Hosta

Small rhombidodecacron

Type	Star polyhedron
Face
Elements	F = 60, E = 120 V = 42 (χ = −18)
Symmetry group	I_h, [5,3], *532
Index references	DU₃₉
dual polyhedron	Small rhombidodecahedron

In geometry, the small rhombidodecacron is a nonconvex isohedral polyhedron. It is the dual of the small rhombidodecahedron. It is visually identical to the Small dodecacronic hexecontahedron. It has 60 intersecting antiparallelogram faces.

Proportions

Each face has two angles of $\arccos({\frac {5}{8}}+{\frac {1}{8}}{\sqrt {5}})\approx 25.242\,832\,961\,52^{\circ }$ and two angles of $\arccos(-{\frac {1}{2}}+{\frac {1}{5}}{\sqrt {5}})\approx 93.025\,844\,508\,96^{\circ }$ . The diagonals of each antiparallelogram intersect at an angle of $\arccos({\frac {1}{4}}+{\frac {1}{10}}{\sqrt {5}})\approx 61.731\,322\,529\,52^{\circ }$ . The ratio between the lengths of the long edges and the short ones equals ${\frac {1}{2}}+{\frac {1}{2}}{\sqrt {5}}$ , which is the golden ratio. The dihedral angle equals $\arccos({\frac {-19-8{\sqrt {5}}}{41}})\approx 154.121\,363\,125\,78^{\circ }$ .

References

Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208

Star-polyhedra navigator
Kepler-Poinsot polyhedra (nonconvex regular polyhedra)	small stellated dodecahedron great dodecahedron great stellated dodecahedron great icosahedron
Uniform truncations of Kepler-Poinsot polyhedra	dodecadodecahedron truncated great dodecahedron rhombidodecadodecahedron truncated dodecadodecahedron snub dodecadodecahedron great icosidodecahedron truncated great icosahedron nonconvex great rhombicosidodecahedron great truncated icosidodecahedron
Nonconvex uniform hemipolyhedra	tetrahemihexahedron cubohemioctahedron octahemioctahedron small dodecahemidodecahedron small icosihemidodecahedron great dodecahemidodecahedron great icosihemidodecahedron great dodecahemicosahedron small dodecahemicosahedron
Duals of nonconvex uniform polyhedra	medial rhombic triacontahedron small stellapentakis dodecahedron medial deltoidal hexecontahedron small rhombidodecacron medial pentagonal hexecontahedron medial disdyakis triacontahedron great rhombic triacontahedron great stellapentakis dodecahedron great deltoidal hexecontahedron great disdyakis triacontahedron great pentagonal hexecontahedron
Duals of nonconvex uniform polyhedra with infinite stellations	tetrahemihexacron hexahemioctacron octahemioctacron small dodecahemidodecacron small icosihemidodecacron great dodecahemidodecacron great icosihemidodecacron great dodecahemicosacron small dodecahemicosacron

Hosta

Proportions

References

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