geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the great dodecicosacron (or great dipteral trisicosahedron) is the dual of the

great dodecicosahedron In geometry, the great dodecicosahedron (or great dodekicosahedron) is a nonconvex uniform polyhedron, indexed as U63. It has 32 faces (20 hexagons and 12 decagrams), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. It ...

(U₆₃). It has 60 intersecting bow-tie-shaped faces.

Proportions

Each face has two angles of

\arccos(\frac+\frac\sqrt)\approx 30.480\,324\,565\,36^

and two angles of

\arccos(-\frac+\frac\sqrt)\approx 81.816\,127\,508\,183^

. The diagonals of each antiparallelogram intersect at an angle of

\arccos(\frac-\frac\sqrt)\approx 67.703\,547\,926\,46^

. The dihedral angle equals

\arccos(\frac)\approx 127.686\,523\,427\,48^

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

\frac+\frac\sqrt

, which is the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

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

References

External links

Uniform polyhedra and duals
Dual uniform polyhedra {{polyhedron-stub