Topology Surfaces Shape Commentary

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.