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