Departure question
Move sameness aroundPort 1 of 8Where two intuitions collide
Composing transformations instead of classifying objects directly brought equations, crystals, and particle symmetries into one structure.
This voyage is an editorial path for understanding, not a claim of direct historical influence or sole invention.
The language mathematics uses for symmetry. Cube rotations, Rubik's Cube moves, and clock addition form groups: the operation is closed and associative, and the structure has an identity and inverses. In the early 19th century, Galois used group structure to explain why the general quintic cannot be solved by radicals.
+ | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
0 | 0 | 1 | 2 | 3 |
1 | 1 | 2 | 3 | 0 |
2 | 2 | 3 | 0 | 1 |
3 | 3 | 0 | 1 | 2 |
The Cayley table of ℤ/4ℤ — clock arithmetic (mod 4). Every element appears exactly once in each row and column, a defining group property. Here 0 is the identity; 1 and 3 are inverses, while 2 is its own inverse.
The mathematics of symmetry — studying what remains unchanged under transformations, from polynomial solvability to crystal structure and particle classification.
Ports in time
Follow the scenes to see problems, notation, standards of proof, and applications changing across different times and places.
Niels Henrik Abel proved that no universal formula using arithmetic operations and radicals solves the general quintic. Particular quintics, numerical roots, and richer functional expressions remain possible.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearBy age 20, Galois had connected permutations of roots with solvability by radicals. His testamentary letter before the duel summarized work already under way.
The recorded place matches a canonical map anchor
Continue this scene on the mapKlein proposed comparing Euclidean, affine, projective, and other geometries through properties invariant under their transformation groups. It organized later research rather than instantly completing every geometry.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearMurray Gell-Mann proposed the Eightfold Way, while Yuval Ne’eman independently developed a related classification. Approximate SU(3) flavor symmetry organized hadrons into multiplets and exposed predictive gaps.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearParticle classification in physics, crystallography, elliptic-curve cryptography, and even solving the Rubik’s Cube — group theory appears wherever symmetry does.
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.
Integer addition is a familiar example of a group with an identity and inverses.
Open the number voyage
Multiplication by unit complex numbers exposes the structure of rotation in a plane.
Open the number voyage
Quaternions calculate three-dimensional rotations whose order changes the result.
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Concept genealogy
Concepts arriving from before
Current port
Group Theory
Concepts opened from here
Only direct editorial links are shown; this is not a complete learning order or historical influence line.
