---
tag: J0K6T02FQQ
kind: concept
name: Bachmann's reflection geometry
msc: [51F15, 51F05]
sources:
  - url: https://en.wikipedia.org/wiki/Absolute_geometry
---

## Summary

Bachmann's reflection geometry builds plane [[absolute geometry|VKFPCSRNDH]] from a [[group|G5S1M9]] generated by reflections.
Lines are replaced by the reflections in them, and points by the half-turns about them.
Incidence and perpendicularity become algebraic relations between involutions, the elements of order $2$ of the group.
The resulting structures are called metric planes, and they include Euclidean, hyperbolic and elliptic planes over many fields.
Friedrich Bachmann developed this approach in his book "Aufbau der Geometrie aus dem Spiegelungsbegriff".

## Explanations

### L1 human

In the Euclidean plane, every line gives a reflection and every point gives a half-turn.
A point lies on a line exactly when the half-turn and the reflection, done one after the other, give a reflection again.
Two lines are perpendicular exactly when the product of their reflections is a half-turn.
Bachmann turned these facts around: start with a group and a set of its involutions, call them lines, and write axioms in the group.
Theorems of geometry then become computations with reflections, as in the three reflections theorem.