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.

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

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

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

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.

• Acute golden rhombohedron.png

  Acute golden rhombohedron

• 

  Obtuse golden rhombohedron

• Bilinski dodecahedron.png

  Bilinski dodecahedron

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  rhombic icosahedron

• 

  rhombic triacontahedron

• Rhombic hexecontahedron.png

  rhombic hexecontahedron

See also

• Golden rectangle
• Golden triangle

References

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

External links

• Weisstein, Eric W., "Golden Rhombus", MathWorld.
• Weisstein, Eric W., "Golden Rhombohedron", MathWorld.