Definition:Erlangen Program
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Definition
The Erlangen program is a method of characterizing systems of geometry based on group theory and projective geometry.
Also known as
The Erlangen program is also known as the Erlanger program.
In non-American English, program is usually spelt programme.
Historical Note
The Erlangen program was published by Felix Christian Klein in $1872$ in his Vergleichende Betrachtungen über neuere geometrische Forschungen.
It is named after the University Erlangen-Nürnberg, where Klein worked.
By $1872$, non-Euclidean geometries had emerged.
However, nothing had been done to determine their hierarchy and relationships.
Klein's approach was as follows:
- Projective geometry was identified as the unifying frame for all other geometries considered by him. Hence the following hierarchy was developed:
- Affine geometry was more restrictive than projective geometry
- Euclidean geometry was more restrictive than affine geometry.
- Klein proposed that group theory was the most useful way of organizing geometrical knowledge, emerging from the discipline of Galois theory.
- Klein specificially identified the appropriate geometrical language in which to define the relevant concepts for each level of the hierarchy.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Erlangen programme or Erlanger programme