Zero-forcing precoding

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

$2\arctan {\frac {1}{\varphi }}\approx 63.43495$ degrees

$2\arctan \varphi \approx 116.56505$ 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

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

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

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

• 22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.
  
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• 22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.
  
  Here is my web site - cottagehillchurch.com