Golden ellipse

A golden ellipse is an ellipse in which the aspect ratio of its two semi-axes $a$ and $b$ corresponds to the golden ratio.

Equivalent characterization

Given is a annulus with outer radius $a$ and inner radius $b$ as well as an ellipse with semi-major axis $a$ and semi-minor axis $b$, where $a$ and $b$ are positive real numbers.

Then the ratio $\frac{a}{b}$ corresponds to the golden ratio $\mathrm{\Phi }$ if and only if the annulus and the ellipse have the same area.

The proof results from the following equivalence chain:

$\pi a^2-\pi b^2=\pi ab\iff a^2-ba-b^2=0\iff a=\frac{b}{2}\pm \sqrt{\frac{b^2}{4}+b^2}\iff a=\frac{1}{2}(1\pm \sqrt{5})\cdot b$

Since only the positive solution is possible, after division by $b$ we get:

$\frac{a}{b}=\frac{1}{2}(1+\sqrt{5})=\mathrm{\Phi }$

Relationship to the golden rectangle

thumb The golden ellipse can be inscribed in a golden rectangle with the side lengths $2a$ and $2b$.

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