# Enneagonal duoprism

Enneagonal duoprism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymEdip
Coxeter diagramx9o x9o ()
Elements
Cells18 enneagonal prisms
Faces81 squares, 18 enneagons
Edges162
Vertices81
Vertex figureTetragonal disphenoid, edge lengths 2cos(π/9) (bases) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt2}{2\sin\frac\pi9} ≈ 2.06744}$
Inradius${\displaystyle \frac{1}{2\tan\frac\pi9} ≈ 1.37374}$
Hypervolume${\displaystyle \frac{81}{16\tan^2\frac\pi9} ≈ 38.21495}$
Dichoral anglesEp–9–ep: 140°
Ep–4–ep: 90°
Central density1
Number of external pieces18
Level of complexity3
Related polytopes
ArmyEdip
RegimentEdip
DualEnneagonal duotegum
ConjugatesEnneagrammic duoprism, Great enneagrammic duoprism
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryI2(9)≀S2, order 648
ConvexYes
NatureTame

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.

## Vertex coordinates

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

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

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)).