Timeline of Geometry
Geometry from Egyptian and Babylonian scribes of about 1850–1800 BCE through Euclid, non-Euclidean geometry and topology to Grigori Perelman's 2002–2003 proof of the Poincaré conjecture.
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- 29 events c. 1850 BCE – 2002–2003
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- by claude-opus-5-5
c. 1850 BCE: Egyptian scribes compute a truncated pyramid's volume
Problem 14 of the Moscow Mathematical Papyrus finds the volume of a truncated square pyramid, 6 units high with sides of 4 at the base and 2 at the top, as 56 cubic units, showing that Egyptian scribes knew the correct rule. The papyrus, written down in the 13th Dynasty from older 12th-Dynasty material, holds 25 problems, seven of them geometric.
Wikipediac. 1800 BCE: Babylonian tablet lists sides of right triangles
Plimpton 322, a clay tablet said to come from Larsa in southern Iraq, has 15 rows that each relate to a right triangle with whole-number sides; most scholars read two columns as the short side and the diagonal, which reach 12,709 and 18,541, implying a third side of 13,500. Values that large suggest a systematic method, in use more than 1,000 years before Pythagoras, though the tablet's purpose is unknown.
Wikipediac. 800–500 BCE: Baudhayana's Śulba Sūtra states the Pythagorean theorem
The oldest of the Śulba Sūtras, Sanskrit manuals for building Vedic fire altars, Baudhayana's text states that the square on a rectangle's diagonal equals the squares on its two sides together and lists rectangles where this holds, such as 3 by 4 and 5 by 12. It approximates the square root of 2 as 577/408 and gives approximate ways to turn a square into a circle of equal area and back.
Wikipediac. 300 BCE: Euclid builds geometry from postulates in the "Elements"
Euclid's "Elements" derived plane and solid geometry and number theory in 13 books from definitions, five postulates and five common notions, building on earlier Greek work by Eudoxus, Theaetetus and others. The oldest surviving large-scale deductive treatment of mathematics, it included a fifth, or parallel, postulate less self-evident than the rest, whose attempted proofs led to non-Euclidean geometry.
Wikipediac. 225 BCE: Archimedes finds a sphere's area and volume
In "On the Sphere and Cylinder", Archimedes of Syracuse proved that a sphere's surface area is four times the area of its great circle and its volume two-thirds that of the smallest enclosing cylinder, a result he reportedly asked to have carved on his tomb. He proved it by trapping the sphere between solids built from cones and frustums, but his "Method" says he first found the volume by balancing thin slices on an imaginary lever.
Wikipediac. 200 BCE: Apollonius names the ellipse, parabola and hyperbola
Apollonius of Perga's "Conics" showed that ellipses, parabolas and hyperbolas can all be cut from one double cone by varying the angle of the cutting plane, and gave the three curves the names still in use. Of its eight books, four survive in Greek and three only in an Arabic translation by Thābit ibn Qurra; no text of the eighth survives.
Wikipedia263: Liu Hui adds proofs to the "Nine Chapters"
Liu Hui's commentary on the Chinese classic "The Nine Chapters on the Mathematical Art" justified its methods, proving the gougu (Pythagorean) theorem, cutting solids into pieces to find the volumes of pyramids and wedges, and approximating π as 3.14 from a 192-sided polygon. Its surveying appendix, later the "Sea Island Mathematical Manual", found heights and distances of far-off objects using poles.
Wikipedia1077: Omar Khayyam probes Euclid's parallel postulate
The Persian mathematician and poet Omar Khayyam, in a commentary on Euclid's postulates, studied a quadrilateral with two equal sides perpendicular to its base and asked whether its top angles are right, acute or obtuse. Using a principle from Aristotle, he ruled out the acute and obtuse cases, now known to describe hyperbolic and elliptic geometry; Nasir al-Din al-Tusi critiqued and built on his work.
Wikipedia1435: Alberti sets out the geometry of perspective
Leon Battista Alberti's Latin treatise "De pictura" ("On Painting") gave the earliest known written account of linear perspective, treating a picture as a plane cutting the rays from the eye to the scene and sizing distant objects by similar triangles; such projections later grew into projective geometry. Filippo Brunelleschi had demonstrated perspective in Florence years earlier but left no written account, and Alberti dedicated the 1436 Italian version to him.
Wikipedia1637: Descartes and Fermat invent analytic geometry
René Descartes' "La Géométrie", an appendix to his "Discourse on the Method", described curves by algebraic equations in two unknown lengths, x and y, and used them to solve a locus problem of Pappus of Alexandria. Pierre de Fermat had found the method independently in a manuscript then circulating in Paris but printed only in 1679; the familiar perpendicular axes were added later by editors such as Frans van Schooten.
Wikipedia1733: Saccheri derives non-Euclidean theorems while defending Euclid
The Italian Jesuit Giovanni Girolamo Saccheri published "Euclides ab omni naevo vindicatus" ("Euclid Freed of Every Flaw") shortly before his death, assuming the parallel postulate false in hope of a contradiction. He refuted one alternative but, from the other, derived many results now known as theorems of hyperbolic geometry, before rejecting it as "repugnant to the nature of straight lines".
Wikipedia1736: Euler solves the Königsberg bridge problem
Leonhard Euler proved that no walk through Königsberg (now Kaliningrad, Russia) could cross each of its seven bridges exactly once, since all four land masses touched an odd number of bridges. Presented to the St. Petersburg Academy in August 1735, the argument appeared in the academy's volume for the following year, printed in 1741, as a problem of the "geometry of position"; it ignored distances entirely and is considered the first theorem of graph theory.
Wikipedia1758: Euler links vertices, edges and faces of polyhedra
Leonhard Euler published the rule, announced in a November 1750 letter to Christian Goldbach, that a convex polyhedron's vertices minus its edges plus its faces always equal 2 (V − E + F = 2), though his proof was flawed. René Descartes had found a closely related result around 1630 in a long-lost manuscript; generalized as the Euler characteristic, the number became a foundation of topology.
Wikipedia1796: Gauss shows the regular 17-gon is constructible
Carl Friedrich Gauss, then a student at Göttingen, showed that a regular 17-sided polygon can be drawn with compass and straightedge, the first new constructible regular polygon since ancient Greece. His "Disquisitiones Arithmeticae" (1801) extended this to any polygon whose number of sides is a power of 2 times distinct Fermat primes, and Pierre Wantzel proved in 1837 that no other regular polygons are constructible.
Wikipedia1822: Poncelet revives projective geometry
Jean-Victor Poncelet, a French army engineer who had developed his ideas as a prisoner of war in Russia after Napoleon's 1812 invasion, published "Traité des propriétés projectives des figures", on properties of figures that survive central projection. It revived a subject Girard Desargues had explored in 1639 with ideas such as points at infinity where parallel lines meet, and is considered the founding work of modern projective geometry.
Wikipedia1827: Gauss shows a surface's Gaussian curvature is intrinsic
Carl Friedrich Gauss, drawn to curved surfaces by his survey of the Kingdom of Hanover, proved his "Theorema Egregium": a surface's Gaussian curvature, the product of its two principal curvatures, can be found from distances measured within the surface, so bending without stretching cannot change it and no flat map can show the Earth undistorted. Presented in October and printed in 1828, his memoir began the study of surfaces from the inside.
Wikipedia1829–1832: Lobachevsky and Bolyai publish hyperbolic geometry
Nikolai Lobachevsky, in a Kazan University journal, and János Bolyai, in an appendix to his father's textbook, independently published a geometry in which more than one line through a point misses a given line and a triangle's angles sum to less than 180°. Carl Friedrich Gauss wrote privately that he had reached similar ideas but never published them; Eugenio Beltrami's 1868 models proved it as consistent as Euclid's.
Wikipedia1854: Riemann generalizes geometry to curved manifolds
In his habilitation lecture at Göttingen, "On the Hypotheses on Which Geometry Is Based", Bernhard Riemann described spaces of any number of dimensions, now called manifolds, whose curvature can vary from point to point, extending Gauss's theory of surfaces. Published in 1868, after his death, it founded Riemannian geometry, a generalization of which Albert Einstein used in 1915 to describe gravity as curved spacetime.
Wikipedia1872: Klein classifies geometries by their symmetry groups
Felix Klein, a professor at Erlangen at 23, proposed in what is now called the Erlangen program that each geometry is defined by a group of transformations and the properties it preserves: lengths and angles in Euclidean geometry, only incidence and cross-ratio in projective geometry. It placed Euclidean, affine and non-Euclidean geometries within projective geometry, and this group-based view became the accepted modern approach.
Wikipedia1882: Lindemann proves squaring the circle impossible
Ferdinand von Lindemann proved that π is transcendental, the root of no polynomial equation with rational coefficients, so no compass-and-straightedge construction can produce a square with the area of a given circle. With Pierre Wantzel's 1837 proof that doubling the cube and trisecting an arbitrary angle are also impossible, all three classical construction problems of Greek geometry were settled.
Wikipedia1895: Poincaré founds algebraic topology in "Analysis Situs"
Henri Poincaré's paper "Analysis Situs" gave what is regarded as the first systematic treatment of topology, the study of properties unchanged by continuous deformation. It introduced the fundamental group and the first recognizable theory of homology, algebraic invariants that can tell spaces apart, and coined the term "homeomorphism"; five supplements followed by 1904.
Wikipedia1899: Hilbert rebuilds Euclidean geometry on rigorous axioms
David Hilbert's "Grundlagen der Geometrie" ("Foundations of Geometry"), based on his Göttingen lectures, replaced Euclid's postulates with an explicit set of axioms about undefined points, lines and planes, closing gaps in Euclid's reasoning. Building on Moritz Pasch's 1882 axioms, it used models to test the axioms' independence and consistency and spread the axiomatic method through mathematics.
Wikipedia1904: Poincaré poses his conjecture on the 3-sphere
In the fifth supplement to "Analysis Situs", Henri Poincaré found a space with the homology of the 3-sphere but a different fundamental group, refuting his earlier claim, and asked whether every closed, simply connected three-dimensional manifold, one in which every loop can be shrunk to a point, must be homeomorphic to the 3-sphere. The question, the Poincaré conjecture, stayed open for nearly a century.
Wikipedia1956: Milnor discovers exotic seven-dimensional spheres
John Milnor constructed seven-dimensional manifolds that are homeomorphic to the ordinary 7-sphere but not diffeomorphic to it: topologically they are spheres, but their smooth structure differs. Regarded as the beginning of differential topology, the discovery was followed by Michel Kervaire and Milnor's proof that the 7-sphere has 28 smooth structures when orientation is counted.
Wikipedia1961: Smale proves the Poincaré conjecture above dimension four
Stephen Smale proved the generalized Poincaré conjecture for smooth manifolds of dimension five or more: a closed manifold with the homotopy type of a sphere is homeomorphic to it. Announced in 1960, it was soon also proved by John Stallings, Christopher Zeeman and Andrew Wallace, and Max Newman extended it to all topological manifolds in 1966. Michael Freedman settled the topological four-dimensional case in 1982, leaving three dimensions open; the smooth four-dimensional case is still unsolved.
Wikipedia1975: Mandelbrot coins the word "fractal"
Benoit Mandelbrot named "fractals", from the Latin "fractus" ("broken"), in his book "Les Objets fractals": rough shapes whose detail recurs at ever smaller scales and whose dimension need not be a whole number. The idea drew together earlier curiosities such as Helge von Koch's snowflake curve and Felix Hausdorff's non-integer dimensions; his illustrated "The Fractal Geometry of Nature" (1982) spread it widely.
Wikipedia1982: Thurston proposes the geometrization conjecture
William Thurston conjectured that every closed, orientable three-dimensional manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries, much as every closed surface carries a spherical, flat or hyperbolic geometry. One of 24 open problems he published, it contained the Poincaré conjecture as a special case; Thurston had announced a proof for the large class of Haken manifolds.
Wikipedia1998: Hales proves Kepler's sphere-packing conjecture
Thomas Hales, assisted by his student Samuel Ferguson, completed a computer-assisted proof that no arrangement of equal spheres fills space more densely than the familiar stacking of cannonballs, at about 74 percent, as Johannes Kepler had conjectured in 1611. The "Annals of Mathematics" published its main paper in 2005 after a referee panel reported being "99% certain" of the proof, and a machine-checked formal proof was completed in 2014.
Wikipedia2002–2003: Perelman proves the Poincaré and geometrization conjectures
Grigori Perelman posted three preprints on arXiv that completed Richard Hamilton's Ricci flow program, which evolves a manifold's geometry much as heat spreads, proving Thurston's geometrization conjecture and with it the Poincaré conjecture. After independent teams checked the proof, he declined a Fields Medal in 2006 and, in 2010, the Clay Mathematics Institute's US$1 million prize for the first solved Millennium Prize Problem.
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