Kepler triangle

A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is where is the golden ratio, and the progression can be written: , or approximately . Squares on the edges of this triangle have areas in another geometric progression, . Alternative definitions of the same triangle characterize it in terms of the three Pythagorean means of two numbers, or via the inradius of isosceles triangles. This triangle is named after Johannes Kepler, but can be found in earlier sources. Although some sources claim that ancient Egyptian pyramids had proportions based on a Kepler triangle, most scholars believe that the golden ratio was not known to Egyptian mathematics and architecture. The Kepler triangle is named after the German mathematician and astronomer Johannes Kepler (1571–1630), who wrote about this shape in a 1597 letter. Two concepts that can be used to analyze this triangle, the Pythagorean theorem and the golden ratio, were both of interest to Kepler, as he wrote elsewhere: Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel. However, Kepler was not the first to describe this triangle. Kepler himself credited it to "a music professor named Magirus". The same triangle appears earlier in a book of Arabic mathematics, the Liber mensurationum of Abû Bekr, known from a 12th-century translation by Gerard of Cremona into Latin, and in the Practica geometriae of Fibonacci (published in 1220–1221), who defined it in a similar way to Kepler. A little earlier than Kepler, Pedro Nunes wrote about it in 1567, and it is "likely to have been widespread in late medieval and Renaissance manuscript traditions". It has also been independently rediscovered several times, later than Kepler.