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geometry

Branch of mathematics concerned with the properties of space, usually in terms of plane (two-dimensional, or 2D) and solid (three-dimensional, or 3D) figures. The subject is usually divided into pure geometry, which embraces roughly the plane and solid geometry dealt with in Greek mathematician Euclid's Stoicheia/Elements, and analytical or coordinate geometry, in which problems are solved using algebraic methods. A third, quite distinct, type includes the non-Euclidean geometries.

Pure geometry

This is chiefly concerned with properties of figures that can be measured, such as lengths, areas, and angles and is therefore of great practical use. An important idea in Euclidean geometry is the idea of congruence. Two figures are said to be congruent if they have the same shape and size (and area). If one figure is imagined as a rigid object that can be picked up, moved and placed on top of the other so that they exactly coincide, then the two figures are congruent. Some simple rules about congruence may be stated: two line segments are congruent if they are of equal length; two triangles are congruent if their corresponding sides are equal in length or if two sides and an angle in one is equal to those in the other; two circles are congruent if they have the same radius; two polygons are congruent if they can be divided into congruent triangles assembled in the same order.

The idea of picking up a rigid object to test congruence can be expressed more precisely in terms of elementary ‘movements’ of figures: a translation (or glide) in which all points move the same distance in the same direction (that is, along parallel lines); a rotation through a defined angle about a fixed point; a reflection (equivalent to turning the figure over).

Two figures are congruent if one can be transformed into the other by a sequence of these elementary movements. In Euclidean geometry a fourth kind of movement is also studied; this is the enlargement in which a figure grows or shrinks in all directions by a uniform scale factor. If one figure can be transformed into another by a combination of translation, rotation, reflection, and enlargement then the two are said to be similar. All circles are similar. All squares are similar. Triangles are similar triangles if corresponding angles are equal.

Coordinate geometry

A system of geometry in which points, lines, shapes, and surfaces are represented by algebraic expressions. In plane (two-dimensional) coordinate geometry, the plane is usually defined by two axes at right angles to each other, the horizontal x-axis and the vertical y-axis, meeting at O, the origin. A point on the plane can be represented by a pair of Cartesian coordinates, which define its position in terms of its distance along the x-axis and along the y-axis from O. These distances are, respectively, the x and y coordinates of the point.

Lines are represented as equations; for example, y = 2x + 1 gives a straight line, and y = 3x² + 2x gives a parabola (a curve). The graphs of varying equations can be drawn by plotting the coordinates of points that satisfy their equations, and joining up the points. One of the advantages of coordinate geometry is that geometrical solutions can be obtained without drawing but by manipulating algebraic expressions. For example, the coordinates of the point of intersection of two straight lines can be determined by finding the unique values of x and y that satisfy both of the equations for the lines, that is, by solving them as a pair of simultaneous equations. The curves studied in simple coordinate geometry are the conic sections (circle, ellipse, parabola, and hyperbola), each of which has a characteristic equation.

Geometry probably originated in ancient Egypt, in land measurements necessitated by the periodic inundations of the River Nile, and was soon extended into surveying and navigation. Early geometers were the Greek mathematicians Thales, Pythagoras, and Euclid. Analytical methods were introduced and developed by the French philosopher René Descartes in the 17th century. From the 19th century, various non-Euclidean geometries were devised by Carl Friedrich Gauss, János Bolyai, and Nikolai Lobachevsky. These were later generalized by Bernhard Riemann and found to have applications in the theory of relativity.

Elements Euclid wrote Elements in 13 books, 9 of which deal with plane and solid geometry and 4 with number theory. The first 6 books deal with plane geometry (points, lines, triangles, squares, parallelograms, circles, and so on), and includes hypotheses such as Pythagoras' theorem, which Euclid generalized, and the theorem that only one line can be drawn through a given point parallel to another line.

Books 11 to 13 discuss solid geometry, ending with the 5 Platonic solids (the tetrahedron, octahedron, cube, icosahedron, and dodecahedron).

affine geometry analytical geometry Beltrami, Eugenio Bolyai, János Lobachevsky, Nikolai Ivanovich

mathematics Riemann, Georg Friedrich Bernhard scalene volume Whitehead, John Henry Constantine