Geometry (1)

Geometry is an ancient branch of mathematics that deals with the properties of space and the size, shape, and relative position of figures. A mathematician who specializes in this field is known as a geometer. For most of its history, geometry was primarily focused on Euclidean geometry, which includes concepts such as points, lines, planes, angles, surfaces, and curves. However, in the 19th century, several significant discoveries expanded the scope of geometry beyond Euclidean geometry. One of these was the Theorema Egregium by Carl Friedrich Gauss, which showed that the curvature of a surface is independent of its embedding in Euclidean space. This led to the development of the theories of manifolds and Riemannian geometry, which allow surfaces to be studied on their own, rather than in relation to a Euclidean space.

Geometry continued to evolve in the 19th century with the discovery that non-Euclidean geometries, or geometries without the parallel postulate, could be developed without contradiction. This led to the application of non-Euclidean geometry in the field of general relativity. In the modern era, the scope of geometry has been greatly expanded, and the field has been divided into various subfields based on the methods used or the properties of Euclidean spaces that are disregarded. These subfields include differential geometry, algebraic geometry, computational geometry, algebraic topology, and discrete geometry. Geometry has a wide range of applications, including in science, art, architecture, and other fields that involve graphics. It also has connections to seemingly unrelated areas of mathematics, such as in Wiles's proof of Fermat's Last Theorem, which used methods of algebraic geometry to solve a problem stated in terms of elementary arithmetic.

The origins of geometry can be traced back to ancient civilizations in Mesopotamia and Egypt, where it was used for practical purposes such as surveying, construction, and astronomy. The earliest known texts on geometry include the Egyptian Rhind Papyrus and Moscow Papyrus, as well as Babylonian clay tablets like Plimpton 322. In the 7th century BC, the Greek mathematician Thales used 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 in geometry and the discovery of four corollaries to Thales's theorem. The Pythagorean School, founded by Pythagoras, is credited with the first proof of the Pythagorean theorem. Eudoxus developed the method of exhaustion, which allowed for the calculation of areas and volumes of curvilinear figures, and a theory of ratios that avoided the problem of incommensurable magnitudes. Around 300 BC, Euclid's Elements revolutionized geometry with its axiomatic method, introducing rigor and logical structure to the subject. The Elements, which is considered the most successful and influential textbook of all time, is still taught in geometry classes today. Archimedes, who lived in the 3rd century BC, used the method of exhaustion to calculate the area under the arc of a parabola and obtained formulas for the volumes of surfaces of revolution.

Indian mathematicians made significant contributions to geometry, as seen in the Shatapatha Brahmana and the Sulba Sutras, which contain rules for ritual geometric constructions and lists of Pythagorean triples. The Bakhshali manuscript includes geometric problems and employs a decimal place value system with a dot for zero. Aryabhata's Aryabhatiya includes calculations of areas and volumes, and Brahmagupta's work includes a theorem on the diagonals of a cyclic quadrilateral and a formula for the area of a cyclic quadrilateral. In medieval Islam, mathematicians such as Al-Mahani, Thabit ibn Qurra, Omar Khayyam, and Nasir al-Din al-Tusi made contributions to algebraic and analytic geometry, and their work on quadrilaterals had a significant influence on the development of non-Euclidean geometry in Europe.

Two important developments in geometry took place in the early 17th century. The first was the creation of analytic geometry, which introduced the use of coordinates and equations in geometry. This was a crucial precursor to the development of calculus and a precise quantitative understanding of physics. The second development was the systematic study of projective geometry by Girard Desargues, which focuses on the properties of shapes that remain unchanged under projections and sections, particularly as they relate to artistic perspective. In the 19th century, the discovery of non-Euclidean geometries by Lobachevsky, Bolyai, and Gauss, as well as the formulation of symmetry as a central consideration in the Erlangen program by Klein, marked significant changes in the way geometry was studied. These changes were influenced by mathematicians such as Riemann, who used tools from mathematical analysis and introduced the concept of Riemann surfaces, and Poincaré, who founded algebraic topology and developed the geometric theory of dynamical systems. These developments expanded the concept of "space" and created a rich and varied foundation for theories such as complex analysis and classical mechanics.

Euclid's approach to geometry, presented in his influential work the Elements, was characterized by its rigor and abstractness. He introduced axioms, or self-evident properties, of points, lines, and planes and used mathematical reasoning to deduce other properties. This approach, known as axiomatic or synthetic geometry, was the foundation of the field until the discovery of non-Euclidean geometries in the 19th century. This led to a renewed interest in geometry, and in the 20th century, David Hilbert used axiomatic reasoning to attempt to provide a modern foundation for the subject.

Points are considered a fundamental element in geometry and are usually defined as either having no parts or as elements of a space set, which is itself axiomatically defined. In synthetic geometry, lines are also considered fundamental objects that are not viewed as sets of points. However, there are modern geometries that do not rely on points as primitive objects or do not include points at all. One example is Whitehead's point-free geometry, which was formulated by Alfred North Whitehead in the early 20th century.

In modern mathematics, the concept of a line is closely connected to the way a geometry is described. In analytic geometry, a line is often defined as a set of points that satisfy a particular linear equation, but in other contexts, such as incidence geometry, a line may be considered an independent object separate from the set of points it contains. In differential geometry, a geodesic is a generalization of the concept of a line to curved spaces. Euclid defined a line as "breadthless length" that "lies equally with respect to the points on itself."

In Euclidean geometry, a plane is a flat, two-dimensional surface that extends indefinitely. Planes are used in various areas of geometry and can be studied in different ways, such as as a topological surface without reference to distances or angles, as an affine space where collinearity and ratios can be studied but not distances, or as the complex plane using techniques of complex analysis. The definitions for other types of geometries are generalizations of the Euclidean definition.

In Euclidean geometry, an angle is defined as the inclination between two lines in a plane that meet at a common endpoint, called the vertex of the angle. In modern terms, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint. Angles are used to study polygons and triangles and also form an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry. In differential geometry and calculus, the angles between plane curves, space curves, or surfaces can be calculated using the derivative.