Mathematics is divided into 2 major branches:
1. Analysis (Continuity, Calculus)
2. Algebra (Set, Discrete numbers, Structure)
In between the two branches, Poincaré invented in 1900s the Topology (拓扑学) – which studies the ‘holes’ (disconnected) in-between, or ‘neighborhood’.
Topology specialised in
– ‘local knowledge’ = Point-Set Topology.
– ”global knowledge’ = Algebraic Topology.
The local data of consumer behavior uses ‘Point-Set Topology’; the global one is ‘BIG Data’ using Algebraic Topology.
Watch this amazing video, mathematically you can turn a sphere inside out, but not a circle:
‘Hairy Ball Theorem’ (Topology)
1. “You can’t comb the hairs on a coconut.”
2. “There exists a calm point on earth where there is no wind.”
Both 1 & 2 are ‘Hairy Ball Theorem’.
If we have a ball with hairs sticking out from each point on it, then impossible to comb the hairs flat with a continuous movement without leaving at least one hair sticking up vertically (typically, at the center of a swirl). Not applicable to doughnut.
=> Wind pattern:
Ball = Earth
Combed down hairs = wind
Length of hairs = wind’s force
=> there must be a place on Earth where there is no wind at all.
Topology (by Poincaré)
Moniker “Rubber-Sheet Geometry“, compared with Geometry’s ‘rigid objects‘.
[Greek]= τοΠοζ(Place) λΟγια(Study)
[Latin]= Analysis Situs (Situation)
1. Remove (invariants) of geometry:
- a. Euclidean (distance)
- b. Affine (//, ratio)
- c. Projective (cross-ratio)
2. Preserve ‘Neighbourhood’ (Nearness)
- define ‘Continuity’ (Analysis)
3. Elastic deformation (stretch, bend, twist)
- a line is no longer a line.