The earliest geometry was not written in books but carved into clay and measured with ropes. In the second millennium BC, scribes in ancient Mesopotamia and Egypt used practical geometry to solve immediate problems of land surveying, construction, and astronomy. The Moscow Papyrus, dating between 2000 and 1800 BC, contains a formula for calculating the volume of a truncated pyramid, a shape known as a frustum. This was not abstract theory but a tool for building. Babylonian clay tablets from 1900 BC, such as Plimpton 322, reveal that astronomers were already using trapezoid procedures to compute the position and motion of Jupiter within time-velocity space. These geometric procedures anticipated the Oxford Calculators and their mean speed theorem by fourteen centuries. South of Egypt, the ancient Nubians established their own system of geometry, including early versions of sun clocks, proving that the study of space was a global endeavor from the very beginning.
The Greek Revolution
In the seventh century BC, the Greek mathematician Thales of Miletus changed the game by using geometry to solve problems like calculating 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 what is now known as Thales's theorem. Pythagoras established the Pythagorean School, which is credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history. The true revolution arrived around 300 BC when Euclid wrote his Elements. This work, widely considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method. Euclid arranged existing knowledge into a single, coherent logical framework of definition, axiom, theorem, and proof. The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse later used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, giving remarkably accurate approximations of pi.
Eastern Wisdom
While Europe slept, Indian mathematicians made profound contributions to the field. The Shatapatha Brahmana from the third century BC contains rules for ritual geometric constructions similar to the Sulba Sutras. These sutras contain the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples, which are particular cases of Diophantine equations, such as 3, 4, 5 and 5, 12, 13. The best conjecture is that these arithmetic rules were part of religious ritual, as a Hindu home was required to have three fires burning at three different altars. The three altars were to be of different shapes, but all three were to have the same area. These conditions led to certain Diophantine problems, a particular case of which is the generation of Pythagorean triples. In the Bakhshali manuscript, there are a handful of geometric problems including problems about volumes of irregular solids. The manuscript also employs a decimal place value system with a dot for zero. Brahmagupta wrote his astronomical work in 628, stating his famous theorem on the diagonals of a cyclic quadrilateral and providing a formula for the area of a cyclic quadrilateral.
In the early 17th century, two important developments in geometry reshaped the discipline. The first was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes and Pierre de Fermat. This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues. Projective geometry studies properties of shapes which are unchanged under projections and sections, especially as they relate to artistic perspective. Earlier, Omar Khayyam, the tent-maker, had written an Algebra that went beyond that of al-Khwarizmi to include equations of third degree. He believed, mistakenly as the 16th century later showed, that arithmetic solutions were impossible for general cubic equations, hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations. One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra.
The Space Unbound
Two developments in geometry in the 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai, and Carl Friedrich Gauss. Gauss's remarkable theorem asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied intrinsically, that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries without the parallel postulate could be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry. Bernhard Riemann, working primarily with tools from mathematical analysis, introduced the Riemann surface, while Henri Poincaré became the founder of algebraic topology and the geometric theory of dynamical systems. As a consequence of these major changes in the conception of geometry, the concept of space became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics.
The Shape of Art
Geometry has found applications in many fields, with art serving as a primary canvas. The theory of perspective showed that there is more to geometry than just the metric properties of figures, as perspective is the origin of projective geometry. Artists have long used concepts of proportion in design, with Vitruvius developing a complicated theory of ideal proportions for the human figure. These concepts have been used and adapted by artists from Michelangelo to modern comic book artists. The golden ratio is a particular proportion that has had a controversial role in art, often claimed to be the most aesthetically pleasing ratio of lengths. Tilings, or tessellations, have been used in art throughout history. Islamic art makes frequent use of tessellations, as did the art of M. C. Escher. Escher's work also made use of hyperbolic geometry. Cézanne advanced the theory that all images can be built up from the sphere, the cone, and the cylinder. This is still used in art theory today, although the exact list of shapes varies from author to author. Architecture relies on geometry to create forced perspective, construct domes using conic sections, and utilize symmetry.
The Hidden Dimensions
Since the late 19th century, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods. Differential geometry uses techniques of calculus and linear algebra to study problems in geometry, and it has applications in physics, econometrics, and bioinformatics. In particular, differential geometry is of importance to mathematical physics due to Albert Einstein's general relativity postulation that the universe is curved. Algebraic geometry became an autonomous subfield of geometry, with a theorem called Hilbert's Nullstellensatz that establishes a strong correspondence between algebraic sets and ideals of polynomial rings. From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by Alexander Grothendieck of scheme theory. Scheme theory allowed to solve many difficult problems not only in geometry, but also in number theory. Wiles's proof of Fermat's Last Theorem is a famous example of a long-standing problem of number theory whose solution uses scheme theory and its extensions such as stack theory. One of seven Millennium Prize problems, the Hodge conjecture, is a question in algebraic geometry.
The Modern Canvas
Contemporary geometry continues to evolve with complex geometry, discrete geometry, and computational geometry. Complex geometry studies the nature of geometric structures modeled on, or arising out of, the complex plane, and has found applications to string theory and mirror symmetry. The primary objects of study in complex geometry are complex manifolds, complex algebraic varieties, and complex analytic varieties, and holomorphic vector bundles and coherent sheaves over these spaces. Special examples of spaces studied in complex geometry include Riemann surfaces, and Calabi-Yau manifolds, and these spaces find uses in string theory. Discrete geometry is a subject that has close connections with convex geometry, concerned mainly with questions of relative position of simple geometric objects, such as points, lines and circles. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects, with important problems historically including the traveling salesman problem, minimum spanning trees, and hidden-line removal. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, and medical imaging.