## Kepler Triangle

A Kepler triangle is a right triangle formed by three squares with areas in geometric progression according to the golden ratio.
So, the sides of such a triangle are in the ratio 1 : √ φ : φ [where φ = ( 1 + √5 )/ 2 is the golden ratio.]

## Inverse Powers of Phi

Summation of Alternating Inverse Powers of Phi…

## The Kepler Triangle, Phi and Pi

A “Kepler triangle” is a right triangle having edge lengths in a geometric progression, in which the common ratio is √ϕ, where ϕ represents the golden ratio.
Well, let’s construct a square with side length √ϕ that inscribes a Kepler triangle, that is, a right triangle with edges 1 : √ϕ : ϕ (or approximately 1 : 1.272 : 1.618), as shown in the picture. Draw then the circumcircle of the Kepler triangle (highlighted in orange in the picture) whose diameter is the hypotenuse of the triangle.
Then, the perimeters of the square (4√ϕ≈5.0884) and the circle (πϕ≈5.083) coincide up to an error less than 0.1%. From this, we can get the approximation coincidence π≈4/√ϕ