Euclidean geometry
Mathematical model of the physical space / From Wikipedia, the free encyclopedia
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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]
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.
The Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was rapidly recognized, with the result that there was little interest in preserving the earlier ones, and they are now nearly all lost.
There are 13 books in the Elements:
Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, for example, "In any triangle, two angles taken together in any manner are less than two right angles." (Book I proposition 17) and the Pythagorean theorem "In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." (Book I, proposition 47)
Books V and VII–X deal with number theory, with numbers treated geometrically as lengths of line segments or areas of surface regions. Notions such as prime numbers and rational and irrational numbers are introduced. It is proved that there are infinitely many prime numbers.
Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. The platonic solids are constructed.
Axioms
Euclidean geometry is an axiomatic system, in which all theorems ("true statements") are derived from a small number of simple axioms. Until the advent of non-Euclidean geometry, these axioms were considered to be obviously true in the physical world, so that all the theorems would be equally true. However, Euclid's reasoning from assumptions to conclusions remains valid independently from the physical reality.[4]
Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath):[5]
- Let the following be postulated:
- To draw a straight line from any point to any point.
- To produce (extend) a finite straight line continuously in a straight line.
- To describe a circle with any centre and distance (radius).
- That all right angles are equal to one another.
- [The parallel postulate]: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
Although Euclid explicitly only asserts the existence of the constructed objects, in his reasoning he also implicitly assumes them to be unique.
The Elements also include the following five "common notions":
- Things that are equal to the same thing are also equal to one another (the transitive property of a Euclidean relation).
- If equals are added to equals, then the wholes are equal (Addition property of equality).
- If equals are subtracted from equals, then the differences are equal (subtraction property of equality).
- Things that coincide with one another are equal to one another (reflexive property).
- The whole is greater than the part.
Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation.[6] Modern treatments use more extensive and complete sets of axioms.
Parallel postulate
To the ancients, the parallel postulate seemed less obvious than the others. They aspired to create a system of absolutely certain propositions, and to them, it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible since one can construct consistent systems of geometry (obeying the other axioms) in which the parallel postulate is true, and others in which it is false.[7] Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the Elements: his first 28 propositions are those that can be proved without it.
Many alternative axioms can be formulated which are logically equivalent to the parallel postulate (in the context of the other axioms). For example, Playfair's axiom states:
- In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the given line.
The "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists.
Methods of proof
Euclidean Geometry is constructive. Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge.[8] In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory.[9] Strictly speaking, the lines on paper are models of the objects defined within the formal system, rather than instances of those objects. For example, a Euclidean straight line has no width, but any real drawn line will have. Though nearly all modern mathematicians consider nonconstructive methods just as sound as constructive ones, Euclid's constructive proofs often supplanted fallacious nonconstructive ones—e.g., some of the Pythagoreans' proofs that involved irrational numbers, which usually required a statement such as "Find the greatest common measure of ..."[10]
Euclid often used proof by contradiction. Euclidean geometry also allows the method of superposition, in which a figure is transferred to another point in space. For example, proposition I.4, side–angle–side congruence of triangles, is proved by moving one of the two triangles so that one of its sides coincides with the other triangle's equal side, and then proving that the other sides coincide as well. Some modern treatments add a sixth postulate, the rigidity of the triangle, which can be used as an alternative to superposition.[11]
Naming of points and figures
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
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
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.
- 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).
- Thales' theorem states that if AC is a diameter, then the angle at B is a right angle.
Pons asinorum
The pons asinorum (bridge of asses) states that in isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.[12] Its name may be attributed to its frequent role as the first real test in the Elements of the intelligence of the reader and as a bridge to the harder propositions that followed. It might also be so named because of the geometrical figure's resemblance to a steep bridge that only a sure-footed donkey could cross.[13]
Congruence of triangles
Triangles are congruent if they have all three sides equal (SSS), two sides and the angle between them equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26). Triangles with three equal angles (AAA) are similar, but not necessarily congruent. Also, triangles with two equal sides and an adjacent angle are not necessarily equal or congruent.
Triangle angle sum
The sum of the angles of a triangle is equal to a straight angle (180 degrees).[14] This causes an equilateral triangle to have three interior angles of 60 degrees. Also, it causes every triangle to have at least two acute angles and up to one obtuse or right angle.
Pythagorean theorem
The celebrated Pythagorean theorem (book I, proposition 47) states that in any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
Thales' theorem
Thales' theorem, named after Thales of Miletus states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Cantor supposed that Thales proved his theorem by means of Euclid Book I, Prop. 32 after the manner of Euclid Book III, Prop. 31.[15][16]
Scaling of area and volume
In modern terminology, the area of a plane figure is proportional to the square of any of its linear dimensions, , and the volume of a solid to the cube, . Euclid proved these results in various special cases such as the area of a circle[17] and the volume of a parallelepipedal solid.[18] Euclid determined some, but not all, of the relevant constants of proportionality. E.g., it was his successor Archimedes who proved that a sphere has 2/3 the volume of the circumscribing cylinder.[19]
Euclidean geometry has two fundamental types of measurements: angle and distance. The angle scale is absolute, and Euclid uses the right angle as his basic unit, so that, for example, a 45-degree angle would be referred to as half of a right angle. The distance scale is relative; one arbitrarily picks a line segment with a certain nonzero length as the unit, and other distances are expressed in relation to it. Addition of distances is represented by a construction in which one line segment is copied onto the end of another line segment to extend its length, and similarly for subtraction.
Measurements of area and volume are derived from distances. For example, a rectangle with a width of 3 and a length of 4 has an area that represents the product, 12. Because this geometrical interpretation of multiplication was limited to three dimensions, there was no direct way of interpreting the product of four or more numbers, and Euclid avoided such products, although they are implied, for example in the proof of book IX, proposition 20.
Euclid refers to a pair of lines, or a pair of planar or solid figures, as "equal" (ἴσος) if their lengths, areas, or volumes are equal respectively, and similarly for angles. The stronger term "congruent" refers to the idea that an entire figure is the same size and shape as another figure. Alternatively, two figures are congruent if one can be moved on top of the other so that it matches up with it exactly. (Flipping it over is allowed.) Thus, for example, a 2x6 rectangle and a 3x4 rectangle are equal but not congruent, and the letter R is congruent to its mirror image. Figures that would be congruent except for their differing sizes are referred to as similar. Corresponding angles in a pair of similar shapes are equal and corresponding sides are in proportion to each other.
Design and Analysis
- Stress Analysis: Stress Analysis - Euclidean geometry is pivotal in determining stress distribution in mechanical components, which is essential for ensuring structural integrity and durability.
- Gear Design: Gear - The design of gears, a crucial element in many mechanical systems, relies heavily on Euclidean geometry to ensure proper tooth shape and engagement for efficient power transmission.
- Heat Exchanger Design: Heat exchanger - In thermal engineering, Euclidean geometry is used to design heat exchangers, where the geometric configuration greatly influences thermal efficiency. See shell-and-tube heat exchangers and plate heat exchangers for more details.
Dynamics
- Vibration Analysis: Vibration - Euclidean geometry is essential in analyzing and understanding the vibrations in mechanical systems, aiding in the design of systems that can withstand or utilize these vibrations effectively.
- Wing Design: Aircraft Wing Design - The application of Euclidean geometry in aerodynamics is evident in aircraft wing design, airfoils, and hydrofoils where geometric shape directly impacts lift and drag characteristics.
- Satellite Orbits: Satellite Orbits - Euclidean geometry helps in calculating and predicting the orbits of satellites, essential for successful space missions and satellite operations. Also see astrodynamics, celestial mechanics, and elliptic orbit.
CAD Systems
- 3D Modeling: In CAD (computer-aided design) systems, Euclidean geometry is fundamental for creating accurate 3D models of mechanical parts. These models are crucial for visualizing and testing designs before manufacturing.
- Design and Manufacturing: Much of CAM (computer-aided manufacturing) relies on Euclidean geometry. The design geometry in CAD/CAM typically consists of shapes bounded by planes, cylinders, cones, tori, and other similar Euclidean forms. Today, CAD/CAM is essential in the design of a wide range of products, from cars and airplanes to ships and smartphones.
- Evolution of Drafting Practices: Historically, advanced Euclidean geometry, including theorems like Pascal's theorem and Brianchon's theorem, was integral to drafting practices. However, with the advent of modern CAD systems, such in-depth knowledge of these theorems is less necessary in contemporary design and manufacturing processes.
Circuit Design
- PCB Layouts: Printed Circuit Board (PCB) Design utilizes Euclidean geometry for the efficient placement and routing of components, ensuring functionality while optimizing space. Efficient layout of electronic components on PCBs is critical for minimizing signal interference and optimizing circuit performance.
Electromagnetic and Fluid Flow Fields
- Antenna Design: Antenna Design - Euclidean geometry of antennas helps in designing antennas, where the spatial arrangement and dimensions directly affect antenna and array performance in transmitting and receiving electromagnetic waves.
- Field Theory: Complex Potential Flow - In the study of inviscid flow fields and electromagnetic fields, Euclidean geometry aids in visualizing and solving potential flow problems. This is essential for understanding fluid velocity field and electromagnetic field interactions in three-dimensional space. The relationship of which is characterized by an irrotational solenoidal field or a conservative vector field.
Controls
- Control System Analysis: Control Systems - The application of Euclidean geometry in control theory helps in the analysis and design of control systems, particularly in understanding and optimizing system stability and response.
- Calculation Tools: Jacobian - Euclidean geometry is integral in using Jacobian matrices for transformations and control systems in both mechanical and electrical engineering fields, providing insights into system behavior and properties. The Jacobian serves as a linearized design matrix in statistical regression and curve fitting; see non-linear least squares. The Jacobian is also used in random matrices, moment, statistics, and diagnostics.
Because of Euclidean geometry's fundamental status in mathematics, it is impractical to give more than a representative sampling of applications here.
- A surveyor uses a level
- Sphere packing applies to a stack of oranges.
- A parabolic mirror brings parallel rays of light to a focus.
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 in art and architecture.
- The water tower consists of a cone, a cylinder, and a hemisphere. Its volume can be calculated using solid geometry.
- Geometry can be used to design origami.
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]