The Kepler Triangle and Its Kin

Takeshi Sugimoto

Kanagawa University, Kanagawa Ward, Yokohama 221-8686, Japan
E-mail address: sugimt01@kanagawa-u.ac.jp

(Received July 29, 2019; Accepted September 30, 2019)

Abstract. The Kepler triangle, also known as the golden right triangle, is the right triangle with its sides of ratios '1:φ^1/2:φ,' where φ denotes the golden ratio. Also known are the silver right triangle and the square-root-three φ right triangle. This study introduces the generalised golden right triangle, which have sides of lengths closely related to φ and the Fibonacci numbers, F_n: '(F_n-2)^1/2:φ^n/2:(F_n)^1/2φ' for any natural number n. This formalism covers all the known φ-related right triangle, i.e., the Kepler triangle and its kin. As n tends to infinity, the ratios of the sides go to 'φ^-1:5^1/4:φ.' Our model plays an important role in the classroom to study the golden ratio, the Fibonacci numbers and the Pythagorean theorem.

Keywords: Golden Ratio, Fibonacci Sequence, Pythagorean Theorem