Geometry

• 1. Introduction

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  Around 1890 BC, an Egyptian scribe wrote down a formula for the volume of a truncated pyramid, a frustum, on what is now called the Moscow Papyrus. Centuries earlier, Babylonian clay tablets like Plimpton 322 had already been inscribed near 1900 BC. These were not abstract exercises. They answered the practical needs of surveying, construction, astronomy, and craft. This is geometry, a branch of mathematics concerned with the distance, shape, size, and relative position of figures. Along with arithmetic, it is one of the oldest branches of mathematics, and the person who works in it carries the name geometer. How did a collection of empirically discovered rules about lengths and volumes become a discipline so vast that its methods underpin the proof of a centuries-old number puzzle and the very shape of the universe in general relativity? And how did the meaning of a single word, space, quietly change underneath it?

• 2. The Word Space

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  Until the 19th century, geometry was almost exclusively Euclidean geometry, built from point, line, plane, distance, angle, surface, and curve as fundamental concepts. The word space meant one thing then. It referred to the three-dimensional space of the physical world, modeled by Euclidean geometry. That settled meaning did not survive the century. Carl Friedrich Gauss proved his Theorema Egregium, the remarkable theorem, asserting roughly that the Gaussian curvature of a surface does not depend on how the surface is embedded in Euclidean space. The consequence was startling. Surfaces could be studied intrinsically, as stand-alone spaces, which grew into the theory of manifolds and Riemannian geometry. Later it became clear that geometries without the parallel postulate, the non-Euclidean geometries, could be developed without any contradiction. By the late 19th century, geometers had split the field into subfields defined by their methods or by which properties of Euclidean space they chose to ignore. Today a geometric space, or simply a space, is a mathematical structure on which some geometry is defined. The physical world became just one case among many.

• 3. From Mesopotamia To Athens

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  In the 7th century BC, Thales of Miletus used geometry to calculate the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, deriving four corollaries to the theorem that bears his name. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history. Eudoxus, who lived from 408 to about 355 BC, developed the method of exhaustion, allowing the calculation of areas and volumes of curvilinear figures. He also produced a theory of ratios that sidestepped the problem of incommensurable magnitudes, clearing the way for later geometers to advance. Around 300 BC, Euclid revolutionized the subject. His Elements is widely considered the most successful and influential textbook of all time. It introduced mathematical rigor through the axiomatic method, the earliest example of the definition, axiom, theorem, and proof format still used today. Most of its contents were already known. Euclid's achievement was arranging them into a single coherent logical framework. The Elements was known to all educated people in the West until the middle of the 20th century. Archimedes of Syracuse, who lived from about 287 to 212 BC, used the method of exhaustion to find the area under the arc of a parabola through the summation of an infinite series. He gave remarkably accurate approximations of pi, studied the spiral that bears his name, and found formulas for the volumes of surfaces of revolution.

• 4. Eastern Contributions

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  The Shatapatha Brahmana of the 3rd century BC contains rules for ritual geometric constructions similar to the Sulba Sutras. The Sulba Sutras are described as containing the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They list Pythagorean triples, particular cases of Diophantine equations. The Bakhshali manuscript holds a handful of geometric problems, including problems about the volumes of irregular solids, and employs a decimal place value system with a dot for zero. Aryabhata's Aryabhatiya, written in 499, includes the computation of areas and volumes. Brahmagupta wrote his astronomical work in 628. Its Chapter 12, containing 66 Sanskrit verses, split into basic operations and practical mathematics. The practical section covered mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain. There Brahmagupta stated his famous theorem on the diagonals of a cyclic quadrilateral, gave a formula for the area of a cyclic quadrilateral that generalizes Heron's formula, and described rational triangles completely. In medieval Islam, Al-Mahani, born in 853, conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thabit ibn Qurra, who lived from 836 to 901, worked with arithmetic operations applied to ratios of geometrical quantities. Omar Khayyam, who lived from 1048 to 1131, found geometric solutions to cubic equations. Work by Ibn al-Haytham, Khayyam, and Nasir al-Din al-Tusi on quadrilaterals fed a long line of research on the parallel postulate, carried on through Vitello, Gersonides, John Wallis, and Giovanni Girolamo Saccheri, that by the 19th century led to hyperbolic geometry.

• 5. Coordinates And Curvature

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  In the early 17th century, Rene Descartes, who lived from 1596 to 1650, and Pierre de Fermat, who lived from 1601 to 1665, created analytic geometry, geometry with coordinates and equations. This was a necessary precursor to calculus and to a precise quantitative science of physics. In the same period, Girard Desargues, who lived from 1591 to 1661, systematically studied projective geometry, which examines properties of shapes unchanged under projections and sections, especially as they relate to artistic perspective. The 19th century changed how geometry was studied. Nikolai Ivanovich Lobachevsky, Janos Bolyai, and Carl Friedrich Gauss discovered the non-Euclidean geometries. Felix Klein then placed symmetry, expressed through the notion of a transformation group, at the center of geometry in his Erlangen program, which generalized the Euclidean and non-Euclidean geometries. Two master geometers worked alongside these shifts. Bernhard Riemann, who lived from 1826 to 1866, worked primarily with tools from mathematical analysis and introduced the Riemann surface. Henri Poincare founded algebraic topology and the geometric theory of dynamical systems. The concept of space became rich and varied, a natural background for theories as different as complex analysis and classical mechanics.

• 6. The Building Blocks

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  Euclid defined a point as that which has no part, and a line as breadthless length which lies equally with respect to the points on itself. In modern mathematics, points are generally defined as elements of a set called a space, itself axiomatically defined, and every geometric shape becomes a set of points. This was not so in synthetic geometry, where a line is a fundamental object not viewed as the set of points it passes through. Some modern geometries dispense with points entirely. Alfred North Whitehead formulated point-free geometry in 1919 to 1920, one of the oldest such geometries. A plane, in Euclidean terms, is a flat two-dimensional surface that extends infinitely. A curve is a 1-dimensional object that may be straight or not; in 2-dimensional space these are plane curves, and in 3-dimensional space they are space curves. A surface is two-dimensional, such as a sphere or paraboloid. A solid is three-dimensional, bounded by a closed surface, as a ball is bounded by a sphere. A manifold generalizes curve and surface: in topology, every point has a neighborhood homeomorphic to Euclidean space. Manifolds appear extensively in physics, including general relativity and string theory. Euclid defined a plane angle as the inclination of two lines that meet but do not lie straight with respect to each other. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry. David Hilbert, who lived from 1862 to 1943, employed axiomatic reasoning to provide a modern foundation of geometry, treating congruence as an undefined term whose properties are fixed by axioms.

• 7. A Field Of Many Geometries

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  Differential geometry uses techniques of calculus and linear algebra, with applications in physics, econometrics, and bioinformatics. It carries weight in mathematical physics because of Albert Einstein's general relativity postulation that the universe is curved. Topology generalizes Euclidean geometry and is often described by the saying that topology is rubber-sheet geometry, since its transformations are homeomorphisms. Algebraic geometry studies algebraic sets, the common zeros of multivariate polynomials, and became an autonomous subfield around 1900 with Hilbert's Nullstellensatz. From the late 1950s through the mid-1970s it underwent major foundational development through Alexander Grothendieck's scheme theory, which brought topological methods into a purely algebraic context. One of seven Millennium Prize problems, the Hodge conjecture, is a question in algebraic geometry. Complex geometry first appeared in Bernhard Riemann's study of Riemann surfaces, with later contemporary treatment beginning in the work of Jean-Pierre Serre, who introduced sheaves to the subject. Superstring theory predicts that the extra 6 dimensions of 10 dimensional spacetime may be modeled by Calabi-Yau manifolds. Discrete geometry studies sphere packings, triangulations, and the Kneser-Poulsen conjecture. Computational geometry tackles algorithms for the travelling salesman problem, minimum spanning trees, and hidden-line removal. Geometric group theory produced Agol's proof of the virtually Haken conjecture, combining Perelman geometrization with cubulation techniques. Convex geometry dates back to antiquity, where Archimedes gave the first known precise definition of convexity.

• 8. Geometry In The World

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  Vitruvius developed a complicated theory of ideal proportions for the human figure, concepts later used and adapted by artists from Michelangelo to modern comic book artists. The golden ratio is a particular proportion with a controversial role in art, often claimed to be the most aesthetically pleasing ratio of lengths, though the most reliable examples were made deliberately by artists aware of the legend. Tessellations recur throughout art history, in Islamic art and in the work of M. C. Escher, whose work also drew on hyperbolic geometry. Cezanne advanced the theory that all images can be built from the sphere, the cone, and the cylinder, an idea still used in art theory today. In architecture, it has been said that geometry lies at the core of architectural design, through forced perspective, conic sections in domes, tessellations, and symmetry. Astronomy, especially the mapping of stars and planets on the celestial sphere, has supplied geometric problems throughout history. The discovery of incommensurable lengths once contradicted the philosophical views of the Pythagoreans, who had considered the role of numbers in geometry. That same tension resolved into power. Today algebraic geometry's methods are fundamental in Wiles's proof of Fermat's Last Theorem, a problem stated in elementary arithmetic that remained unsolved for several centuries, its solution reaching all the way back to scheme theory and its extensions such as stack theory.