\left(\cos\left(\frac{k\pi}9\right),±\sin\left(\frac{k\pi}9\right),1,0\right),$
• $\left(\cos\left(\frac{k\pi}9\right),±\sin\left(\frac{k\pi}9\right),\cos\left(\frac{j\pi}9\right),±\sin\left(\frac{j\pi}9\right)\right),$
• $\left(\cos\left(\frac{k\pi}9\right),±\sin\left(\frac{k\pi}9\right),-\frac12,±\frac{\sqrt3}2\right),$
• $\left(-\frac12,±\frac{\sqrt3}2,1,0\right),$
• $\left(-\frac12,±\frac{\sqrt3}2,\cos\left(\frac{j\pi}9\right),±\sin\left(\frac{j\pi}9\right)\right),$
• $\left(-\frac12,±\frac{\sqrt3}2,-\frac12,±\frac{\sqrt3}2\right),$

where j, k = 2, 4, 8.

• (1, 0, 1, 0),
• (1, 0, cos(2π/9), ±sin(2π/9)),
• (1, 0, cos(4π/9), ±sin(4π/9)),
• (1, 0, –1/2, ±√3/2),
• (1, 0, cos(8π/9), ±sin(8π/9)),
• (cos(2π/9), ±sin(2π/9), 1, 0),
• (cos(2π/9), ±sin(2π/9), cos(2π/9), ±sin(2π/9)),
• (cos(2π/9), ±sin(2π/9), cos(4π/9), ±sin(4π/9)),
• (cos(2π/9), ±sin(2π/9), –1/2, ±√3/2),
• (cos(2π/9), ±sin(2π/9), cos(8π/9), ±sin(8π/9)),
• (cos(4π/9), ±sin(4π/9), 1, 0),
• (cos(4π/9), ±sin(4π/9), cos(2π/9), ±sin(2π/9)),
• (cos(4π/9), ±sin(4π/9), cos(4π/9), ±sin(4π/9)),
• (cos(4π/9), ±sin(4π/9), –1/2, ±√3/2),
• (cos(4π/9), ±sin(4π/9), cos(8π/9), ±sin(8π/9)),
• (–1/2, ±√3/2, 1, 0),
• (–1/2, ±√3/2, cos(2π/9), ±sin(2π/9)),
• (–1/2, ±√3/2, cos(4π/9), ±sin(4π/9)),
• (–1/2, ±√3/2, –1/2, ±√3/2),
• (–1/2, ±√3/2, cos(8π/9), ±sin(8π/9)),
• (cos(8π/9), ±sin(8π/9), 1, 0),
• (cos(8π/9), ±sin(8π/9), cos(2π/9), ±sin(2π/9)),
• (cos(8π/9), ±sin(8π/9), cos(4π/9), ±sin(4π/9)),
• (cos(8π/9), ±sin(8π/9), –1/2, ±√3/2),
• (cos(8π/9), ±sin(8π/9), cos(8π/9), ±sin(8π/9)).

External links

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

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

• Klitzing, Richard. "n-n-dip".