*** Welcome to piglix ***

Kepler triangle


A Kepler triangle is a right triangle with edge lengths in geometric progression. The ratio of the edges of a Kepler triangle is linked to the golden ratio

and can be written: , or approximately 1 : 1.272 : 1.618. The squares of the edges of this triangle (see figure) are in geometric progression according to the golden ratio.

Triangles with such ratios are named after the German mathematician and astronomer Johannes Kepler (1571–1630), who first demonstrated that this triangle is characterised by a ratio between short side and hypotenuse equal to the golden ratio. Kepler triangles combine two key mathematical concepts—the Pythagorean theorem and the golden ratio—that fascinated Kepler deeply, as he expressed in this quotation:

Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into mean and extreme ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.

Some sources claim that a triangle with dimensions closely approximating a Kepler triangle can be recognized in the Great Pyramid of Giza.

The fact that a triangle with edges , and , forms a right triangle follows directly from rewriting the defining quadratic polynomial for the golden ratio :


...
Wikipedia

...