Departure question
Read the world behind shapePort 11 of 13Where two intuitions collide
Discarding a map's exact shape made bridges, communication networks, and social ties readable in one language.
This voyage is an editorial path for understanding, not a claim of direct historical influence or sole invention.
Abstract objects as vertices and relations as edges. Euler's 1736 paper on the Königsberg bridges was an early milestone because it kept only connectivity to decide whether a walk was possible. Social, communication, transport, and biological networks can be modeled as graphs, while real flows, directions, capacities, and time require additional structure.
A | B | C | D | |
|---|---|---|---|---|
A | 0 | 1 | 1 | 0 |
B | 1 | 0 | 1 | 1 |
C | 1 | 1 | 0 | 1 |
D | 0 | 1 | 1 | 0 |
Adjacency matrix — A↔B, A↔C, B↔C, B↔D, C↔D. Each vertex’s degree equals its row sum (A:2, B:3, C:3, D:2). Total 10 = 5 edges × 2 ✓ by the Handshaking Lemma.
The mathematics of connection — born from a Sunday walk puzzle, now the language of social networks and the brain.
Ports in time
Follow the scenes to see problems, notation, standards of proof, and applications changing across different times and places.
Euler proved that no walk could cross each of Königsberg’s seven bridges exactly once, turning a city puzzle into the birth of graph theory.
The recorded place matches a canonical map anchor
Continue this scene on the mapCould every planar map be colored with only four colors so that neighboring regions differ? The 1852 question was finally proved with computer assistance in 1976.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearEdsger Dijkstra devised an efficient method for finding shortest paths in a weighted graph. It became a foundation for routing, navigation, and network optimization.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearGoogle’s founders modeled web pages as nodes and hyperlinks as edges, then used the resulting graph to estimate importance and transform web search.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearGoogle PageRank, social-network recommendations, GPS routing, computer networks, graph neural networks, and contact tracing — graphs appear wherever things connect.
Curated sources and problems. Bring one discovery back from OEIS, Project Euler, MathOverflow, or arXiv.
No concept belongs to one person
These are not inventor credits. They are different ports: opening a problem, sharpening a language, or carrying it into another world.
Number lenses
These numbers are editorial lenses for the voyage, not required prerequisites.
Begin with lengths and angles, then move toward less visible properties of space: connection, holes, and dimension.
Open the number voyage
Begin with lengths and angles, then move toward less visible properties of space: connection, holes, and dimension.
Open the number voyage
Begin with lengths and angles, then move toward less visible properties of space: connection, holes, and dimension.
Open the number voyage
Vertex degrees and path lengths expose connection structure through natural numbers.
Open the number voyage
Concept genealogy
Concepts arriving from before
Current port
Graph Theory
Concepts opened from here
Only direct editorial links are shown; this is not a complete learning order or historical influence line.
