A Study on Extention of the Pythagorean Theorem from R to R Using the Pythagorean Triples

The study investigates theoretically the transition of the Pythagorean theorem from \(\mathbb{R}^2\) to \(\mathbb{R}^3\) (three-dimensional space). The research shows that for any Pythagorean triples a, b and c then \(a^3+b^3+c^3=d^3\) holds. The investigation further highlights the interplay between volumes of three geometrical constructs. In other words, volume a\(^3\) plus volume b\(^3\) plus volume c\(^3\) equals volume d\(^3\).