# Enneagonal-truncated cubic duoprism

Enneagonal-truncated cubic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEtic
Coxeter diagramx9o x4x3o ()
Elements
Tera8 triangular-enneagonal duoprisms, 9 truncated cubic prisms, 6 octagonal-enneagonal duoprisms
Cells72 triangular prisms, 54 octagonal prisms, 12+24 enneagonal prisms, 9 truncated cubes
Faces72 triangles, 108+216 squares, 54 octagons, 24 enneagons
Edges108+216+216
Vertices216
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, 2+2, 2+2 (base triangle), cos(π/9) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7+4{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 2.30247}$
Hypervolume${\displaystyle {\frac {21(3+2{\sqrt {2}})}{4\tan {\frac {\pi }{9}}}}\approx 84.07073}$
Diteral anglesTiccup–tic–ticcup: 140°
Tedip–ep–oedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Tedip–trip–ticcup: 90°
Oedip–op–ticcup: 90°
Oedip–ep–oedip: 90°
Central density1
Number of external pieces23
Level of complexity30
Related polytopes
ArmyEtic
RegimentEtic
DualEnneagonal-triakis octahedral duotegum
ConjugatesEnneagrammic-truncated cubic duoprism, Great enneagrammic-truncated cubic duoprism, Enneagonal-quasitruncated hexahedral duoprism, Enneagrammic-quasitruncated hexahedral duoprism, Great enneagrammic-quasitruncated hexahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(9), order 864
ConvexYes
NatureTame

The enneagonal-truncated cubic duoprism or etic is a convex uniform duoprism that consists of 9 truncated cubic prisms, 6 octagonal-enneagonal duoprisms, and 8 triangular-enneagonal duoprisms. Each vertex joins 2 truncated cubic prisms, 1 triangular-enneagonal duoprism, and 2 octagonal-enneagonal duoprisms.

## Vertex coordinates

The vertices of an enneagonal-truncated cubic duoprism of edge length 2sin(π/9) are given by all permutations of the last three coordinates of:

• ${\displaystyle \left(1,\,0,\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{9}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{9}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{9}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),}$

where j = 2, 4, 8.