Golden rhombus

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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[edit]

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 golden rhombohedron.png
Acute form
Flat golden rhombohedron.png
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.

Rhombictriacontahedron.svgRhombic icosahedron.png

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

Rhombic hexecontahedron.png

Element[edit]

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}

See also[edit]

References[edit]

  • M. Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number, New York: Broadway Books, p. 206, 2002.

External links[edit]