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 in wikipedia: