geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, 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 The bow tie is a type of necktie. A modern bow tie is tied using a common shoelace knot, which is also called the bow knot for that reason. It consists of a ribbon of fabric tied around the collar of a shirt in a symmetrical manner so that ...

-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 A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...

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 sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

. 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