# 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 two distinct convex golden rhombohedra constructed from six golden rhombi as a trigonal trapezohedron. These are sometimes called acute or prolate and the obtuse or oblate golden rhombohedron.

 Acute form Obtuse form

The rhombic triacontahedron is 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 rhombic icosahedron is also constructed with golden rhombi.

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

## Element

The internal angles of the rhombus are

$2\arctan\frac{1}{\varphi} = \arctan{2}\approx63.43495$ degrees
$2\arctan\varphi = \arctan{1} + \arctan{3} \approx116.56505$ degrees, which is also the dihedral angle of the dodecahedron

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

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