Mathematics of Relativity

A 100-year arc of mathematics describing the shape of the universe — from Gauss's surfaces, to Riemann's manifolds, to Minkowski's spacetime, to Einstein's general relativity.

About 35 min·8 nodes

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The path across the map

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STEP 1 · Mathematician1827Göttingen· Main activity

Carl Friedrich Gauss

The Prince of Mathematicians. He made revolutionary contributions to nearly every field — number theory, statistics, differential geometry, electromagnetism.

↓3 years later

STEP 2 · Concept~1830Göttingen· Discovery

Non-Euclidean Geometry

Centuries of failed attempts to prove Euclid's fifth postulate revealed that changing it can produce consistent alternative geometries, opening hyperbolic, elliptic, and more general curved spaces.

↓24 years later

STEP 3 · Mathematician1854Göttingen· Discovery

Bernhard Riemann

A concise, transformative author in complex analysis, geometry, and number theory. His 1854 lecture opened higher-dimensional metric geometry; a six-page 1859 paper posed the still-open hypothesis on zeta zeros.

↓Same year

STEP 4 · Concept1854Göttingen· Discovery

Manifold

Spaces that look flat locally but curve globally — from Earth's surface to spacetime.

↓54 years later

STEP 5 · Mathematician1908Göttingen· Main activity

Hermann Minkowski

Einstein's teacher at Zürich ETH. In 1908 he declared that space and time become shadows, and only their union retains an independent reality — birthing Minkowski spacetime, the mathematical foundation of general relativity. Died at 44 from appendicitis.

↓55 years later

STEP 6 · Mathematician1963Oxford· Main activity

Michael Atiyah

A mathematician who built a bridge between a number counted by differential equations and a number counted by topology. With Isadore Singer, Atiyah developed the theorem equating the analytic and topological indices of an elliptic differential operator. The index is not simply the dimension of a solution space: it is the difference between the dimensions of a kernel and cokernel. The theorem places results in the Riemann–Roch and Gauss–Bonnet families in one framework and was central to his 1966 Fields Medal. Atiyah and Hirzebruch developed topological K-theory, whose periodicity rests on Bott’s theorem: complex K-theory is 2-periodic and real K-theory 8-periodic. He was Savilian Professor of Geometry at Oxford from 1963 to 1969 and later led research and institutions at Oxford, the IAS, and Cambridge. His work connecting gauge theory and geometry formed important background for four-manifold topology, but Donaldson–Thomas and Seiberg–Witten theory cannot be reduced to one Atiyah–Witten collaboration.

↓27 years later

STEP 7 · Mathematician1990Princeton· Main activity

Edward Witten

Edward Witten has expanded the boundary between mathematics and theoretical physics. After studying history at Brandeis, he earned a physics PhD at Princeton and joined the Institute for Advanced Study faculty in 1987. His work connected gauge theory with topology, interpreted the Jones polynomial through Chern-Simons theory, and developed string-theory dualities and the M-theory proposal. He became the first theoretical physicist to receive a Fields Medal in 1990.

↓Next step

STEP 8 · RiverGöttingen· Journey start

The River of Relativity Mathematics

Gauss's surfaces → Riemann's manifolds → Minkowski's spacetime → Einstein's general relativity.

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