Enneagonal-decagonal duoprism Rank 4 Type Uniform Notation Bowers style acronym Edidip Coxeter diagram x9o x10o ( ) Elements Cells 10 enneagonal prisms , 9 decagonal prisms Faces 90 squares , 10 enneagons , 9 decagons Edges 90+90 Vertices 90 Vertex figure Digonal disphenoid , edge lengths 2cos(π/9) (base 1), √(5+√5 )/2 (base 2), and √2 (sides)Measures (edge length 1) Circumradius
3
+
5
2
+
1
4
sin
2
π
9
≈
2.18064
{\displaystyle {\sqrt {{\frac {3+{\sqrt {5}}}{2}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{9}}}}}}\approx 2.18064}
Hypervolume
45
5
+
2
5
8
tan
π
9
≈
47.56425
{\displaystyle {\frac {45{\sqrt {5+2{\sqrt {5}}}}}{8\tan {\frac {\pi }{9}}}}\approx 47.56425}
Dichoral angles Ep–9–ep: 144° Dip–10–dip: 140° Ep–4–dip: 90° Central density 1 Number of external pieces 19 Level of complexity 6 Related polytopes Army Edidip Regiment Edidip Dual Enneagonal-decagonal duotegum Conjugates Enneagonal-decagrammic duoprism , Enneagrammic-decagonal duoprism , Enneagrammic-decagrammic duoprism , Great enneagrammic-decagonal duoprism , Great enneagrammic-decagrammic duoprism Abstract & topological properties Euler characteristic 0 Orientable Yes Properties Symmetry I2 (9)×I2 (10) , order 360Convex Yes Nature Tame
The enneagonal-decagonal duoprism or edidip , also known as the 9-10 duoprism , is a uniform duoprism that consists of 9 decagonal prisms and 10 enneagonal prisms , with two of each joining at each vertex.
The coordinates of an enneagonal-decagonal 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
,
(
1
+
5
)
sin
π
9
,
0
)
,
{\displaystyle \left(1,0,\left(1+{\sqrt {5}}\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
)
,
(
1
+
5
)
sin
π
9
,
0
)
,
{\displaystyle \left(\cos \left({\frac {j\pi }{9}}\right),\pm \sin \left({\frac {j\pi }{9}}\right),\left(1+{\sqrt {5}}\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
,
(
1
+
5
)
sin
π
9
,
0
)
,
{\displaystyle \left(-{\frac {1}{2}},\pm {\frac {\sqrt {3}}{2}},\left(1+{\sqrt {5}}\right)\sin {\frac {\pi }{9}},0\right),}
where j = 2, 4, 8.
An enneagonal-decagonal duoprism has the following Coxeter diagrams :
x9o x10o (full symmetry)
x5x x9o ( ) (decagons as dipentagons)