The Metallic Right-Triangles

Takeshi Sugimoto

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

(Received June 24, 2020; Accepted October 19, 2020)

Abstract. The Kepler triangle and its kin are systematically described by the unique formalism (Sugimoto, 2020). The present study affords the super-set of the formalism covering the above-mentioned triangles and other new triangles based on the metallic means and the generalised Fibonacci sequences adjoined to those means. The super-set is given by (the short leg, the long leg, the hypotenuse) = (G_n-2^1/2(m), m^1/2Φ^n/2(m), G_n^1/2(m)Φ(m)), where G_n(m) and Φ(m) designate the nth generalised Fibonacci sequence and the metal mean of the metal number 'm', respectively. The initial members of the golden, silver and bronze right-triangles are presented.

Keywords: Metallic Means, Generalised Fibonacci Sequences, Pythagorean Theorem