Small ditrigonal dodecacronic hexecontahedron

Polyhedron with 60 faces
Small ditrigonal dodecacronic hexecontahedron
Type Star polyhedron
Face
Elements F = 60, E = 120
V = 44 (χ = −16)
Symmetry group Ih, [5,3], *532
Index references DU43
dual polyhedron Small ditrigonal dodecicosidodecahedron
3D model of a small ditrigonal dodecacronic hexecontahedron

In geometry, the small ditrigonal dodecacronic hexecontahedron (or fat star) is a nonconvex isohedral polyhedron. It is the dual of the uniform small ditrigonal dodecicosidodecahedron. It is visually identical to the small dodecicosacron. Its faces are darts. A part of each dart lies inside the solid, hence is invisible in solid models.

Proportions

Faces have two angles of arccos ⁡ ( 5 12 + 1 4 5 ) ≈ 12.661 078 804 43 ∘ {\displaystyle \arccos({\frac {5}{12}}+{\frac {1}{4}}{\sqrt {5}})\approx 12.661\,078\,804\,43^{\circ }} , one of arccos ⁡ ( − 5 12 − 1 60 5 ) ≈ 116.996 396 851 70 ∘ {\displaystyle \arccos(-{\frac {5}{12}}-{\frac {1}{60}}{\sqrt {5}})\approx 116.996\,396\,851\,70^{\circ }} and one of 360 ∘ − arccos ⁡ ( − 1 12 − 19 60 5 ) ≈ 217.681 445 539 45 ∘ {\displaystyle 360^{\circ }-\arccos(-{\frac {1}{12}}-{\frac {19}{60}}{\sqrt {5}})\approx 217.681\,445\,539\,45^{\circ }} . Its dihedral angles equal arccos ⁡ ( − 44 − 3 5 61 ) ≈ 146.230 659 755 53 ∘ {\displaystyle \arccos({\frac {-44-3{\sqrt {5}}}{61}})\approx 146.230\,659\,755\,53^{\circ }} . The ratio between the lengths of the long and short edges is 31 + 5 5 38 ≈ 1.110 008 944 41 {\displaystyle {\frac {31+5{\sqrt {5}}}{38}}\approx 1.110\,008\,944\,41} .

References

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

External links

  • Weisstein, Eric W. "Small ditrigonal dodecacronic hexecontahedron". MathWorld.
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