In geometry, the great rhombidodecacron (or Great dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

of the

great rhombidodecahedron In geometry, the great rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U73. It has 42 faces (30 squares, 12 decagram (geometry), decagrams), 120 edges and 60 vertices. Its vertex figure is a antiparallelogram, crossed quadrilater ...

. It is visually identical to the

great deltoidal hexecontahedron In geometry, the great deltoidal hexecontahedron (or great sagittal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the nonconvex great rhombicosidodecahedron. It is visually identical to the great rhombidodecacron. It h ...

. Its faces are antiparallelograms.

Proportions

Each antiparallelogram has two angles of

\arccos(\frac+\frac\sqrt)\approx 18.699\,407\,085\,15^

and two angles of

\arccos(-\frac+\frac\sqrt)\approx 110.211\,801\,805\,89^

. The diagonals of each antiparallelogram intersect at an angle of

\arccos(\frac+\frac)\approx 51.088\,791\,108\,96^

. The dihedral angle equals

\arccos(\frac)\approx 91.553\,403\,672\,16^

. The ratio between the lengths of the long edges and the short ones equals

\frac+\frac\sqrt

, which is the golden ratio. Part of each face lies inside the solid, hence is invisible in solid models.

References

* p. 88

External links

* Dual uniform polyhedra {{polyhedron-stub