## 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

## Fibonacci Right Triangle

The sum of the squares of consecutive Fibonacci numbers is another Fibonacci number.

## Sprouts Game

All possible ways a game of “Sprouts” with two initial dots can evolve. Sprouts is a paper-and-pencil game that can be enjoyed simply by both adults and children.

## Equi-extended and isoperimetric non-congruent triangles

The picture below shows the ONLY one pair of triangles with the following properties:
· One triangle is a right triangle and one is isosceles,
· All side lengths of both triangles are rational numbers, and
· The perimeters and areas of both triangles are equal.

## Icosahedron with golden ratio cross-sections

3 intersecting golden rectangles (1 : φ) will create the vertices of an icosahedron.