# Golden rhombus

The golden rhombus.

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

## Golden rhombohedra

There are 2 convex golden rhombohedra, one constructed from six golden rhombi as a trigonal trapezohedron, a cube that has been stretched along one of its diagonal axes. This is also called the acute golden rhombohedron.

The other is the rhombic triacontahedron, constructed with 30 golden rhombic faces, alternating 3 and 5 around every vertex. The dihedral angle between adjacent rhombi of the rhombic triacontahedron is 144°, which can be constructed by placing the short sides of two golden triangles back-to-back.

The nonconvex rhombic hexecontahedron can be constructed by 20 acute golden rhombohedron. It also represents a stellation of the rhombic triacontahedron.

## Element

The internal angles of the rhombus are

$2\arctan\frac{1}{\varphi}\approx63.43495$ degrees
$2\arctan\varphi\approx116.56505$ degrees

The edge length of the golden rhombus with short diagonal $q=1$ is

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

A golden rhombus with unit edge length has diagonal lengths

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