Kepler triangle

A Kepler triangle is a right triangle whose side lengths are in a geometric progression, that is, the ratio between the two legs is the same as the ratio between the longer leg and the hypotenuse. It should not be a surprise that the common ratio of this progression is related to the golden ratio and is exactly $\sqrt{𝜙}$.

Proof that the common ratio is $\sqrt{𝜙}$

To find the common ratio of the triangle’s side’s geometric progression, we can use the Pythagorean theorem. By setting the shorter leg to have a length of $1$, and the longer leg to have a length of $x$, the ratio we are looking for is also just $x$. The triangle’s hypotenuse is therefore $\sqrt{1^2+x^2}$.

Since the side lengths need to be in a geometric progression, we have the following equation expressing that the ratios of the longer to the shorter leg and the hypotenuse to the longer leg are the same:

$\frac{x}{1}=\frac{\sqrt{1^2+x^2}}{\mathrm{x}}$

Multiplying both sides by $x$ and then squaring both sides, we have:

$x^4=1+x^2$

Substituting $x^2$ with $𝜙$ results in the familiar golden quadratic equation. This means that $x$, the common ratio we are looking for, is the square root of the golden ratio.

Etymology

The Kepler triangle is named after Johannes Kepler because he was the first known to describe this special right triangle and its relationship to the golden ratio. The Kepler triangle also combines two things in geometry that Kepler declares as “great treasures”: the Pythagorean theorem and the golden ratio.

“Geometry has two great treasures: one is the Theorem of Pythagoras, the other the division of a line in extreme and mean ratio. The first we can compare to a mass of gold; the other we may call a precious jewel.” —Johannes Kepler