Herleitung von Skalarprodukten aus Symmetrieprinzipien

TL;DR

This paper derives the scalar product in three-dimensional space from symmetry principles, leading to a precise form of Pythagoras theorem without assuming the scalar product beforehand.

Contribution

It introduces a novel approach to model ambient space using symmetry principles, resulting in the derivation of the scalar product and Pythagoras theorem.

Findings

Existence of an invariant scalar product from symmetry considerations

Derivation of Pythagoras theorem based on the scalar product

Modeling space via automorphism groups rather than predefined metrics

Abstract

This is an attempt to model ambient space as a three-dimensional real affine space with a distinguished group of automorphisms containing the translations and acting freely and transitively on pairs consisting of a half-plane together with a half-line on its boundary. From there the existence of an invariant scalar product is deduced, which then also implies Pythagoras theorem in a quite precise form. This is in contrast to the usual procedure to model ambient space by asking for a distinguished scalar product and using Pythagoras theorem as known from high school to connect with reality.