Geometry

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Photo by: ojka

The term geometry is derived from the Greek word geometria, meaning "to measure the Earth." In its most basic sense, then, geometry was a branch of mathematics originally developed and used to measure common features of Earth. Most people today know what those features are: lines, circles, angles, triangles, squares, trapezoids, spheres, cones, cylinders, and the like.

Humans have probably used concepts from geometry as long as civilization has existed. But the subject did not become a real science until about the sixth century B.C. At that point, Greek philosophers began to express the principles of geometry in formal terms. The one person whose name is most closely associated with the development of geometry is Euclid (c. 325–270 B.C. ), who wrote a book called Elements. This work was the standard textbook in the field for more than 2,000 years, and the basic ideas of geometry are still referred to as Euclidean geometry.

Elements of geometry

Statements. Statements in geometry take one of two forms: axioms and propositions. An axiom is a statement that mathematicians accept as being true without demanding proof. An axiom is also called a postulate. Actually, mathematicians prefer not to accept any statement without proof. But one has to start somewhere, and Euclid began by listing certain statements as axioms because they seemed so obvious to him that he couldn't see how anyone would disagree.

One axiom is that a single straight line, and only one, can be drawn through two points. Another axiom is that two parallel lines (lines running next to each other like train tracks) will never meet, no matter how far they are extended into space. Indeed, mathematicians accepted these statements as true without trying to prove them for 2,000 years. Statements such as these form the basis of Euclidean geometry.

However, the vast majority of statements in geometry are not axioms but propositions. A proposition is a statement that can be proved or disproved. In fact, it is not too much of a stretch to say that geometry is a branch of mathematics committed to proving propositions.

Proofs. A proof in geometry requires a series of steps. That series may consist of only one step, or it may contain hundreds or thousands of steps. In every case, the proof begins with an axiom or with some proposition that has already been proved. The mathematician then proceeds from the known fact by a series of logical steps to show that the given proposition is true (or not true).

Constructions. A fundamental part of geometric proofs involves constructions. A construction in geometry is a drawing that can be made with the simplest of tools. Euclid permitted the use of a straight edge and a compass only. An example of a straight edge would be a meter stick that contained no markings on it. A compass is permitted in order to determine the size of angles used in a construction.

Many propositions in geometry can be proved by making certain kinds of constructions. For example, Euclid's first proposition was to show that, given a line segment AB, one can construct an equilateral triangle ABC. (An equilateral triangle is one with three equal angles.)

Plane and solid geometry

Euclidean geometry dealt originally with two general kinds of figures: those that can be represented in two dimensions (plane geometry) and those that can be represented in three dimensions (solid geometry). The simplest geometric figure of all is the point. A point is a figure with no dimensions at all. The points we draw on a piece of paper while studying geometry do have a dimension, of course, but that condition is due to the fact that the point must be made with a pencil, whose tip has real dimensions. From a mathematical standpoint, however, the point has no measurable size.

Perhaps the next simplest geometric figure is a line. A line is a series of points. It has dimensions in one direction (length) but in no other. A line can also be defined as the shortest distance between two points. Lines are used to construct all other figures in plane geometry, including angles, triangles, squares, trapezoids, circles, and so on. Since a line has no beginning or end, most of the "lines" one deals with in geometry are actually line segments—portions of a line that do have a limited length.

In general, lines can have one of three relationships to each other. They can be parallel, perpendicular, or at an angle to each other. According to Euclidean geometry, two lines are parallel to each other if they never meet, no matter how far they are extended. Perpendicular lines are lines that form an angle of 90 degrees (a right angle, as in a square or aT) to each other. And two lines that cross each other at any angle other than 90 degrees are simply said to form an angle with each other.

Closed figures. Lines also form closed figures, such as circles, triangles, and quadrilaterals. A circle is a closed figure in which every part of the figure is equidistant (at an equal distance) from some given point called the center of the circle. A triangle is a closed figure consisting of three lines. Triangles are classified according to the sizes of the angles formed by the three lines. A quadrilateral is a figure with four sides. Some common quadrilaterals are the square (in which all four sides are equal), the trapezoid (which has two parallel sides), the parallelogram (which has two pairs of parallel sides), the rhombus (a parallelogram with four equal sides), and the rectangle (a parallelogram with four right- or 90-degree angles).

Solid figures. The basic figures in solid geometry can be visualized as plane figures being rotated through space. Imagine that a circle is caused to rotate around its center. The figure produced is a sphere. Or imagine that a right triangle is rotated around its right angle. The figure produced is a cone.

Area and volume

The fundamental principles of geometry involve statements about the properties of points, lines, and other figures. But one can go beyond those fundamental principles to express certain measurements about such figures. The most common measurements are the length of a line, the area of a plane figure, or the volume of a solid figure. In the real world, length can be determined using a meter stick or yard stick. However, the field of analytic geometry provides a way to determine the length of a line by using principles adapted from geometry.

Mathematical formulas are available for determining the area of any figures in geometry, such as rectangles, squares, various kinds of triangles, and circles. For example, the area of a rectangle is given by the formula A = l · h, where l is the length of the rectangle and h is its height. One can find the areas of portions of solid figures as well. For example, the base of a cone is a circle. The area of the base, then, is A = π · r ² , where π is a constant whose value is approximately 3.1416 and r is the radius of the base. (Pi [π] is the ratio of the circumference of a circle to its diameter, and it is always the same, no matter the size of the circle. The circumference of a circle is its total length around; its diameter is the length of a line segment that passes through the center of the circle from one side to the other. A radius is a line from the center to any point on the circle.)

Formulas for the volume of geometric figures also are available. For example, the volume of a cube (a three-dimensional square) is given by the formula V = s ³ , where s is equal to the length of one side of the cube.

Other geometries

With the growth of the modern science of mathematics, scholars began to ask whether Euclid's initial axioms were necessarily true. That is, would it be possible to imagine a world in which more than one straight line could be drawn through two points. Such ideas often sound bizarre at first. For example, can you imagine two parallel lines that do eventually meet at some point far in the distance? If so, what does the term parallel really mean?

Yet, such ideas have turned out to be very productive for the study of certain special kinds of spaces. They have been given the name non-Euclidean geometries and are used to study certain kinds of mathematical, scientific, and technical problems.