Golden rhombus


In geometry, a golden rhombus is a rhombus whose diagonals are in the golden ratio:
Equivalently, it is the Varignon parallelogram formed from the edge midpoints of a golden rectangle.
Rhombi with this shape form the faces of several notable polyhedra.
The golden rhombus should be distinguished from the two rhombi of the Penrose tiling, which are both related in other ways to the golden ratio but have different shapes than the golden rhombus.

Angles

The internal supplementary angles of the golden rhombus are:
  • Acute angle: ;
  • Obtuse angle:

    Edge and diagonals

By using the parallelogram law :
The edge length of the golden rhombus in terms of the diagonal length is:
  • Hence:
The diagonal lengths of the golden rhombus in terms of the edge length are:

Area

  • By using the area formula of the general rhombus in terms of its diagonal lengths and :
  • By using the area formula of the general rhombus in terms of its edge length :
Note: , hence:

As the faces of polyhedra

Several notable polyhedra have golden rhombi as their faces.
They include the two golden rhombohedra, the Bilinski dodecahedron,
the rhombic icosahedron,
the rhombic triacontahedron, and
the nonconvex rhombic hexecontahedron. The first five of these are the only convex polyhedra with golden rhomb faces, but there exist infinitely many nonconvex polyhedra having this shape for all of their faces.