Theorem of three squares across three geometries

TL;DR

This paper investigates how the Pythagorean theorem, expressed through three squares in Euclidean geometry, can be extended or adapted to spherical and hyperbolic geometries.

Contribution

It introduces analogous expressions of the Pythagorean theorem across three different geometries, expanding understanding of geometric relationships beyond Euclidean space.

Findings

Identifies Pythagorean-like relations in spherical geometry.

Derives hyperbolic geometry analogs of the theorem.

Provides a unified framework for three geometries.

Abstract

In Euclidean geometry, the Pythagorean theorem is presented as an equation involving three squares. This paper explores how analogous expressions may be identified in spherical and hyperbolic geometries.