Enneagonal-dodecagonal duoprismatic prism Rank 5 Type Uniform Notation Bowers style acronym Etwip Coxeter diagram x x9o x12o ( ) Elements Tera 12 square-enneagonal duoprisms , 9 square-dodecagonal duoprisms , 2 enneagonal-dodecagonal duoprisms Cells 108 cubes , 9+18 dodecagonal prisms , 12+24 enneagonal prisms Faces 108+108+216 squares , 24 enneagons , 18 dodecagons Edges 108+216+216 Vertices 216 Vertex figure Digonal disphenoidal pyramid , edge lengths 2cos(π/9) (disphenoid base 1), √2+√3 (disphenoid base 2), √2 (remaining edges)Measures (edge length 1) Circumradius ${\frac {\sqrt {9+4{\sqrt {3}}+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 2.47370$ Hypervolume $27{\frac {2+{\sqrt {3}}}{4\tan {\frac {\pi }{9}}}}\approx 69.21265$ Diteral angles Sendip–ep–sendip: 150° Sitwadip–twip–sitwadip: 140° Sitwadip–cube–sendip: 90° Etwadip–ep–sendip: 90° Sitwadip–twip–etwadip: 90° Height 1 Central density 1 Number of external pieces 23 Level of complexity 30 Related polytopes Army Etwip Regiment Etwip Dual Enneagonal-dodecagonal duotegmatic tegum Conjugates Enneagonal-dodecagrammic duoprismatic prism , Enneagrammic-dodecagonal duoprismatic prism , Enneagrammic-dodecagrammic duoprismatic prism , Great enneagrammic-dodecagonal duoprismatic prism , Great enneagrammic-dodecagrammic duoprismatic prism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry I_{2} (9)×I_{2} (12)×A_{1} , order 864Convex Yes Nature Tame

The enneagonal-dodecagonal duoprismatic prism or etwip , also known as the enneagonal-dodecagonal prismatic duoprism , is a convex uniform duoprism that consists of 2 enneagonal-dodecagonal duoprisms , 9 square-dodecagonal duoprisms , and 12 square-enneagonal duoprisms . Each vertex joins 2 square-enneagonal duoprisms, 2 square-dodecagonal duoprisms, and 1 enneagonal-dodecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron .

The vertices of an enneagonal-dodecagonal duoprismatic prism of edge length 2sin(π/9) are given by all permutations of the third and fourth coordinates of:

$\left(0,\,1,\,\pm (1+{\sqrt {3}})\sin {\frac {\pi }{9}},\,\pm (1+{\sqrt {3}})\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
$\left(0,\,1,\,\pm \sin {\frac {\pi }{9}},\,\pm (2+{\sqrt {3}})\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
$\left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm (1+{\sqrt {3}})\sin {\frac {\pi }{9}},\,\pm (1+{\sqrt {3}})\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
$\left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm \sin {\frac {\pi }{9}},\,\pm (2+{\sqrt {3}})\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
$\left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm (1+{\sqrt {3}})\sin {\frac {\pi }{9}},\,\pm (1+{\sqrt {3}})\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
$\left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{9}},\,\pm (2+{\sqrt {3}})\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
where j = 2, 4, 8.

An enneagonal-dodecagonal duoprismatic prism has the following Coxeter diagrams :

x x9o x12o ( ) (full symmetry)
x x6x x9o ( ) (G_{2} ×I_{2} (9)×A_{1} symmetry, dodecagons as dihexagons)
xx9oo xx12oo&#x (enneagonal-dodecagonal duoprism atop enneagonal-dodecagonal duoprism)
xx9oo xx6xx&#x