Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier,[1] Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems.[2]
"Plane geometry" redirects here. For other uses, see Plane geometry (disambiguation).
The Elements begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.[1]
For more than two thousand years, the adjective "Euclidean" was unnecessary because
Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that theorems proved from them were deemed absolutely true, and thus no other sorts of geometry were possible. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances (relative to the strength of the gravitational field).[3]
Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects. This is in contrast to analytic geometry, introduced almost 2,000 years later by René Descartes, which uses coordinates to express geometric properties by means of algebraic formulas.
Notation and terminology[edit]
Naming of points and figures[edit]
Points are customarily named using capital letters of the alphabet. Other figures, such as lines, triangles, or circles, are named by listing a sufficient number of points to pick them out unambiguously from the relevant figure, e.g., triangle ABC would typically be a triangle with vertices at points A, B, and C.
Complementary and supplementary angles[edit]
Angles whose sum is a right angle are called complementary. Complementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the right angle. The number of rays in between the two original rays is infinite.
Angles whose sum is a straight angle are supplementary. Supplementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the straight angle (180 degree angle). The number of rays in between the two original rays is infinite.
Modern versions of Euclid's notation[edit]
In modern terminology, angles would normally be measured in degrees or radians.
Modern school textbooks often define separate figures called lines (infinite), rays (semi-infinite), and line segments (of finite length). Euclid, rather than discussing a ray as an object that extends to infinity in one direction, would normally use locutions such as "if the line is extended to a sufficient length", although he occasionally referred to "infinite lines". A "line" in Euclid could be either straight or curved, and he used the more specific term "straight line" when necessary.
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Because of Euclidean geometry's fundamental status in mathematics, it is impractical to give more than a representative sampling of applications here.
As suggested by the etymology of the word, one of the earliest reasons for interest in and also one of the most common current uses of geometry is surveying.[20] In addition it has been used in the cognitive and computational approaches to visual perception of objects. Certain practical results from Euclidean geometry (such as the right-angle property of the 3-4-5 triangle) were used long before they were proved formally.[21] The fundamental types of measurements in Euclidean geometry are distances and angles, both of which can be measured directly by a surveyor. Historically, distances were often measured by chains, such as Gunter's chain, and angles using graduated circles and, later, the theodolite.
An application of Euclidean solid geometry is the determination of packing arrangements, such as the problem of finding the most efficient packing of spheres in n dimensions. This problem has applications in error detection and correction.
Geometry is used extensively in architecture.
Geometry can be used to design origami. Some classical construction problems of geometry are impossible using compass and straightedge, but can be solved using origami.[22]
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The triangle angle sum theorem states that the sum of the three angles of any triangle, in this case angles α, β, and γ, will always equal 180 degrees.
The Pythagorean theorem states that the sum of the areas of the two squares on the legs (a and b) of a right triangle equals the area of the square on the hypotenuse (c).
A parabolic mirror brings parallel rays of light to a focus.
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As a description of the structure of space[edit]
Euclid believed that his axioms were self-evident statements about physical reality. Euclid's proofs depend upon assumptions perhaps not obvious in Euclid's fundamental axioms,[37] in particular that certain movements of figures do not change their geometrical properties such as the lengths of sides and interior angles, the so-called Euclidean motions, which include translations, reflections and rotations of figures.[38] Taken as a physical description of space, postulate 2 (extending a line) asserts that space does not have holes or boundaries; postulate 4 (equality of right angles) says that space is isotropic and figures may be moved to any location while maintaining congruence; and postulate 5 (the parallel postulate) that space is flat (has no intrinsic curvature).[39]
As discussed above, Albert Einstein's theory of relativity significantly modifies this view.
The ambiguous character of the axioms as originally formulated by Euclid makes it possible for different commentators to disagree about some of their other implications for the structure of space, such as whether or not it is infinite[40] (see below) and what its topology is. Modern, more rigorous reformulations of the system[41] typically aim for a cleaner separation of these issues. Interpreting Euclid's axioms in the spirit of this more modern approach, axioms 1–4 are consistent with either infinite or finite space (as in elliptic geometry), and all five axioms are consistent with a variety of topologies (e.g., a plane, a cylinder, or a torus for two-dimensional Euclidean geometry).
Treatment of infinity[edit]
Infinite objects[edit]
Euclid sometimes distinguished explicitly between "finite lines" (e.g., Postulate 2) and "infinite lines" (book I, proposition 12). However, he typically did not make such distinctions unless they were necessary. The postulates do not explicitly refer to infinite lines, although for example some commentators interpret postulate 3, existence of a circle with any radius, as implying that space is infinite.[40]
The notion of infinitesimal quantities had previously been discussed extensively by the Eleatic School, but nobody had been able to put them on a firm logical basis, with paradoxes such as Zeno's paradox occurring that had not been resolved to universal satisfaction. Euclid used the method of exhaustion rather than infinitesimals.[42]
Later ancient commentators, such as Proclus (410–485 CE), treated many questions about infinity as issues demanding proof and, e.g., Proclus claimed to prove the infinite divisibility of a line, based on a proof by contradiction in which he considered the cases of even and odd numbers of points constituting it.[43]
At the turn of the 20th century, Otto Stolz, Paul du Bois-Reymond, Giuseppe Veronese, and others produced controversial work on non-Archimedean models of Euclidean geometry, in which the distance between two points may be infinite or infinitesimal, in the Newton–Leibniz sense.[44] Fifty years later, Abraham Robinson provided a rigorous logical foundation for Veronese's work.[45]
Infinite processes[edit]
Ancient geometers may have considered the parallel postulate – that two parallel lines do not ever intersect – less certain than the others because it makes a statement about infinitely remote regions of space, and so cannot be physically verified.[46]
The modern formulation of proof by induction was not developed until the 17th century, but some later commentators consider it implicit in some of Euclid's proofs, e.g., the proof of the infinitude of primes.[47]
Supposed paradoxes involving infinite series, such as Zeno's paradox, predated Euclid. Euclid avoided such discussions, giving, for example, the expression for the partial sums of the geometric series in IX.35 without commenting on the possibility of letting the number of terms become infinite.
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