Golden rhombus

Gold rhombus - a rhombus whose diagonals relate to each other as $\varphi$ ${\ displaystyle \ varphi}$ $\ varphi$ where $\varphi \approx 1,618$ ${\ displaystyle \ varphi \ approx 1,618}$ ${\ displaystyle \ varphi \ approx 1,618}$ ( golden ratio ).

Golden rhombus

Properties

The angles of the golden diamond are:

Acute angles: $2\arctan {\frac {1}{\varphi }}=\arctan 2\approx 63,43495$ ${\ displaystyle 2 \ arctan {\ frac {1} {\ varphi}} = \ arctan 2 \ approx 63,43495}$ °
Dull corners: $2\arctan \varphi =\arctan 1+\arctan 3\approx 116,56505$ ${\ displaystyle 2 \ arctan \ varphi = \ arctan 1+ \ arctan 3 \ approx 116,56505}$ °, which coincide with the dihedral angle of the dodecahedron .

The ratio of the side of the golden rhombus to its short diagonal is ${\frac {1}{2}}{\sqrt {1+\varphi ^{2}}}={\frac {1}{4}}{\sqrt {10+2{\sqrt {5}}}}\approx 0.95106$ ${\ displaystyle {\ frac {1} {2}} {\ sqrt {1+ \ varphi ^ {2}}} = {\ frac {1} {4}} {\ sqrt {10 + 2 {\ sqrt {5 }}}} \ approx 0.95106}$ .

The lengths of the diagonals of a gold rhombus with a length of 1 are equal to:

p={\frac {2+2{\sqrt {5}}}{\sqrt {10+2{\sqrt {5}}}}}\approx 1.70130

{\ displaystyle p = {\ frac {2 + 2 {\ sqrt {5}}} {\ sqrt {10 + 2 {\ sqrt {5}}}}} approx 1.70130}

q={\frac {4}{\sqrt {10+2{\sqrt {5}}}}}\approx 1.05146

{\ displaystyle q = {\ frac {4} {\ sqrt {10 + 2 {\ sqrt {5}}}} \ approx 1.05146}

The radius of the inscribed circle of the golden rhombus is ${\frac {p\varphi }{\sqrt {2(5+{\sqrt {5}})}}}$ ${\ displaystyle {\ frac {p \ varphi} {\ sqrt {2 (5 + {\ sqrt {5}})}}}}$ .

The area of the golden diamond is ${\frac {p^{2}}{1+{\sqrt {5}}}}$ ${\ displaystyle {\ frac {p ^ {2}} {1 + {\ sqrt {5}}}}}$ . ^[one]

A golden rectangle can be described around a golden rhombus.

Notes

↑ Weisstein, Eric W. Golden Rhombus . mathworld.wolfram.com. Date of treatment December 29, 2016.

[1] Weisstein, Eric W. Golden Rhombus . mathworld.wolfram.com. Date of treatment December 29, 2016.

Golden rhombus

Properties

See also

Notes

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