Euclidean Geometry PDF

Euclidean Geometry The term "Geometry" originated from the Greek word geometrein ("geo" means "Earth", and "metrein" means "to measure"). Euclid of Alexandria is regarded as the father of Geometry due to his work The Elements, which is a thirteen-book compilation of ground- breaking discoveries, the...

Euclidean Geometry The term "Geometry" originated from the Greek word geometrein ("geo" means "Earth", and "metrein" means "to measure"). Euclid of Alexandria is regarded as the father of Geometry due to his work The Elements, which is a thirteen-book compilation of ground- breaking discoveries, theorems, and postulates contributing a lot to the Geometry we study in the present times. Many aspects of The Elements have helped greatly in solving mathematical problems in the past and in the present, most especially the definitions and postulates provided by Euclid. The Definitions A point is that which has no part. A line is breadthless length. The extremities of a line are points. A straight line is a line which lies evenly with the points on itself. A surface is that which has length and breadth only. The extremities of a surface are lines. A plane surface is a surface which lies evenly with the straight lines on itself. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. And when the lines containing the -angle are straight, the angle is called rectilinear. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. An obtuse angle is an angle greater than a right angle. An acute angle is an angle less than a right angle. A boundary is that which is an extremity of anything. A figure is that which is contained by any boundary or boundaries. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another. And the point is called the center of the circle. A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle. Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute. Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right- angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia. The Postulates To draw a straight line from any point to any point. To produce a finite straight line continuously in a straight line. To describe a circle with any center and distance. That all right angles are equal to one another. 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 are the angles less than the two right angles (Alt: for any line and any point not on that line, there is exactly one line parallel to the given line that passes through the given point). The Common Notions Things which are equal to the same thing are also equal to one another. If equals be added to equals, the wholes are equal. If equals be subtracted from equals, the remainders are equal. Things which coincide with one another are equal to one another. The whole is greater than the part. Non-Euclidean Geometry Non-Euclidean geometry refers literally to any geometry that is not the same as Euclidean geometry. Although the term is frequently used to refer only to hyperbolic geometry, common usage includes those few geometries (hyperbolic and spherical) that differ from but are very close to Euclidean geometry. The non-Euclidean geometries developed along two different historical threads. The first thread started with the search to understand the movement of stars and planets in the apparently hemispherical sky. For example, Euclid wrote about spherical geometry in his astronomical work Phaenomena. In addition to looking to the heavens, the ancients attempted to understand the shape of the Earth and to use this understanding to solve problems in navigation over long distances (and later for large-scale surveying). These activities are aspects of spherical geometry. The second thread started with the fifth postulate in Euclid's Elements. For 2,000 years following Euclid, mathematicians attempted either to prove the postulate as a theorem (based on the other postulates) or to modify it in various ways. These attempts culminated when the Russian Nikolai Lobachevsky (1829) and the Hungarian Jånos Bolyai (1831) independently published a description of a geometry that, except for the parallel postulate, satisfied all of Euclid's postulates and common notions. It is this geometry that is called hyperbolic geometry. Spherical Geometry From early times, people noticed that the shortest distance between two points on Earth were great circle routes. Great circles are the "straight lines" of spherical geometry. This is a consequence of the properties of a sphere, in which the shortest distances on the surface are great circle routes. Such curves are said to be "intrinsically" straight. (Note, however, that intrinsically straight and shortest are not necessarily identical.) Three intersecting great circle arcs form a spherical triangle; while a spherical triangle must be distorted to fit on another sphere with a different radius, the difference is only one of scale. Hyperbolic Geometry The first description of hyperbolic geometry was given in the context of Euclid's postulates, and it was soon proved that all hyperbolic geometries differ only in scale (in the same sense that spheres only differ in size). In 1901 the German mathematician David Hilbert proved that it is impossible to define a complete hyperbolic surface using real analytic functions. However, in 1955 the Dutch mathematician Nicolaas Kuiper proved the existence of a complete hyperbolic surface, and in the 1970s the American mathematician William Thurston described the construction of a hyperbolic surface. Elliptic Geometry The consequences of the hyperbolic axiom had been thoroughly explored before the systematic study of elliptic geometry began. As with hyperbolic geometry, an axiomatic system for elliptic geometry is obtained from Euclid's geometry by replacing the fifth postulate (in the form of Playfair's axiom) with a negation. In this case the negation is known as the elliptic axiom: 'two lines always intersect.' Playfair's Axiom In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate). Euclidean Geometry: Through a given point not on a given line, exactly one parallel can be drawn to a given line. Thus, Euclid's geometry can be said to be based on Postulates 1 through 4 and Playfair's axiom. Playfair's Axiom for Non-Euclidean Geometries Non-Euclidean geometry, on the other hand, is based on Euclid's Postulates 1 through 4 and a negation of Playfair's axiom. The two possible negations of Playfair's axiom given here lead to two vastly different non-Euclidean geometries: hyperbolic and elliptic (for elliptic geometry, a modification of Postulate 2 must also be made). Hyperbolic Axiom: Through a given point, not on a given line, at least two lines can be drawn that do not intersect the given line. Elliptic Axiom: Two lines always intersect. Corollaries for Hyperbolic Geometry A corollary is a proposition that follows from (and is often appended to) one already proved. There are several corollaries for hyperbolic geometry: Two lines with a common perpendicular are ultra parallel. The base and summit of a Saccheri quadrilateral are ultraparallel. The sum of the angles of a quadrilateral is less than four right angles. Two lines cannot have more than one common perpendicular. There do not exist lines that are everywhere equidistant. Properties of Single Elliptic A line does not separate the plane. There is at least one line through each pair of points. Each pair of lines meets in exactly one point. On a given line, corresponding to each point there is an opposite point on the line. All lines have the same length Tk. All lines perpendicular to any given line go through the same point. The distance from this point to any point of the given line is TIC/ 2. The point is called the pole of the given line and the line is called the polar of the point. All the lines through a point are perpendicular to the polar of that point. There exists a unique perpendicular to a given line through a given point if and only if the point is not the pole of the given line. The summit angles of a Saccheri quadrilateral are congruent and obtuse. The angle sum of every triangle exceeds 1800. The area of a triangle is given by: area(AABC) = k2(mLABC + mLBCA + mLCAB - 1800) Properties of Double Elliptic A line separates the plane. There is at least one line through each pair of points. Each pair of lines meets in exactly two points. There is a positive constant k such that the distance between two points never exceeds 7Tk. Two points at the maximum distance are called opposite points. All lines have the same length, 27Tk. Corresponding to each point there is a unique opposite point. Two points lie on a unique line if and only if the points are not opposite. All the lines through a given point also pass through the point opposite the given point. All the lines perpendicular to any given line meet in the same pair of opposite points. The distance from each of these points to any point of the given line is TTk/2. Two points that divide a line into equal segments are called opposite points. These two opposite points are called poles of the given line and the line is called the polar of the two points. All the lines through a point are perpendicular to the polar of that point. There exists a unique perpendicular to a given line through a given point if and only if the point is not the pole of the line. The summit angles of a Saccheri quadrilateral are congruent and obtuse. The angle sum of every triangle exceeds 1800. The area of a triangle is given by: area(AABC) = k2(mLABC + mLBCA + mLCAB - 1800) Finite Geometry Geometries which have few axioms and theorems and a definite number of elements that can be named by a counting number are called finite geometries. The first finite geometry to be considered was a three-dimensional geometry. All of the finite geometries in this chapter have point and line as undefined terms. However, the connotation of line is not the same in finite geometry as in ordinary Euclidean geometry since a line in finite geometry cannot have an infinite number of points on it. Projective Geometry Projective geometry is a branch of mathematics that deals with the relationships between geometric figures and the images, or mappings, that result from projecting them onto another surface. Common examples of projections are the shadows cast by opaque objects and motion pictures displayed on a screen. Axioms for Three-point Finite Geometry The first simple finite geometry to be investigated, called a three- point geometry, has only four axioms: There exist exactly three distinct points in the geometry. Two distinct points are on exactly one line. Not all the points of the geometry are on the same line. Two distinct lines are on at least one point. Theorem 1.1: Two distinct lines are on exactly one point. By axiom 4, two distinct lines are on at least one point. Assume that two lines are on more than one point. If two lines m and n lie on points P and Q, then axiom 2 is contradicted, because points P and Q would be on two distinct lines. Theorem 1.2: The three-point geometry has exactly three lines. From axiom 2, each pair of points is on exactly one line. Each possible pair of points is on a distinct line, so the geometry has at least three lines. Suppose there is a fourth line. From axiom 1, there are only three points in the geometry. This fourth line must also be on two of the three points, but this contradicts axiom 2 and theorem 1.1. Axioms for Four-line Finite Geometry The total number of lines is four. Each pair of lines has exactly one point in common. Each point is on exactly two lines. Theorem 1.3: The four-line geometry has exactly six points. By axiom 1, there are 6 pairs of lines. The number six is obtained as the combination of four things taken two at a time. By axiom 2, each pair of lines has exactly one point in common. Suppose that two of these six points are not distinct. That would be a contradiction of axiom 3, because each point would be on more than two lines. Also, by axiom 3, no other point could exist in the geometry other than those six on the pairs of lines. Theorem 1.4: Each line of the four-line geometry has exactly three points on it. By axiom 2, each line of the geometry has a distinct point in common with each of the other three lines, and all three of these points are on the given line. Then by axiom 3, it must also be on one of the other lines. But this is impossible, because the other three lines already determine exactly one point with the given line, and by axiom 2, they can only determine one. Thus, each line of the geometry has exactly three points on it. Axioms for Four-point Finite Geometry The four-point geometry is the dual of four-line geometry. The total number of points in this geometry is four. Each pair of points has exactly one line in common. Each line is on exactly two points. Theorem 1.5: The four-point geometry has exactly six lines. Theorem 1.6: Each point of the four-point geometry has exactly three lines on it. Notice that for the given configurations, interchanging the terms points and lines make the figures still valid. Finite GeometHes of Fano and Young The original finite geometry of Fano was a three-dimensional geometry, but the cross section that is formed by a plane that passed through it yields a plane finite geometry, also called Fano's Geometry. The set of axioms follows: There exist at least one line. Every line of the geometry has exactly three points on it. Not all points of the geometry are on the same line. For two distinct points, there exist exactly one line on both of them. Each two lines have at least one point in common. Theorem 1.7: Each two lines have exactly one point in common. By axiom 5, two lines have at least one point in common. The assumption that they have two distinct points in common violates Axiom 4, because then the two distinct points would have two lines containing both of them. From axioms 1 and 2, there are at least three points in the geometry, while from axiom 3 there is at least a fourth point. By axiom 4, there must be lines joining this fourth point and each of the existing points, and by axioms 4 and 5, there must be lines joining all triple combination of points. Thus, the geometry of Fano contains at least seven points and seven lines. Theorem 1.8: Fano's geometry consists of exactly seven points and seven lines. A different finite geometry can be obtained from Fano by a modification of the last axiom. Young's geometry has the first four axioms of Fano's geometry along with the following substitute axiom for axiom 5. There exist at least one line. Every line of the geometry has exactly three points on it. Not all points of the geometry are on the same line. For two distinct points, there exist exactly one line on both of them. If a point does not lie on a given line, then there is exists exactly one line on that point that does not intersect the given line. The following theorem illustrate Young's geometry: For every point, there is a line not on that point. For every point, there are exactly 4 lines on that point. There are three lines, no two of which intersect. Each line is parallel to exactly two lines. There are exactly 12 lines. There are exactly 9 points. Finite Geometries of Pappus and Desargues Axioms for Finite Geometry of Pappus There exists at least one line. Every line has exactly three points. Not all points are on the same line. There exists exactly one line through a point not on a line that is parallel to the given line. If P is a point not on a line, there exists exactly one point P' on the line such that no line joins P and P'. With the exemption in axiom 5, if P and Q are distinct points, then exactly one line contains both of them. Theorem 1.9 — Theorem of Pappas: If A, B, and C are three distinct points on one line and if A', B', and C' are three different distinct points on a second line, then the intersections of lines AC' and CA', AB' and BA', and BC' and CB' are collinear. Theorem 1.10: Each point in the geometry of Pappus lies on exactly three lines. The Desargues configuration may be generated by drawing two triangles ABC and A'B'C' which are both perspective from a point —that is, there is a point of concurrency where all lines passing through corresponding vertices meet at this point. According to Desargues, two triangles perspective from a point, are also perspective from a line. If two triangles are perspective from a line, then the corresponding sides of the triangles meet at points on this line. Given the figure, determine the point of concurrency and line of perspectivity. Axioms for Finite Geometry of Desargues There exists at least one point. Each point has at least one polar. Every line has at least one pole. Two distinct points are on at most one line. Every line has at least three distinct points on it. Ifa line does not contain a certain point, then there is a point on both the line and any polar of the point. Theorem 1.11: Every line of the geometry of Desargues has exactly one pole. Theorem 1.11: Every point of the geometry of Desargues has exactly one polar. 12 1, 6th Century BCE ~ Pythagoras of Samos: Pythagoras, after studying in Egypt, founded a philosophical school in Magna Gracia (modern-day southern Italy. He viewed geometry as a way to understand the universe's perfection but was more focused on its philosophical implications rather than creating a unified mathematical system 2. **ath Century BCE ~ Euclid* Euclid, a pivotal figure in mathematics, compiled *The Elements", a monumental 13-book work that synthesized al known geometric knowledge of the time. His work emphasized logical rigor, Starting with basic definitions and postulates, and bulding 3 complete system. *The Elements* influenced geametry, algebra, and calculus and remains the second-most republished book after the Bible. A key issue in his work was the **5th Postulate**, which described parallel lines and was much more complex than the others. Euclid avoided using it unless absolutely necessary. 3, **5th Century CE ~ Proclus**: Proclus, a philosopher in Athens, attempted to solve the Sth Postulate but failed, marking the beginning of 2,000 years of attempts to simplify or prove it. His failure occurred during time when Western Europe was collapsing, with the fall of the Western Roman Empire and the loss of educational institutions, 4. **7th-12th Centuries CE — Islamic Golden Age": During this period, Islamic scholars preserved and expanded upon Greek knowledge. Mathematicians like **Omar Khayyam** and = contributions: **AlKhwarizmi** formalized algebra and solved equations geometrically, intertwining algebra and geometry. +Omar Khayyam** grappled with the Sth Postulate and wrote extensively about I, but his efforts were mare a reworcing of Eucid's original - The Islamic mathematicians advanced number theory by Incorporating irrational numbers nto geometry 5. **12th Century ~ Adelard of Bath™": After the Crusades, **Adelard**, an English scholar, translated Euclid's “Elements from Arabic Into Latin, bringing this knowledge back to Europe. This translation became the foundation for Western mathematical education for the next 600 years **Renaissance and Beyond": The rediscovery of Euclid's works during the Renaissance played a significant role in European intellectual history, setting the stage for the eventual development of non- Euclidean geometry. Later thinkers built on the unresolved Sth Postulate, which was essential in the evolution of modern geometry, Overall, the history of Euclidean geometry spans from ancient Greece through the Islamic Golden Age to Renaissance Europe, with the unresolved Issue of the Sth Postulate pushing centuries of mathematical inquiry The text highlights the contributions of **Pythagoras, Euclid, Proclus, al-Khwarizmi, Omar Khayyam®, and **Adelard of Bath**, all of whom advanced geometry and shaped the field over millennia al-Khwarizmi** made important 34 1. *+12th Century - Adelard of Bath**: Adelard reintroduced *The Elements* to Europe, where it became widely popular among the literate elite. By the mid-15¢h century, the printing press spread Euclid’s work more widely, becoming the second-most printed book after the Bible. In 1570, John Dee's English translation included a pop. up 30 section for polyhedra, making Euclid more accessible to the emerging merchant lass. 17th Century - René Descartes* *: Descartes, after receiving a Jesuit education, pursued adventure and fought a5 a mercenary, learning advanced geometry along the way. His major contributions include the **Cartesian coordinate system®*, which inked algebra and geometry, and his **deductive reasoning®* method, foundational to modern science. Descartes’ new approach to geometry enabled future mathematicians to address natural phenomena using algebraic methods. 17th Century - 1saac Newton®*: Newton but on Descartes’ work and Euclidean geometry to develop calculus, allowing him to solve the long-standing problem of **squaring the circle®. His method of **integration®* breaking a curve into infinite, infinitesimally small rectangles —was a revolutionary approach to calculating areas. under curves. Newton's calculus and his laws of motion (in *Principia*) reshaped physics and mathematics, leading t0 breakthroughs in atomic theory and astronomy. 19th Century - J4nos Bolyai and Nikolay Lobachevsky*": Bolyai and Lobachevsky challenged **Eucid's Sth Postulate*", exploring alternative geometries where parallel lines could curve away from each other. This development was a major departure from Euclidean geometry, introducing * “hyperbolic geometry, where the angles of a triangle sum to less than 180°, and space is not flat but curved. 5. **10th Century - Bernhard Riemann**: Riemann expanded on Bolyal and Lobachevsky's work, proposing that ~*infinitely many non-Euclidean geometries * exist, with curved spaces that could be hyperbolic or spherical His lecture, attended by the legendary Carl Friedrich Gauss, laid the foundation for unifying various geometries, providing asystem to explore spaces where Euclidean laws no longer applied. Riemann's insights became pivotal to modern physics, especialy in the study of curved space, influencing later developments like Einstein's theory of general relativity. “Key Insights" Euclidean geometry, popularized through the printing press, aid the groundwork for mathematical advancements nthe Renaissance and beyond. ~ Descartes and Newton extended geometry into new realms, creating calculus and the Cartesian plane, crucial for scientific progress The challenge to Euclid's Sth Postulate by Bolyai, Lobachevsky, and later Riemann, broke the mold, leading to non Euclidean geometries, with far-reaching Implications for modern mathematics and physics.