Departure question
Read the world behind shapePort 9 of 13“Through Homology: What survives when shapes change, and which rules divide one world from another?”
Begin with lengths and angles, then move toward less visible properties of space: connection, holes, and dimension.
This voyage is an editorial path for understanding, not a claim of direct historical influence or sole invention.
Understand it in one breath
"Stretch all you like with rubber — the number of holes does not change." A donut (genus 1) has H₁ = ℤ²; a sphere has H₁ = 0. A closed loop that is not the boundary of a filled-in region records a one-dimensional hole. In the 21st century, persistent homology turns this idea into tools for analyzing the shape of data.
At a glance
Concept
A way to record holes in a space with algebra. H₀ detects connected components, H₁ detects loops that do not bound a filled region, and H₂ detects higher-dimensional enclosed structure. Rather than merely counting holes, homology uses groups and boundary maps to distinguish topology that survives stretching and bending.
Key formula
Modern applications
Protein-folding analysis, topological data analysis, and coverage guarantees in sensor networks.
Beyond MathVoyage
Curated sources and problems. Bring one discovery back from OEIS, Project Euler, MathOverflow, or arXiv.
- Wikipedia
- Wolfram MathWorld
No concept belongs to one person
Follow people who played different roles
These are not inventor credits. They are different ports: opening a problem, sharpening a language, or carrying it into another world.
Number lenses
A concept looks different when its world of numbers changes
These numbers are editorial lenses for the voyage, not required prerequisites.
Rational Numbers
Begin with lengths and angles, then move toward less visible properties of space: connection, holes, and dimension.
Open the number voyage
Irrational Numbers
Begin with lengths and angles, then move toward less visible properties of space: connection, holes, and dimension.
Open the number voyage
Completion of the Reals
Begin with lengths and angles, then move toward less visible properties of space: connection, holes, and dimension.
Open the number voyage
Concept genealogy
What supports it, and what does it open?
Concepts arriving from before
Current port
Homology
Concepts opened from here
No direct successor port is curated yet.
Only direct editorial links are shown; this is not a complete learning order or historical influence line.