Note: the following has been abstracted from the Grolier Encyclopedia.

Geometry

Geometry is a mathematical system that is usually concerned with points, lines, surfaces, and solids. All mathematical systems are based on undefined elements, assumed relations, unproved statements (postulates and assumptions), and proved statements (theorems). Different sets of assumptions give rise to different geometries.

Historical Development

Geometric figures first appeared over 15,000 years ago in both practical and decorative forms such as shapes of buildings and pottery, cave paintings, and decorations on pottery. The word geometry is derived from the Greek words for "Earth" and "measure." In this context geometry was originally used at least 5,000 years ago by Egyptian surveyors who tried to reestablish the boundaries of fields that were obliterated by the annual flooding of the Nile River. Prior to the work (c.600 BC) of the early Greek philosophers, geometry consisted of rules that produced useful, approximate results--but not always accurate by modern standards.

Ancient reports that Thales of Miletus (fl. c.600 BC) developed the first general theorems for geometry are now in dispute. Pythagoras of Samos (fl. c.540 BC) tried to explain all aspects of the universe in terms of counting numbers, which he frequently represented by sets of objects arranged in geometric shapes. For example, there were the triangular numbers 1, 3, 6, 10, 15, . . . , (1/2)n (n + 1); square numbers 1, 4, 9, . . . , nn; and so forth. Magnitudes--measures of quantities that could not be represented by counting numbers--were represented by lengths of line segments. Line segments were used to develop a geometric algebra of magnitudes. Geometric procedures corresponding to most of the usual laws of algebra were developed to solve for the positive roots of linear and quadratic equations with positive coefficients.

Plato (fl. c.400 BC) emphasized geometry in his Academy and used the five regular Polyhedrons to explain the scientific phenomena of the universe. Aristotle, a pupil of Plato, developed the laws of logical reasoning. The mathematics taught in Plato's Academy was structured by Euclid (fl. c.300 BC) into a logical system. For 2,000 years the geometry of Euclid's Elements was assumed to be the one true geometry. Despite doubts about Euclid's parallel postulate, all attempts to derive it from his other four postulates or to develop other geometries without it were futile until the 19th century. The tremendous productivity of mathematicians in the 19th and 20th centuries has led to the development of several types of geometries, some of which are identified below.

Types of Geometries

Euclidean Geometry

The most common geometry is Euclidean Geometry, which appears to explain the universe in which humans live. Applications of it are found in nearly all aspects of daily activity as well as in the development of all industrial and scientific products. One of its postulates assumes the existence of one and only one line that is both parallel to a given line m and contains a given point that is not a point of the line m. The two non- Euclidean geometries were developed from efforts to prove Euclid's parallel postulate and are based on alternatives to it.

Other Approaches

Spherical geometry, the geometry of points on a sphere, can be easily visualized. Because the Earth is approximately spherical in shape, this geometry has practical applications in navigation and surveying. The surface of a given sphere is a two-dimensional space because each point may be located (identified) by two coordinates. For example, points (positions) on the Earth may be identified by latitude and longitude. On this two-dimensional surface the shortest distance between two points is measured along a spherical line called a Great Circle--that is, a circle obtained as the intersection of the sphere and a plane that contains the center of the sphere. Spherical geometry is a study of points and spherical lines under the assumptions imposed by considering the two-dimensional spherical surface as a part of three- dimensional Euclidean space.

Angles formed by spherical lines have the same measures as the angles formed by the planes that determine the spherical lines in the model. Spherical Trigonometry is the study of triangles on a sphere, or spherical triangles.

Transformations of a Euclidean plane onto itself are often considered in terms of translations (sliding motions) and rotations (about a point). Only four two-dimensional geometries exist that have two distinct types of transformations and have lines such that line segments have the properties of intervals of real numbers. If a geometry requires that two distinct lines have at most one common point, then spherical geometry is excluded, and the only geometries possible are Euclidean geometry and the two Non-Euclidean Geometries (elliptic geometry and hyperbolic geometry). As in the case of spherical geometry, each of the non-Euclidean geometries may be studied by using a model in Euclidean space. These models and theorems of Euclidean geometry may be used to derive theorems of each of the non-Euclidean geometries. Each of the three geometries (Euclidean, elliptic, and hyperbolic) is consistent if the set of real numbers is consistent. Also, any one of the three geometries may be the actual geometry of the entire universe.

Coordinates, which were first used by Apollonius of Perga (fl. c.225 BC) to identify points on a conic and developed further by Oresme (fl. c.1360), Rene Descartes (1596-1650), and Pierre de Fermat (1601-65), are used extensively today in the study of geometries. They are usually called Cartesian coordinates, in recognition of Descartes' work on coordinate systems. Because ordered sets of coordinates can be handled algebraically for any number of coordinates, geometries of any number of dimensions may be considered algebraically. Algebraic Geometry is the study of geometry in terms of coordinates and algebraic representations of figures.

After Renaissance artists began to use Perspective in their work, artists and mathematicians used sequences of perspectivities to obtain projectivities and develop Projective Geometry. Efforts to represent three-dimensional figures by plane figures led to the development of Descriptive Geometry.

Archimedes (287-212 BC) used small elements of volume to derive a formula for the volume of a sphere. The use of such small elements (infinitesimals) led to the development of analysis (calculus) and Differential Geometry.

Many geometries may be considered studies of properties that are invariant (unchanged) under a group of transformations. A hierarchy of geometries may be obtained by considering the effect of removing various assumptions from Euclidean geometry. From this point of view, Euclidean geometry is a study of Congruent Figures. Any two congruent figures have the same area and the same shape. If only the requirement that area be preserved is removed (while retaining the requirement that the shape remain the same), the geometry of similar figures (corresponding figures having the same shape) is obtained. If only the requirement that shape be preserved is removed (while requiring that the area remain the same), equiareal geometry (the study of corresponding figures having the same area) is obtained. If both requirements (area and shape) are removed, affine geometry is obtained.

Affine geometry is the one used by artists when a horizon (ideal) line is added to the Euclidean plane. In Euclidean geometry, lines correspond to lines, and parallel lines correspond to parallel lines. In affine geometry, any two lines that have a point of the horizon line in common are regarded as parallel. If this special property is removed and the horizon line is treated as indistinguishable from any other line, no parallel lines exist, and the subject is called projective geometry.

The study of lines that are allowed to correspond to linear continua (curves) is called Topology. Two-dimensional topology is sometimes loosely referred to as "rubber sheet geometry" because it does not allow cutting apart or sealing together, but deformations are permitted. This representation is correct only for a special case of topology but may help to clarify the subject matter of topology. Formally, topology is the study of properties that are invariant under bicontinuous and biunique transformations--that is, the transformations are continuous and unique, and the inverse of each transformation is continuous and unique. Topology is the most general geometry in the hierarchy of the branches of geometries shown in the accompanying figure.

Logical Structure

Each of the 13 books of Euclid's Elements opens with a statement of the definitions required in that book. In the first book the definitions are followed by the assumptions (postulates and common notions) that are to be used. There follows a set of propositions (theorems) with proofs. The logical structure of the exposition of the proofs has influenced all scientific thinking since Euclid's time. This logical structure is essentially as follows: (1) A statement of the proposition. (2) A statement of the given data (usually with a diagram). (3) An indication of the use that is to be made of the data. (4) A construction of any additional lines or figures. (5) A synthetic proof. (6) A conclusion stating what has been done.

Modern refinements of logical procedure have long since shown that Euclid's work needs modification. The recently developed fractal geometry, for example, requires a more abstract, general definition of dimension than Euclid's. Furthermore, mathematicians today think of such Euclidean statements as "a point is that which has no part" and "a line is breadthless length" as descriptions rather than definitions, and they recognize that any logical system must contain some undefined terms. Besides modifying Euclidean definitions they may also come to question basic postulates, or assumptions, of Euclidean geometry. For example, the theories in physics known as relativity theory and quantum mechanics involve complex mathematical structures. In trying to deal with these structures mathematicians have developed a geometry, called noncommutative geometry, in which the basic Commutative Laws of addition and multiplication (a + b = b + a, and a x b = b x a) are not observed. Unlike in Euclidean geometry, that is, the order in which operations take place is of critical importance in noncommutative geometry. The reasons for this lie in the nature of space-time and cannot be gone into here, but noncommutative geometry provides a further example of the ways in which the logical structures of Euclidean geometry are being modified.

More broadly still, mathematicians are examining the basic principles underlying all the different geometries and their particular postulates and theorems. One attempt at an inclusive geometry of this nature is now known as symplectic geometry. The need for modern refinements of Euclidean geometry, however, does not detract from the significance of Euclid's work and its profound influence on the development of mathematics.