Test Your Knowledge of Geometry

Geometry: A Branch of Mathematics

• Geometry is a branch of mathematics that studies the properties of space, including distance, shape, size, and relative position of figures.

• Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, was almost exclusively studied until the 19th century.

• Geometry has applications in almost all sciences, art, architecture, and other activities related to graphics, and also in areas of mathematics that are apparently unrelated.

• The scope of geometry has been greatly expanded since the 19th century, and the field has been split into many subfields, such as differential geometry, algebraic geometry, computational geometry, and others.

• The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC.

• Euclid's Elements introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today.

• In the Middle Ages, mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry.

• The 17th century saw the creation of analytic geometry, or geometry with coordinates and equations, and the systematic study of projective geometry.

• The 19th century saw the discovery of non-Euclidean geometries and the formulation of symmetry as the central consideration in the Erlangen programme of Felix Klein.

• Objects such as points, lines, planes, angles, curves, and surfaces are fundamental to building geometry.

• Euclidean geometry is described as a flat, two-dimensional surface that extends infinitely, while other geometries are generalizations of that.

• Surfaces can be described by patches that are assembled by diffeomorphisms or homeomorphisms, or by polynomial equations in algebraic geometry.Geometry: A Comprehensive Overview

• Manifolds are a generalization of the concepts of curve and surface and are used extensively in physics, including in general relativity and string theory.

• Length, area, and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively.

• Metrics measure the distance between points in a space while measures assign a size or measure to sets, following rules similar to those of classical area and volume.

• Congruence and similarity describe when two shapes have similar characteristics and are generalized in transformation geometry.

• Classical geometers paid special attention to constructing geometric objects that had been described in some other way, often using the compass and straightedge.

• Mathematicians and physicists have used higher dimensions for nearly two centuries, including in the configuration space of a physical system, which has a dimension equal to the system's degrees of freedom.

• Symmetry in geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines.

• Topology is the field concerned with the properties of continuous mappings and can be considered a generalization of Euclidean geometry.

• Algebraic geometry studies geometry through the use of concepts in commutative algebra and has applications in many areas, including cryptography and string theory.

• 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.

• Discrete geometry is 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 and has many applications in computer vision, image processing, and computer-aided design.

• Geometric group theory uses large-scale geometric techniques to study finitely generated groups and is closely connected to low-dimensional topology.