Extension of the Pythagorean Theorem from R^2 to R^3 and R^1

Abstract

Pythagorean theorem has been and remains a fundamental pillar in many spheres of mathematics, physics and engineering and many other fields such as carbon dating and oceanography. The present study will theoretically study the transition of the Pythagorean theorem from  to (three dimensional space) and thereafter from  to (one dimensional space). Initially We shall then expand from  to  where we shall show that for any Pythagorean triples a, b and c then  We shall then highlight the interplay between volumes of three geometrical constructs. In other words volume  plus volume  plus volume equals volume. Finally, we shall transition from  to  where we shall show that . Both of the above transitions are not taken care of by the classical Pythagorean theorem in . in so doing we will investigate valuable insights into the challenges and considerations posed by the 3 dimensional scope.

Additionally, we shall engage the Pythagorean Theorem to a rigorous test involving integers and decimals. We shall then cast our eyes onto the various practical applications in the realm of mathematics, Physics, engineering and architecture unveiling the theorems potential to optimize design and construction processes.

Finally, the study will make recommendation and prospects for further generalizations to higher dimensions and the various generalizations to higher dimensions. This investigation will contribute to the enrichment of spatial mathematics and the broad spectrum of practical implications in diverse disciplines.