# Golden rhombus

The golden rhombus.

In geometry, a golden rhombus is a rhombus whose diagonals are in the ratio ${\displaystyle {\frac {p}{q}}=\varphi \!}$, where ${\displaystyle \varphi \!}$ is the golden ratio.

## Elements

The internal angles of the rhombus are

${\displaystyle 2\arctan {\frac {1}{\varphi }}=\arctan {2}\approx 63.43495}$ degrees
${\displaystyle 2\arctan \varphi =\arctan {1}+\arctan {3}\approx 116.56505}$ degrees, which is also the dihedral angle of the dodecahedron

The edge length of the golden rhombus with short diagonal ${\displaystyle q=1}$ is

${\displaystyle {\begin{array}{rcl}e&=&{\tfrac {1}{2}}{\sqrt {p^{2}+q^{2}}}\\&=&{\tfrac {1}{2}}{\sqrt {1+\varphi ^{2}}}\\&=&{\tfrac {1}{4}}{\sqrt {10+2{\sqrt {5}}}}\\&\approx &0.95106\end{array}}}$

A golden rhombus with unit edge length has diagonal lengths

${\displaystyle {\begin{array}{rcl}p&=&{\frac {\varphi }{e}}\\&=&2{\frac {1+{\sqrt {5}}}{\sqrt {10+2{\sqrt {5}}}}}\\&\approx &1.70130\end{array}}}$
${\displaystyle {\begin{array}{rcl}q&=&{\frac {1}{e}}\\&=&4{\frac {1}{\sqrt {10+2{\sqrt {5}}}}}\\&\approx &1.05146\end{array}}}$

## Polyhedra

Several notable polyhedra have golden rhombi as their faces. They include the two golden rhombohedra (with six faces each), the Bilinski dodecahedron (with 12 faces), the rhombic icosahedron (with 20 faces), the rhombic triacontahedron (with 30 faces), and the nonconvex rhombic hexecontahedron (with 60 faces). 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 its faces.[1]