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 rhombic triacontahedron is a nonconvex

isohedral In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...

, isotoxal

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

. It is the dual of the

great icosidodecahedron In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r. It is the rectification of the great stell ...

(U54). Like the convex

rhombic triacontahedron In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Cata ...

it has 30 rhombic faces, 60 edges and 32 vertices (also 20 on 3-fold and 12 on 5-fold axes). It can be constructed from the convex solid by expanding the faces by factor of

\varphi^3 \approx 4.236

, where

\varphi\!

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 ( ...

. This solid is to the

compound of great icosahedron and great stellated dodecahedron There are two different compounds of great icosahedron and great stellated dodecahedron: one is a dual compound and a stellation of the great icosidodecahedron, the other is a stellation of the icosidodecahedron. Dual compound It can be seen as ...

what the convex one is to the

compound of dodecahedron and icosahedron In geometry, this polyhedron can be seen as either a polyhedral stellation or a compound. As a compound It can be seen as the compound of an icosahedron and dodecahedron. It is one of four compounds constructed from a Platonic solid or Ke ...

: The crossing edges in the

dual compound In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connected ...

are the diagonals of the rhombs. What resembles an "excavated" rhombic triacontahedron (compare

excavated dodecahedron In geometry, the excavated dodecahedron is a star polyhedron that looks like a dodecahedron with concave pentagonal pyramids in place of its faces. Its exterior surface represents the Ef1g1 stellation of the icosahedron. It appears in Magnus We ...

and excavated icosahedron) can be seen within the middle of this compound. The rest of the polyhedron strikingly resembles a

rhombic hexecontahedron In geometry, a rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry. It was described mathematically in 1940 by Helmut Unkelbach. It is topologically ident ...

. The rhombs have two angles of

\arccos(\frac\sqrt)\approx 63.434\,948\,822\,92^

, and two of

\arccos(-\frac\sqrt)\approx 116.565\,051\,177\,08^

. Its

dihedral angles 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 ...

equal

\arccos(-\frac+\frac\sqrt)= 72^

. Part of each rhomb lies inside the solid, hence is invisible in solid models. The ratio between the lengths of the long and short diagonal of the rhombs equals the golden ratio

\varphi

References

External links

* * David I. McCooey
animation and measurements

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