Geometry, << jee AHM uh tree, >> is a branch of mathematics that involves studying the shape, size, and position of lines, angles, curves, and figures. The figures studied in geometry include plane (flat) figures, such as circles, triangles, and rectangles; and solid (three-dimensional) figures, such as cubes, cones, and spheres. The name geometry comes from two Greek words meaning earth and to measure. The earliest uses of geometry included measuring lengths and areas of land. Most scholars believe that the ancient Egyptians and Mesopotamians were the first people to use geometry extensively.
Geometric shapes fill the world around us. For example, honey bees build their honeycombs in a pattern of hexagons, and earthworms are shaped like cylinders. Most houses and buildings have walls shaped like rectangles, and many bridges have supports shaped like triangles. By knowing something about geometry, we can better understand and appreciate our world.
Geometry has many practical uses. To construct stable and attractive buildings, architects and carpenters, for example, must understand the characteristics of geometric objects. Navigators of airplanes, ships, and spacecraft rely on geometric ideas to chart and follow a set course. Artists, designers, engineers, and photographers also use geometric principles in their work.
The study of geometry
Mathematicians have developed geometry as a logical system. In such a system, people use a kind of thinking called deductive reasoning to develop new principles based on previous ideas. The principles of geometry form a set of rules and conclusions called an axiomatic system.
Deductive reasoning
begins with statements that are already accepted as true. These truths are then combined in a logical way to reach a conclusion. When the original statements are true, correct deductive reasoning always leads to true conclusions.
Suppose we wish to know the sum of the measures of the angles of a quadrilateral (four-sided figure). We might already know that two statements are true: (1) Any quadrilateral can be divided into two triangles, and (2) The sum of the measures of the angles of any triangle is 180°. Using deductive reasoning, we can combine the two facts to show that the sum of the measures of the angles of any quadrilateral must equal 360°—that is, twice the sum of the measures of the angles of a triangle, or 2 × 180°. In quadrilateral ABCD below, the measure of angle 1 (symbol, m∠1) + m∠2 + m∠3 = 180°, and m∠4 + m∠5 + m∠6 = 180°. Thus, m∠1 + m∠2 + m∠3 + m∠4 + m∠5 + m∠6 = 360°. Because we have reasoned logically from known facts, we can be certain that our conclusion is true.
Deductive reasoning is the kind of thinking most frequently used in geometry. Less often, another type of reasoning, called inductive reasoning, can help mathematicians develop new ideas about geometry. A person using the inductive method starts by observing particular facts and tries to determine a general rule or principle that explains all the facts. For an explanation of inductive reasoning, see Inductive method .
Axiomatic system.
Geometry is based on statements called axioms that are accepted as true. From these truths, we can use deductive reasoning to prove statements about classes of things. In geometry, these statements, called theorems, concern geometric figures. Terms, axioms, and theorems form the basis of the study of geometry.
Terms
enable people to understand and express geometric ideas. Scholars divide geometric terms into two categories: undefined and defined.
Undefined terms, such as point, line, and plane, form the basic building blocks of geometry. We can imagine these structures in our minds, but we cannot adequately describe them using pictures or other geometric terms. For example, in geometry, a point has a position in space, but no size. A line has length, but no width. On paper, however, a dot representing a point must have some size, and even the finest line has width. Similarly, a plane has length and width, but no depth. Even the thinnest piece of paper, however, has depth.
Defined terms are terms that can be described using undefined terms. For example, line segment AB (symbol line over letters AB), shown below, can be defined as the set of points A and B and all points between A and B on line AB (symbol left-right arrow over letters AB). Similarly, ray AB (symbol right arrow over letters AB) can be defined as the part of line AB that contains point A and all points on the same side of the line as B.
Axioms,
also called postulates, are statements that are assumed to be true and are therefore accepted without proof. An example of an axiom is the statement For every pair of distinct (different) points, there is exactly one line that contains them. The Greek mathematician Euclid developed and organized the first set of geometric axioms about 300 B.C. in his book called the Elements. Since then, mathematicians have developed improved sets of axioms. Today, most high school geometry textbooks use axioms based on a set introduced in 1932 by George D. Birkhoff, an American mathematician.
Theorems
are statements that can be proved true using deductive reasoning. People who study geometry use a step-by-step procedure to prove a theorem. Each step involves a reference to a definition, an axiom, a previously proven theorem, or some other information already given.
An example of a theorem is the statement, discussed previously, that the sum of the measures of the angles of a quadrilateral is 360°. In proving this theorem, we would refer to the axiom Any quadrilateral can be divided into two triangles and to the previously proven theorem The sum of the measures of the angles of any triangle is 180 degrees.
Properties of geometric figures
Geometric axioms and theorems help people understand the characteristics or properties of geometric figures. Two of the most commonly studied properties, congruence and similarity, describe a special relationship between two distinct figures.
Congruence.
Many important axioms and theorems in geometry deal with facts about congruent figures. Congruent figures are figures that have the same size and shape. Congruence serves as an important concept in many areas of our lives. In the mass production of automobiles, for example, manufacturers must make sure that all the front bumpers for a certain model of car are congruent. If they are not congruent, workers cannot efficiently assemble the front of the automobiles.
The simplest types of congruent figures include congruent line segments and congruent angles. Because all line segments have the same shape, congruent line segments are defined as those that have the same length. Two angles are congruent if they have the same measure. In the illustration below, angle CED is congruent to angle XYZ because each angle measures 45°. To express this congruence symbolically, we write ∠CED ≅ ∠XYZ.
To establish the congruence of two triangles, we must set up a correspondence between them—that is, we must match them up so that we can compare their sides and angles. We can do this by pairing the triangles’ sides and vertexes, the points where two sides meet, in such a way that the matching angles and sides are congruent. Suppose we know that in triangle ABC (symbol ΔABC) and ΔEDF, ∠B ≅ ∠D, ∠C ≅ ∠F, ∠A ≅ ∠E, line over letters BC ≅ line over letters DF, line over letters AB ≅ line over letters ED, and line over letters AC ≅ line over letters EF. We could then conclude that triangles ABC and EDF are congruent (ΔABC ≅ ΔEDF).
We do not always need to know that all the pairs of sides and angles of two triangles are congruent to prove that the triangles are congruent. Certain axioms and theorems specify conditions that are sufficient to prove that the two triangles are congruent. For example, the side-angle-side axiom states that If two sides of one triangle and the angle included between them are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
People can use geometry to describe the congruence of figures other than triangles. However, most work with congruence in geometry focuses on triangles.
Similarity
is another way that two geometric figures can be related. Mathematicians consider two figures to be similar if their corresponding angles are congruent and their corresponding sides are proportional. The sides of two figures are said to be proportional when the ratios of the lengths of their corresponding sides are equal. Mathematicians call the value of this ratio the constant of proportionality.
The triangles ABC and HJK below are examples of similar figures. Notice that line over letters HJ measures 10 units, and is twice as long as line over letters AB, which measures 5 units. Symbolically, we write this relationship m(line over letters HJ) = 2m(line over letters AB). Furthermore, m(line over letters HK) = 2m(line over letters AC) and m(line over letters JK) = 2m(line over letters BC). Finally, ∠A ≅ ∠H, ∠C ≅ ∠K, and ∠B ≅ ∠J. In other words, the corresponding angles are congruent, and the corresponding sides are proportional. To express the similarity between triangles ABC and HJK, we write ΔABC ∼ ΔHJK. For triangles ABC and HJK, the constant of proportionality is 2.
The notion of similarity has many important uses. For example, maps and scale drawings are based on the idea of similarity, as are the reduction and enlargement of drawings and photographs.
Some basic geometric constructions
Since the beginnings of geometry, people have studied methods of constructing geometric figures. The ancient Greeks established the practice of using only a compass and an unmarked ruler or straightedge. We can make the following constructions using only those instruments.
Bisecting an angle.
Suppose you want to bisect (divide into two equal parts) angle AOC, below. Place the point of a compass on point O and draw an arc that intersects the sides of the angle at points X and Y. Extend the width of the compass to a length greater than half the distance from X to Y. Place the compass point on X and draw an arc inside ∠AOC. Repeat, with the compass point on Y. The point P where the two arcs cross is called the point of intersection. Draw ray OP. This ray bisects ∠AOC, creating two congruent angles, ∠POA and ∠POC.
Bisecting a line segment.
To bisect line segment AB, below, set the compass width at greater than half the length of line over letters AB. Put the compass point on point A and draw an arc above and below line over letters AB. Move the compass point to point B and repeat the drawing of two arcs to create the points of intersection X and Y. Draw line XY. The point where left-right arrow over letters XY intersects line over letters AB, called point Z, bisects line over letters AB. Furthermore, left-right arrow over letters XY is perpendicular to line over letters AB.
Drawing a perpendicular to a line.
Suppose you want to construct a perpendicular to line AB from a point on the line (point P). Put the compass point on P and draw arcs that intersect left-right arrow over letters AB on either side of P (at points E and F). Using E and F as end points, bisect line over letters EF according to the directions given above. The bisecting line, left-right arrow over letters XY, is perpendicular to left-right arrow over letters AB at point P.
To construct a perpendicular to line AB from a point not on left-right arrow over letters AB (point Q), place the compass point on Q and draw an arc that intersects left-right arrow over letters AB (see the diagram below). The arc will intersect left-right arrow over letters AB at two points (E and F). Next, bisect line over letters EF to find the midpoint (M) of line over letters EF. left-right arrow over letters QM is perpendicular to left-right arrow over letters AB.
Types of geometry
The study of geometry can be approached in a number of ways. Mathematicians call geometry based on Euclid’s axioms Euclidean geometry. Non-Euclidean geometries rely on different sets of axioms. Analytic geometry uses the same axioms as Euclidean geometry, but it employs algebraic methods in working with geometric figures. Mathematicians call all geometries that do not use algebraic methods synthetic geometries.
Euclidean geometry
employs the axioms developed by Euclid in the Elements and theorems later derived from Euclid’s axioms. Euclidean geometry includes plane geometry and solid geometry.
Plane geometry
involves the study of two-dimensional figures, such as lines, angles, circles, and polygons. A polygon is a closed figure with three or more sides made up of line segments. Triangles, quadrilaterals, and hexagons are all polygons.
Plane geometry deals largely with axioms and theorems that describe the properties of triangles, quadrilaterals, and circles. For example, the method we used to construct a perpendicular through a point on a given line relies on the properties of a kind of quadrilateral called a rhombus. A rhombus is a quadrilateral with four sides of equal length. If we draw line segments connecting, in order, E, X, F, and Y, we can see that these points form the vertexes of a rhombus. A theorem of plane geometry states The diagonals of a rhombus are perpendicular to each other. Diagonals are line segments that connect the opposite vertexes of a quadrilateral. Line segments EF and XY form the diagonals of rhombus EXFY. Because we know that a rhombus’s diagonals are perpendicular to each other, we can be sure that we have constructed perpendicular lines.
A knowledge of plane geometry helps when building or making things. Imagine you are making a picture frame from four boards, and you want to make sure it is a perfect rectangle. You might start by drawing a picture of a rectangle like rectangle WXYZ shown below.
Plane geometry defines a rectangle as a quadrilateral whose pairs of opposite sides are congruent and parallel and whose four angles all measure 90°. This means that line over letters WX ≅ line over letters ZY and line over letters XY ≅ line over letters WZ. Likewise, since all four angles measure 90°, ∠X ≅ ∠Y. This information enables us to use the side-angle-side axiom to show that ΔWXY ≅ ΔZYX. And, because the corresponding parts of congruent triangles are congruent, we know that line over letters ZX ≅ line over letters WY.
We have used deductive reasoning to prove the theorem The diagonals of a rectangle are congruent. So, we can check the picture frame by measuring the distance between its pairs of opposite corners. If both distances are equal, we can be sure that the frame is a perfect rectangle. In fact, when carpenters are constructing a rectangular shape with four boards, they sometimes measure the diagonals to confirm that the angles at the corners measure 90°.
Solid geometry
involves the study of geometric figures in three dimensions. These figures may be two-dimensional figures, such as lines and planes, that have been arranged in three-dimensional ways, or they can be three-dimensional objects, such as spheres and cubes.
Solid geometry uses special terms to describe two- dimensional figures arranged in three-dimensional ways. For example, intersecting planes meet at an angle called a dihedral angle. The line where the two planes meet is referred to as the edge of the dihedral angle. The edge resembles the vertex of an angle in plane geometry. In the figure below, planes I and J intersect to form a dihedral angle with edge left-right arrow over letters WY.
To measure the dihedral angle, we must first select a ray in each plane that is perpendicular to the edge. Both rays must share a common end point along the edge. The measure of the angle between these rays is the measure of the dihedral angle. right arrow over letters YP lies in plane J and is perpendicular to edge left-right arrow over letters WY. right arrow over letters YZ lies in plane I, is perpendicular to edge left-right arrow over letters WY, and shares end point Y with right arrow over letters YP. Thus, the measure of this dihedral angle is equal to m∠ZYP.
Solid geometry also deals with the study of closed three-dimensional objects, such as pyramids, prisms, spheres, cylinders, and cones. Mathematicians call these objects solids because each one could be made out of a solid piece of a material, such as wood or metal. Many solids, such as prisms and pyramids, have surfaces made up entirely of flat polygons. These solids are known as polyhedrons. Each of the polygons is called a face. Like intersecting planes, two faces meet at an edge. Mathematicians call any point where three or more faces meet a vertex of the polyhedron.
In the 1700’s, the Swiss mathematician Leonhard Euler discovered an interesting relationship between the number of edges (E), vertexes (V), and faces (F) of any polyhedron. This relationship, sometimes called Euler’s polyhedral formula, states that V + F − E = 2. For example, a cube has 8 vertexes, 6 faces, and 12 edges, satisfying the formula (8 + 6 − 12 = 2).
Some of the theorems of solid geometry deal with a type of solids called regular polyhedrons. Mathematicians consider a polyhedron regular if all of its faces are the same regular polygon, all of its edges are congruent, and the same number of polygons meet at each of its vertexes. For example, the icosahedron is a regular polyhedron with 20 faces. Each face is a triangle, all of the edges are congruent, and five triangles meet at each vertex. Two other regular polyhedrons have triangular faces: the tetrahedron, which has four faces, and the octahedron, which has eight faces. The most familiar regular polyhedron, the cube, has six square faces. The 12 faces of a dodecahedron are pentagons (five-sided polygons). The five regular polyhedrons are often called Platonic solids because the ancient Greek philosopher Plato studied them extensively.
The parallel postulate.
All Euclidean geometries use the parallel postulate, also known as Euclid’s fifth postulate or the parallel axiom. One way of stating the parallel axiom is Through a point not on a given line, only one line can be drawn parallel to the given line. For example, in the illustration below, line I is the only line parallel to line AB that can be drawn through point P.
Non-Euclidean geometry.
During Euclid’s time, and for centuries thereafter, mathematicians attempted to prove that the parallel postulate could be derived from Euclid’s other axioms. In the 1800’s, however, mathematicians discovered that the parallel postulate cannot be proved from the other axioms. This discovery led to the creation of non-Euclidean systems of geometry. Mathematicians devised these systems by replacing the parallel postulate with various other axioms.
Hyperbolic geometry replaces the parallel postulate with the axiom Through a point not on a given line, more than one line may be drawn parallel to the given line. One model of hyperbolic geometry defines a plane as a set of points that lie in the interior of a circle. Line is defined as a chord of the circle—that is, any line that divides the circle into two pieces. Parallel lines are defined as lines that never intersect. In the diagram below, therefore, lines l, m, and n are all considered parallel to line AB, even though they all pass through the same point, P. The Russian mathematician Nikolai Lobachevsky is often credited with developing hyperbolic geometry in the early 1800’s. Mathematicians sometimes call hyperbolic geometry Lobachevskian geometry.
Another basic type of non-Euclidean geometry, elliptical geometry, replaces the parallel postulate with the statement Through a point not on a given line, there are no lines that do not intersect the given line. In other words, in elliptical geometry, parallel lines do not exist. One model of elliptical geometry defines line as the great circle of a sphere. A great circle is any circle that divides a sphere into equal halves. Any two such circles on a sphere must intersect. In the sphere at the right, the great circle ABCD intersects the great circle PCQA. The German mathematician Georg Friedrich Bernhard Riemann developed elliptical geometry in the mid-1800’s. Scholars sometimes call this system Riemannian geometry.
Because one important use of geometry is to describe the physical world, we might ask which type of geometry, Euclidean or non-Euclidean, provides the best model of reality. Euclidean geometry better describes familiar geometric figures and principles, such as those used in building, engineering, and surveying. Non-Euclidean geometry, on the other hand, has important applications in the study of math and physics. For example, the German-born American physicist Albert Einstein’s theory of general relativity describes the universe as a four-dimensional structure that can be curved. Non-Euclidean geometry can help physicists understand such a structure. See Relativity .
Analytic geometry
uses the techniques of algebra to study the properties of geometric figures. Analytic geometry deals with the same subjects as Euclidean geometry but provides simpler ways of proving many theorems. It also plays a major role in the study of other mathematical fields, such as trigonometry and calculus.
Analytic geometry uses a set of numbers called coordinates to describe the position of a point on a plane. A point’s coordinates are determined by comparing its position to two number lines. The x-coordinate, called the abscissa, gives the location of the point along the x-axis (horizontal number line). The y-coordinate, called the ordinate, locates the point along the y-axis (vertical number line). This coordinate system is sometimes called the rectangular system or Cartesian system.
A point’s coordinates are written in the form (x,y). For example, the paired coordinates for point A in the figure below are (2,1). This means that point A is two units to the right of the y-axis, and one unit directly above the x-axis. In addition, the figure shows several other points—B, C, and D—and their coordinates.
Using coordinates enables us to write algebraic equations that represent the points that make up geometric figures. For example, the equation 2x + y = 2 has many solutions of the form (x,y), such as (–2,6), (–1,4), (0,2), (1,0), and (2,–2). If we plot these points on a coordinate graph and connect them, we find that they lie on a straight line. A graph of solutions of the equation is shown below. Any point (x,y) that lies on the line has coordinates that satisfy the equation 2x + y = 2, and any pair of numbers (x,y) that satisfy the equation will be a point on the line. Other plane geometric figures have their own equations and can be graphed on a coordinate system. These figures include conic sections. Conic sections are types of curves formed by the intersection of planes and cones. They include the circle, a type of oval called an ellipse, and a special curve called a parabola.
History
Historians do not know the exact origins of geometry. However, records of the ancient Egyptians and Mesopotamians indicate that they knew of some geometric principles as long as 5,000 years ago. The Egyptians developed geometric ideas that could be used to reestablish land boundaries after the annual flooding of the Nile River. They also used geometry to build the pyramids.
The Mesopotamians lived in what is now Iraq. Like the Egyptians, the Mesopotamians focused on the practical uses of geometry, such as the methods of measurement needed for building and surveying. The Mesopotamians also studied ideas later expressed in certain geometric theorems, including the Pythagorean theorem, which relates the length of the sides of a right triangle (see Pythagorean theorem ).
Greek geometry.
Two Greek philosophers who lived during the 500’s B.C.—Thales and Pythagoras—influenced the development of geometry. Historians credit Thales with creating the first deductive proof of a theorem. Pythagoras, often called the father of mathematics, founded a school in which math was studied extensively. He formulated the Pythagorean theorem.
In the 300’s B.C., the Greeks became the first people to study mathematics from a theoretical point of view, and not simply for its practical applications. This change in emphasis came largely from the influence of the philosopher Plato and the students at his Academy. Plato insisted on the use of deductive reasoning in proving geometric theorems. He argued that because the senses can be fooled, a person should use reason, instead of physical diagrams, to prove geometric theorems. Also during the 300’s B.C., the Greek philosopher Aristotle clearly laid out the foundations for an axiomatic system and deductive reasoning.
The ancient Greek mathematician most often associated with geometry is Euclid. Euclid proved many new theorems, but his greatest contribution to geometry was in organizing geometric ideas that already existed. About 300 B.C., he provided in his Elements the classic example of an axiomatic system by defining terms, listing axioms, and then using the axioms to prove hundreds of theorems. During the 200’s B.C., the Greek mathematician Archimedes discovered methods for finding the area and volume of conic figures.
Beginnings of modern geometry.
Historians trace the beginning of modern geometry to the 1600’s. At that time, greater communication arose among mathematicians than there had been since the time of Plato. Two Frenchmen, Rene Descartes and Pierre de Fermat, began working on what later became known as analytic geometry. Descartes spelled out the foundations of analytic geometry in his book Geometrie (1637). Fermat’s approach to geometry more closely resembles modern analytic geometry than Descartes’s approach does. However, because Fermat did not publish any of his work, most scholars credit Descartes with the discovery of analytic geometry.
Rise of non-Euclidean geometry.
In the early 1800’s, three mathematicians—Carl Friedrich Gauss of Germany, Janos Bolyai of Hungary, and Nikolai Lobachevsky of Russia—discovered non-Euclidean geometry independently of one another. In attempting to prove Euclid’s parallel postulate, each of these mathematicians concluded that no such proof was possible. Each then introduced hyperbolic geometry, the first non-Euclidean geometry. However, historians generally credit Lobachevsky for the discovery because of his published work, especially his article “On the Principles of Geometry” (1829).
Non-Euclidean geometry remained separate from traditional geometry until the mid-1800’s. At that time, Georg Friedrich Bernhard Riemann of Germany began treating non-Euclidean geometry as part of the general subject of geometry. In a lecture in 1854, Riemann argued that geometry should be viewed as a study of unspecified objects of any number of dimensions in any number of spaces. His view of geometry as the general study of “curved spaces” later enabled Albert Einstein to develop his theory of general relativity.
Mathematical findings in the 1800’s also led to the development of other approaches to geometry. One of them, transformation geometry, investigates geometric properties that remain unchanged when a geometric figure undergoes certain transformations (changes in position). A type of transformation geometry called topology involves the study of geometric properties that do not change when figures are deformed by bending, stretching, or molding (see Topology ).
In the early 1900’s, research in geometry as an isolated subject began to slow. Instead, research then and throughout the 1900’s increasingly focused on the similarities between geometry and other mathematical disciplines. This approach brought about advances both in geometry and in other fields of mathematics. For example, the field of algebraic geometry emerged as an extension of analytic geometry. While analytic geometry had simply placed geometric objects on the coordinate system, algebraic geometry made connections between classes of algebraic functions and classes of geometric objects. This enabled researchers to study geometric figures by working with their algebraic equivalents.
In the late 1900’s and early 2000’s, research continued to focus on the connections between geometry and other disciplines. Mathematicians combined topology and algebraic geometry with scientific fields, such as particle physics, to gain new insights. Few researchers worked with classic Euclidean geometry, but it remained an important part of mathematics education.
