the.com/topology
the math where a coffee mug and a donut are secretly the same shape
means The branch of mathematics studying the properties of shapes that survive stretching, bending, and twisting—but not tearing or gluing—so it cares about holes and connectedness rather than exact measurements.
from From the Greek 'topos' (place) plus '-logia' (study of), so literally 'the study of place.' The term entered mathematics in the 19th century; an earlier name was the Latin 'analysis situs,' meaning 'analysis of position.' The same 'topos' gives us 'topic' and 'utopia' (literally 'no-place').
- hole counting
- Cares about holes, not distances or angles
- famous proof
- Perelman solved Poincaré, refused a million dollars
- hairy ball
- You cannot comb a hairy sphere flat
- rubber sheet
- Stretch and bend freely, just never tear
- map coloring
- Four colors suffice for any flat map
klein bottle — non-orientable 4-dimensional surface that cannot exist in 3d space without self-intersection, named after felix klein 1882
torus — doughnut-shaped surface of revolution, fundamental object in topology with genus 1