Hendecagonal-dodecagonal duoprism Rank 4 Type Uniform Notation Bowers style acronym Hen tw ad ip Coxeter diagram x11o x12o ( ) Elements Cells 12 hendecagonal prisms , 11 dodecagonal prisms Faces 132 squares , 12 hendecagons , 11 dodecagons Edges 132+132 Vertices 132 Vertex figure Digonal disphenoid , edge lengths 2cos(π/11) (base 1), (√2 +√6 )/2 (base 2), and √2 (sides)Measures (edge length 1) Circumradius
2
+
3
+
1
4
sin
2
π
11
≈
2.62330
{\displaystyle {\sqrt {2+{\sqrt {3}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{11}}}}}}\approx 2.62330}
Hypervolume
33
(
2
+
3
)
4
tan
π
11
≈
104.85913
{\displaystyle {\frac {33(2+{\sqrt {3}})}{4\tan {\frac {\pi }{11}}}}\approx 104.85913}
Dichoral angle Henp–11–henp: 150° Twip–12–twip:
9
π
11
≈
147.27273
∘
{\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}
Henp–4–twip: 90° Central density 1 Number of external pieces 23 Level of complexity 6 Related polytopes Army Hentwadip Regiment Hentwadip Dual Hendecagonal-dodecagonal duotegum Conjugates Hendecagonal-dodecagrammic duoprism , Small hendecagrammic-dodecagonal duoprism , Small hendecagrammic-dodecagrammic duoprism , Hendecagrammic-dodecagonal duoprism , Hendecagrammic-dodecagrammic duoprism , Great hendecagrammic-dodecagonal duoprism , Great hendecagrammic-dodecagrammic duoprism , Grand hendecagrammic-dodecagonal duoprism , Grand hendecagrammic-dodecagrammic duoprism Abstract & topological properties Flag count3168 Euler characteristic 0 Orientable Yes Properties Symmetry I2 (11)×I2 (12) , order 528Flag orbits 6 Convex Yes Nature Tame
The hendecagonal-dodecagonal duoprism or hentwadip , also known as the 11-12 duoprism , is a uniform duoprism that consists of 11 dodecagonal prisms and 12 hendecagonal prisms , with two of each joining at each vertex.
The coordinates of a hendecagonal-dodecagonal duoprism, centered at the origin and with edge length 2sin(π/11), are given by:
(
1
,
0
,
±
(
1
+
3
)
sin
π
11
,
±
(
1
+
3
)
sin
π
11
)
{\displaystyle \left(1,0,\pm \left(1+{\sqrt {3}}\right)\sin {\frac {\pi }{11}},\pm \left(1+{\sqrt {3}}\right)\sin {\frac {\pi }{11}}\right)}
,
(
1
,
0
,
±
sin
π
11
,
±
(
2
+
3
)
sin
π
11
)
{\displaystyle \left(1,0,\pm \sin {\frac {\pi }{11}},\pm \left(2+{\sqrt {3}}\right)\sin {\frac {\pi }{11}}\right)}
,
(
1
,
0
,
±
(
2
+
3
)
sin
π
11
,
±
sin
π
11
)
{\displaystyle \left(1,0,\pm \left(2+{\sqrt {3}}\right)\sin {\frac {\pi }{11}},\pm \sin {\frac {\pi }{11}}\right)}
,
(
cos
(
j
π
11
)
,
±
sin
(
j
π
11
)
,
±
(
1
+
3
)
sin
π
11
,
±
(
1
+
3
)
sin
π
11
)
{\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\pm \sin \left({\frac {j\pi }{11}}\right),\pm \left(1+{\sqrt {3}}\right)\sin {\frac {\pi }{11}},\pm \left(1+{\sqrt {3}}\right)\sin {\frac {\pi }{11}}\right)}
,
(
cos
(
j
π
11
)
,
±
sin
(
j
π
11
)
,
±
sin
π
11
,
±
(
2
+
3
)
sin
π
11
)
{\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\pm \sin \left({\frac {j\pi }{11}}\right),\pm \sin {\frac {\pi }{11}},\pm \left(2+{\sqrt {3}}\right)\sin {\frac {\pi }{11}}\right)}
,
(
cos
(
j
π
11
)
,
±
sin
(
j
π
11
)
,
±
(
2
+
3
)
sin
π
11
,
±
sin
π
11
)
{\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\pm \sin \left({\frac {j\pi }{11}}\right),\pm \left(2+{\sqrt {3}}\right)\sin {\frac {\pi }{11}},\pm \sin {\frac {\pi }{11}}\right)}
,
where j = 2, 4, 6, 8, 10.
A hendecagonal-dodecagonal duoprism has the following Coxeter diagrams :
x11o x12o ( ) (full symmetry)
x6x x11o ( ) (G2 ×I2 (11) symmetry, dodecagons as dihexagons)