Automorphism
/ˌɔtəˈmɔrfɪzəm/noun
A structure-preserving map from a mathematical object to itself that can be reversed.
The Greek roots
Literally: “self form”
The story of the word
An isomorphism of a thing with itself sounds like an empty idea until you notice what it collects: all the ways an object can be rearranged without changing anything that matters, which is to say its symmetries. The automorphisms of any object form a group under composition. Galois turned the question of solving equations into a question about the automorphisms of a field, and most of modern algebra followed him through that door.
- How it travelled
- Ancient Greek → German → English
- First recorded
- 19th century
The same roots elsewhere
Other languages reached for exactly the same Greek pieces.
Same family
In a sentence
Complex conjugation is an automorphism of the complex numbers fixing the reals.
Counting the automorphisms told us the graph was more symmetric than it looked.