Enneagonal-decagrammic duoprism Rank 4 Type Uniform Notation Bowers style acronym E stad edip Coxeter diagram x9o x10/3o ( Elements Cells 10 enneagonal prisms , 9 decagrammic prisms Faces 90 squares , 10 enneagons , 9 decagrams Edges 90+90 Vertices 90 Vertex figure Digonal disphenoid , edge lengths 2cos(π/9) (base 1), √(5–√5 )/2 (base 2), √2 (sides)Measures (edge length 1) Circumradius
3
−
5
2
+
1
4
sin
2
π
9
≈
1.58717
{\displaystyle {\sqrt {{\frac {3-{\sqrt {5}}}{2}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{9}}}}}}\approx 1.58717}
Hypervolume
45
5
−
2
5
8
tan
π
9
≈
11.22840
{\displaystyle 45{\frac {\sqrt {5-2{\sqrt {5}}}}{8\tan {\frac {\pi }{9}}}}\approx 11.22840}
Dichoral angles Stiddip–10/3–stiddip: 140° Ep–4–stiddip: 90° Ep–9–ep: 72° Central density 3 Number of external pieces 29 Level of complexity 12 Related polytopes Army Semi-uniform edidip Regiment Estadedip Dual Enneagonal-decagrammic duotegum Conjugates Enneagonal-decagonal duoprism , Enneagrammic-decagonal duoprism , Enneagrammic-decagrammic duoprism , Great enneagrammic-decagonal duoprism , Great enneagrammic-decagrammic duoprism Abstract & topological properties Flag count2160 Euler characteristic 0 Orientable Yes Properties Symmetry I2 (9)×I2 (10) , order 360Convex No Nature Tame
The enneagonal-decagrammic duoprism , also known as estadedip or the 9-10/3 duoprism , is a uniform duoprism that consists of 10 enneagonal prisms and 9 decagrammic prisms , with 2 of each at each vertex.
The coordinates of a enneagonal-decagrammic duoprism, centered at the origin and with edge length 2sin(π/9), are given by:
(
1
,
0
,
±
sin
π
9
,
±
5
−
2
5
sin
π
9
)
{\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{9}},\,\pm {\sqrt {5-2{\sqrt {5}}}}\sin {\frac {\pi }{9}}\right)}
(
1
,
0
,
±
3
−
5
2
sin
π
9
,
±
5
−
5
2
sin
π
9
)
{\displaystyle \left(1,\,0,\,\pm {\frac {3-{\sqrt {5}}}{2}}\sin {\frac {\pi }{9}},\,\pm {\sqrt {\frac {5-{\sqrt {5}}}{2}}}\sin {\frac {\pi }{9}}\right)}
(
1
,
0
,
±
(
5
−
1
)
sin
π
9
,
0
)
{\displaystyle \left(1,\,0,\,\pm \left({\sqrt {5}}-1\right)\sin {\frac {\pi }{9}},\,0\right)}
(
cos
(
j
π
9
)
,
±
sin
(
j
π
9
)
,
±
sin
π
9
,
±
5
−
2
5
sin
π
9
)
{\displaystyle \left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm \sin {\frac {\pi }{9}},\,\pm {\sqrt {5-2{\sqrt {5}}}}\sin {\frac {\pi }{9}}\right)}
(
cos
(
j
π
9
)
,
±
sin
(
j
π
9
)
,
±
3
−
5
2
sin
π
9
,
±
5
−
5
2
sin
π
9
)
{\displaystyle \left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm {\frac {3-{\sqrt {5}}}{2}}\sin {\frac {\pi }{9}},\,\pm {\sqrt {\frac {5-{\sqrt {5}}}{2}}}\sin {\frac {\pi }{9}}\right)}
(
cos
(
j
π
9
)
,
±
sin
(
j
π
9
)
,
±
(
5
−
1
)
sin
π
9
,
0
)
{\displaystyle \left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm \left({\sqrt {5}}-1\right)\sin {\frac {\pi }{9}},\,0\right)}
(
−
1
2
,
±
3
2
,
±
sin
π
9
,
±
5
−
2
5
sin
π
9
)
{\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{9}},\,\pm {\sqrt {5-2{\sqrt {5}}}}\sin {\frac {\pi }{9}}\right)}
(
−
1
2
,
±
3
2
,
±
3
−
5
2
sin
π
9
,
±
5
−
5
2
sin
π
9
)
{\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {3-{\sqrt {5}}}{2}}\sin {\frac {\pi }{9}},\,\pm {\sqrt {\frac {5-{\sqrt {5}}}{2}}}\sin {\frac {\pi }{9}}\right)}
(
−
1
2
,
±
3
2
,
±
(
5
−
1
)
sin
π
9
,
0
)
{\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \left({\sqrt {5}}-1\right)\sin {\frac {\pi }{9}},\,0\right)}
where j = 2, 4, 8.
An enneagonal-decagrammic duoprism duoprism has the following Coxeter diagrams :
