A golden ellipse is an ellipse in which the aspect ratio of its two semi-axes a {\displaystyle a} and b {\displaystyle b} corresponds to the golden ratio.
Equivalent characterization
Given is a annulus with outer radius a {\displaystyle a} and inner radius b {\displaystyle b} as well as an ellipse with semi-major axis a {\displaystyle a} and semi-minor axis b {\displaystyle b} , where a {\displaystyle a} and b {\displaystyle b} are positive real numbers. Then the ratio a b {\displaystyle {\frac {a}{b}}} corresponds to the golden ratio Φ {\displaystyle \Phi } if and only if the annulus and the ellipse have the same area. The proof results from the following equivalence chain:
π a 2 − π b 2 = π a b ⇔ a 2 − b a − b 2 = 0 ⇔ a = b 2 ± b 2 4 + b 2 ⇔ a = 1 2 ( 1 ± 5 ) ⋅ b {\displaystyle \pi a^{2}-\pi b^{2}=\pi ab\Leftrightarrow a^{2}-ba-b^{2}=0\Leftrightarrow a={\frac {b}{2}}\ \pm {\sqrt {{\frac {b^{2}}{4}}+b^{2}}}\Leftrightarrow a={\frac {1}{2}}(1\pm {\sqrt {5}})\cdot b}
Since only the positive solution is possible, after division by b {\displaystyle b} we get:
a b = 1 2 ( 1 + 5 ) = Φ {\displaystyle {\frac {a}{b}}={\frac {1}{2}}(1+{\sqrt {5}})=\Phi }
Relationship to the golden rectangle
The golden ellipse can be inscribed in a golden rectangle with the side lengths 2 a {\displaystyle 2a} and 2 b {\displaystyle 2b} .
References
Further reading Anthony David Rawlins: A note on the golden ratio. Mathematical Gazette, 79, (1995), page 104
External links Daniel Favre: Golden ratio (Sectio Aurea) in the Elliptical Honeycomb ResearchGate, January 2016 Tadeusz E. Dorozinski: Goldene Ellipse on 3doro.de