The Language of Symmetry — From Groups to Galois
What is symmetry? From rotations of an equilateral triangle to why the general quintic cannot be solved by radicals. Galois's existing research and his roughly seven-page final summary letter reveal the bridge between equations and symmetry.
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Évariste Galois
A mathematician who connected solvability by radicals with the symmetries of polynomial roots. By age 20, Galois had developed a group-based way to study relations preserved when roots are permuted. His approximately seven-page testamentary letter to Auguste Chevalier, written before his fatal duel in 1832, summarized existing results and requested that his manuscripts be reviewed; it was not the night on which he created the whole theory from scratch. He died from his wound the next day. Chevalier and Galois’s brother preserved the papers, and Liouville announced their importance in 1843 before publishing major manuscripts in 1846. This work became the starting point of what is now called Galois theory.
William Rowan Hamilton
An Irish mathematician and astronomer who developed characteristic functions in optics and mechanics. After years of trying to extend complex numbers to triples, he recalled discovering the four-component quaternion relation at Broom Bridge in 1843 and carving it into the stone. Quaternions later became one useful tool for representing rotations.
Arthur Cayley
A British mathematician who combined legal practice with research for roughly fourteen years. Sylvester coined “matrix,” and earlier work existed on arrays and transformations; Cayley’s 1858 memoir was a major step in organizing matrix addition, multiplication, inverses, and characteristic equations into an algebraic theory.
Camille Jordan
A mathematician who moved between pictures that look obvious and algebraic forms made computable. Jordan’s name remains attached to the curve theorem, canonical form, measure, and Jordan–Hölder theorem. His 1870 Treatise on Substitutions and Algebraic Equations organized the theory of permutation groups and equations developed after Galois and greatly helped group theory circulate. Calling it the solitary first textbook made directly from Galois’s scattered notes would erase the intervening research network. The attribution and formulation of Jordan form also involve earlier and later work. In 1887 he stated that a simple closed plane curve separates an inside from an outside, but the rigor of his argument was disputed; Veblen supplied a new proof in 1905. The story shows why an obvious-looking claim may still demand difficult definitions and proof.
Felix Klein
Builder of the Göttingen school. His 1872 Erlangen Program defined geometry as the study of invariants under a group of transformations — unifying Euclidean, projective, and non-Euclidean geometry under one banner. The Klein bottle is his contribution to topology.
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