Geometry

Geometry is a major branch of mathematics focused on the properties of space. It examines the distance, shape, size, and relative position of various figures.

A mathematician who specializes in this field is known as a geometer. While it began as a tool to model the physical world, it now supports almost every science. It also plays a vital role in art, architecture, and computer graphics. Beyond these uses, geometry connects to seemingly unrelated mathematical areas. For example, algebraic geometry was essential to Andrew Wiles's proof of Fermat's Last Theorem. This famous problem was stated using simple arithmetic but remained unsolved for centuries.

The core mechanism of classical geometry involves building complex ideas from simple rules. These rules are called axioms, or postulates.

An axiom is a self-evident property that is accepted as true without proof. In his influential work, *Elements*, Euclid used these axioms to create a system of mathematical rigor. He used deductive reasoning to derive theorems from these starting points. This method of definition, axiom, theorem, and proof established the standard format for modern mathematics. By following this logical framework, geometers can prove the properties of points, lines, and planes.

Geometry is divided into many distinct subfields based on their specific methods. Some branches focus on the properties of Euclidean spaces, while others ignore them. For instance, projective geometry looks at the alignment of points but ignores distance and parallelism. Affine geometry is another type that omits the concepts of angle and distance. Finite geometry differs by omitting the concept of continuity. Other specialized areas include differential geometry, algebraic geometry, and computational geometry. There is also algebraic topology and discrete geometry, which is sometimes called combinatorial geometry. This diversity has changed the definition of "space" itself. It no longer just means the three-dimensional physical world, but any mathematical structure where geometry is defined.

The history of geometry stretches back to the 2nd millennium BC in Mesopotamia and Egypt.

Early geometry was a collection of empirical principles used for surveying and construction. The Egyptian Rhind Papyrus and Moscow Papyrus are among the earliest known texts. The Moscow Papyrus even contains a formula for the volume of a truncated pyramid, also called a frustum. In Greece, Thales of Miletus used geometry to calculate the height of pyramids. Pythagoras established a school that is credited with the first proof of the Pythagorean theorem. Later, Euclid revolutionized the field around 300 BC by organizing mathematical knowledge into a coherent logical system.

Many different cultures contributed to the growth of geometric knowledge. Indian mathematicians produced the Śulba Sūtras, which contained early verbal expressions of the Pythagorean Theorem. In the year 628, Brahmagupta wrote about plane figures and the stacking of bricks. During the Middle Ages, Islamic mathematicians made significant progress in algebraic geometry. Al-Mahani worked on reducing geometric problems to algebraic ones. Omar Khayyam discovered geometric solutions to cubic equations. In the 17th century, René Descartes and Pierre de Fermat created analytic geometry. This system used coordinates and equations to describe shapes, which helped lead to the development of calculus.

Major discoveries in the 19th century dramatically expanded the scope of the field. Carl Friedrich Gauss published his *Theorema Egregium*, or "remarkable theorem." This discovery showed that the curvature of a surface is independent of how it is placed in space. This meant that surfaces could be studied intrinsically as stand-alone spaces. This idea led to the theory of manifolds and Riemannian geometry. During this same era, mathematicians like Nikolai Lobachevsky and János Bolyai discovered non-Euclidean geometries. They showed that geometries without the parallel postulate could exist without contradiction.

Today, the applications of geometry are vast and deeply integrated into modern science. Non-Euclidean geometry provides the mathematical foundation for the theory of general relativity. The field has become a bridge between different mathematical disciplines. It connects complex analysis and classical mechanics through the study of various geometric spaces. Whether studying the curvature of a surface or the properties of a digital shape, geometry remains essential. It continues to evolve as new mathematical structures are defined and explored.