Leonhard Euler, Antoine-Jean Lhuilier, August Ferdinand Möbius, Johann Benedict Listing, Henri Poincaré mathematics |

## Topology

- The beginning of topology
- were Euler’s method of proving
- no solution existed for the problem
- of the Seven Bridges of Königsberg
- and the
*Euler characteristic*relating - polyhedra’s vertices, edges, and faces.
- *
- Lhuilier showed the Euler characteristic
- wasn’t true of solids with holes through them.
- Möbius showed a Möbius strip is not orientable;
- it can’t be moved into its mirror image.
- Listing discovered the Möbius strip before Möbius,
- and coined the name
*topology.* - Poincaré introduced the concepts
*homotopy*and*homology* - and rigorously defined the new mathematics.
- *
- Topology became the study
- of how spatial properties are preserved
- when a surface is bent and stretched
- without causing discontinuities or new connections.

## Surfaces

- Some bridges must be crossed more than once.
- Hairy balls cannot be combed without cowlicks.
- A knot is a closed loop that cannot be simplified.
- Möbius strips and Klein bottles
- are non-orientable surfaces.
- Doughnuts can be stretched into coffee mugs.

## Shape

- I’m out of shape,
- not bent out of shape,
- by bending the rules.
- I’m working things out
- without working out,
- and it’s working out.

Many mathematicians, most not named here, have contributed to topology, so that it has so many branches today that it needs a topology of topology.

See also in

The book of science:Graph theory—Leonhard EulerEuler characteristic—Leonhard EulerKnot theory—Alexandre-Théophile Vandermonde, Peter Guthrie TaitReadings on wikipedia:

Other readings: