Lesson 1: Non-Euclidean Geometries PDF

CSTC COLLEGE OF SCIENCES TECHNOLOGY AND COMMUNICATION, INC. c CSTC College Bldg. Gen. Luna St. Maharlika Hi-way, Pob. 3, Arellano Sub. Sariaya Province of Quezon R4A...

CSTC COLLEGE OF SCIENCES TECHNOLOGY AND COMMUNICATION, INC. c CSTC College Bldg. Gen. Luna St. Maharlika Hi-way, Pob. 3, Arellano Sub. Sariaya Province of Quezon R4A Registrar’s Office: 042 3290850 / 042 7192818 CSTC IT Center: 042 7192805 Atimonan Contact Number: 042 7171420 SCHOOL OF TEACHER EDUCATION Instructional Module in HISTORY OF MATHEMATICS Preliminaries I. Lesson Number II. Lesson Title Non-Euclidean Geometries III. Brief Non-Euclidean geometry is a mathematical discipline that Introduction of explores geometrical systems which do not adhere to the rules the Lesson and principles laid out by Euclid in his seminal work, "Elements." It offers alternative ways of understanding space and geometry, where concepts like parallel lines and the sum of angles in a triangle may differ from traditional Euclidean geometry. Non-Euclidean geometry has important applications in modern physics, especially in understanding the curved spacetime of Einstein's theory of general relativity. IV. Lesson Objectives At the end of this lesson, the student should be able to: 1. Understand the historical context and development of non-Euclidean geometry as a departure from Euclid's classical geometric principles. 2. Differentiate between Euclidean, hyperbolic, and elliptic geometries, and grasp the fundamental differences in their axioms and properties. 3. Explore the concept of curvature in non-Euclidean geometries and its implications for parallel lines, angles, and shapes. Lesson Proper I. Getting Started Analyze the picture if it is a Euclidean Geometry,Spherical Geometry and Hyperbolic Geometry.  II. Discussion Euclidean Geometry The geometry with which we are most familiar is called Euclidean geometry. Euclidean geometry was named after Euclid, a Greek mathematician who lived in 300 BC. Most of the theorems which are taught in high schools today can be found in Euclid's 2000 year old book "THE ELEMENTS". The Elements begins with definitions and five postulates. The first three postulates are postulates of construction, for example the first postulate states that it is possible to draw a straight line between any two points. These postulates also implicitly assume the existence of points, lines and circles and then the existence of other geometric objects are deduced from the fact that these exist. There are other assumptions in the postulates which are not explicit. For example it is assumed that there is a unique line joining any two points. Similarly postulates two and three, on producing straight lines and drawing circles, respectively, assume the uniqueness of the objects and the possibility of whose construction is being postulated. The fourth and fifth postulates are of a different nature. Postulate four states that all right angles are equal. This may seem "obvious" but it actually assumes that space is homogeneous - by this we mean that a figure will be independent of the position in space in which it is placed. The famous fifth, or parallel, postulate states that one and only one line can be drawn through a point parallel to a given line. Euclid's decision to make this a postulate led to Euclidean geometry. It was not until the 19th century that this postulate was dropped and non-euclidean geometries were studied. Euclid's five postulates: 1. It is possible To draw a straight line from any point to any point. 2. It is possible To produce (extend) a finite line continuously in any straight line. 3. It is possible To describe a circle with any center and radius. 4. That all right angles equal one another. 5. That if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on the side on which the angles are less than the two right angles. Non-Euclidean Geometry Non-Euclidean Geometry, literally 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. Gauss and Bolyai The first person to understand the problem of the fifth postulate was Gauss. In 1817, after looking at the problem for many years, he had become convinced it was independent of the other four. Gauss then began to look at the consequences of a geometry where this fifth postulate was not necessarily true. He never published his work due to pressure of time, perhaps illustrating Kant’s statement that Euclidean geometry requires the inevitable necessity of thought. As is often the case in mathematics, similar ideas were developed independently by Janos Bolyai. His father, Wolfgang Bolyai, friend of Gauss, had once told Janos, Janos ignored his father’s impassioned plea, however, and worked on the problem himself. Like Gauss, he looked at the consequences of the fifth postulate not being necessary. His major breakthrough was not his work, which had already been done by Gauss, but the fact that he believed that this ‘other’ geometry actually existed. Despite the revolutionary new ideas that were being put forward, there was little public recognition to be had. Lobachevsky Another mathematician, Lobachevsky, worked on the same problems as Gauss and Bolyai but again, despite working at the same time, he knew nothing of their work. Lobachevsky also assumed the fifth postulate was not necessary and from this formed a new geometry. In 1840, he explained how this new geometry would work. Riemann and Klein The next example of what we could now call a ‘non-euclidean’ geometry was given by Riemann. A lecture he gave which was published in 1868, two years after his death, speaks of a ‘spherical’ geometry in which every line through a point P not on a line AB meets the line AB. Here, no parallels are possible. Also, in 1868, Eugenio Beltrami wrote a paper in which he puts forward a model called a ‘pseudo-sphere’. The importance of this model is that it gave an example of the first four postulates holding but not the fifth. From this, it can be seen that non-euclidean geometry is just as consistent as euclidean geometry. In 1871, Klein completed the ideas of non euclidean geometry and gave the solid underpinnings to the subject. He shows that there are essentially three types of geometry: that proposed by Bolyai and Lobachevsky, where straight lines have two infinitely distant points, the Riemann ‘spherical’ geometry, where lines have no infinitely distant points, and Euclidean geometry, where for each line there are two coincident infinitely distant points. Euclid's fifth postulate is c). Saccheri proved that the hypothesis of the obtuse angle implied the fifth postulate, so obtaining a contradiction. Saccheri then studied the hypothesis of the acute angle and derived many theorems of non-Euclidean geometry without realising what he was doing. However he eventually 'proved' that the hypothesis of the acute angle led to a contradiction by assuming that there is a 'point at infinity' which lies on a plane. In 1766 Lambert followed a similar line to Saccheri. However he did not fall into the trap that Saccheri fell into and investigated the hypothesis of the acute angle without obtaining a contradiction. Lambert noticed that, in this new geometry, the angle sum of a triangle increased as the area of the triangle decreased. Legendre spent 40 years of his life working on the parallel postulate and the work appears in appendices to various editions of his highly successful geometry book Eléments de Géométrie Ⓣ. Legendre proved that Euclid's fifth postulate is equivalent to:- The sum of the angles of a triangle is equal to two right angles. Legendre showed, as Saccheri had over 100 years earlier, that the sum of the angles of a triangle cannot be greater than two right angles. This, again like Saccheri, rested on the fact that straight lines were infinite. In trying to show that the angle sum cannot be less than 180° Legendre assumed that through any point in the interior of an angle it is always possible to draw a line which meets both sides of the angle. This turns out to be another equivalent form of the fifth postulate, but Legendre never realised his error himself. Elementary geometry was by this time engulfed in the problems of the parallel postulate. D'Alembert, in 1767, called it the scandal of elementary geometry. The first person to really come to understand the problem of the parallels was Gauss. He began work on the fifth postulate in 1792 while only 15 years old, at first attempting to prove the parallels postulate from the other four. By 1813 he had made little progress and wrote: In the theory of parallels we are even now not further than Euclid. This is a shameful part of mathematics... However by 1817 Gauss had become convinced that the fifth postulate was independent of the other four postulates. He began to work out the consequences of a geometry in which more than one line can be drawn through a given point parallel to a given line. Perhaps most surprisingly of all Gauss never published this work but kept it a secret. At this time thinking was dominated by Kant who had stated that Euclidean geometry is the inevitable necessity of thought and Gauss disliked controversy. Gauss discussed the theory of parallels with his friend, the mathematician Farkas Bolyai who made several false proofs of the parallel postulate. Farkas Bolyai taught his son, János Bolyai, mathematics but, despite advising his son not to waste one hour's time on that problem of the problem of the fifth postulate, János Bolyai did work on the problem. In 1823 János Bolyai wrote to his father saying I have discovered things so wonderful that I was astounded... out of nothing I have created a strange new world. However it took Bolyai a further two years before it was all written down and he published his strange new world as a 24 page appendix to his father's book, although just to confuse future generations the appendix was published before the book itself. Gauss, after reading the 24 pages, described János Bolyai in these words while writing to a friend: I regard this young geometer Bolyai as a genius of the first order. However in some sense Bolyai only assumed that the new geometry was possible. He then followed the consequences in a not too dissimilar fashion from those who had chosen to assume the fifth postulate was false and seek a contradiction. However the real breakthrough was the belief that the new geometry was possible. Gauss, however impressed he sounded in the above quote with Bolyai, rather devastated Bolyai by telling him that he (Gauss) had discovered all this earlier but had not published. Although this must undoubtedly have been true, it detracts in no way from Bolyai's incredible breakthrough. Nor is Bolyai's work diminished because Lobachevsky published a work on non-Euclidean geometry in 1829. Neither Bolyai nor Gauss knew of Lobachevsky's work, mainly because it was only published in Russian in the Kazan Messenger a local university publication. Lobachevsky's attempt to reach a wider audience had failed when his paper was rejected by Ostrogradski. In fact Lobachevsky fared no better than Bolyai in gaining public recognition for his momentous work. He published Geometrical investigations on the theory of parallels in 1840 which, in its 61 pages, gives the clearest account of Lobachevsky's work. The publication of an account in French in Crelle's Journal in 1837 brought his work on non-Euclidean geometry to a wide audience but the mathematical community was not ready to accept ideas so revolutionary.Euclid's fifth postulate is c). Saccheri proved that the hypothesis of the obtuse angle implied the fifth postulate, so obtaining a contradiction. Saccheri then studied the hypothesis of the acute angle and derived many theorems of non-Euclidean geometry without realising what he was doing. However he eventually 'proved' that the hypothesis of the acute angle led to a contradiction by assuming that there is a 'point at infinity' which lies on a plane. In 1766 Lambert followed a similar line to Saccheri. However he did not fall into the trap that Saccheri fell into and investigated the hypothesis of the acute angle without obtaining a contradiction. Lambert noticed that, in this new geometry, the angle sum of a triangle increased as the area of the triangle decreased. Legendre spent 40 years of his life working on the parallel postulate and the work appears in appendices to various editions of his highly successful geometry book Eléments de Géométrie Ⓣ. Legendre proved that Euclid's fifth postulate is equivalent to:- The sum of the angles of a triangle is equal to two right angles. Legendre showed, as Saccheri had over 100 years earlier, that the sum of the angles of a triangle cannot be greater than two right angles. This, again like Saccheri, rested on the fact that straight lines were infinite. In trying to show that the angle sum cannot be less than 180° Legendre assumed that through any point in the interior of an angle it is always possible to draw a line which meets both sides of the angle. This turns out to be another equivalent form of the fifth postulate, but Legendre never realised his error himself. Elementary geometry was by this time engulfed in the problems of the parallel postulate. D'Alembert, in 1767, called it the scandal of elementary geometry. The first person to really come to understand the problem of the parallels was Gauss. He began work on the fifth postulate in 1792 while only 15 years old, at first attempting to prove the parallels postulate from the other four. By 1813 he had made little progress and wrote: In the theory of parallels we are even now not further than Euclid. This is a shameful part of mathematics... However by 1817 Gauss had become convinced that the fifth postulate was independent of the other four postulates. He began to work out the consequences of a geometry in which more than one line can be drawn through a given point parallel to a given line. Perhaps most surprisingly of all Gauss never published this work but kept it a secret. At this time thinking was dominated by Kant who had stated that Euclidean geometry is the inevitable necessity of thought and Gauss disliked controversy. Gauss discussed the theory of parallels with his friend, the mathematician Farkas Bolyai who made several false proofs of the parallel postulate. Farkas Bolyai taught his son, János Bolyai, mathematics but, despite advising his son not to waste one hour's time on that problem of the problem of the fifth postulate, János Bolyai did work on the problem. In 1823 János Bolyai wrote to his father saying I have discovered things so wonderful that I was astounded... out of nothing I have created a strange new world. However it took Bolyai a further two years before it was all written down and he published his strange new world as a 24 page appendix to his father's book, although just to confuse future generations the appendix was published before the book itself. Gauss, after reading the 24 pages, described János Bolyai in these words while writing to a friend: I regard this young geometer Bolyai as a genius of the first order. However in some sense Bolyai only assumed that the new geometry was possible. He then followed the consequences in a not too dissimilar fashion from those who had chosen to assume the fifth postulate was false and seek a contradiction. However the real breakthrough was the belief that the new geometry was possible. Gauss, however impressed he sounded in the above quote with Bolyai, rather devastated Bolyai by telling him that he (Gauss) had discovered all this earlier but had not published. Although this must undoubtedly have been true, it detracts in no way from Bolyai's incredible breakthrough. Nor is Bolyai's work diminished because Lobachevsky published a work on non-Euclidean geometry in 1829. Neither Bolyai nor Gauss knew of Lobachevsky's work, mainly because it was only published in Russian in the Kazan Messenger a local university publication. Lobachevsky's attempt to reach a wider audience had failed when his paper was rejected by Ostrogradski. In fact Lobachevsky fared no better than Bolyai in gaining public recognition for his momentous work. He published Geometrical investigations on the theory of parallels in 1840 which, in its 61 pages, gives the clearest account of Lobachevsky's work. The publication of an account in French in Crelle's Journal in 1837 brought his work on non-Euclidean geometry to a wide audience but the mathematical community was not ready to accept ideas so revolutionary. Spherical geometry From early times, people noticed that the shortest distance between two points on Earth were great circle routes. For example, the Greek astronomer Ptolemy wrote in Geography (c. 150 CE): It has been demonstrated by mathematics that the surface of the land and water is in its entirety a sphere…and that any plane which passes through the centre makes at its surface, that is, at the surface of the Earth and of the sky, great circles. 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, as shown in the figure.) Three intersecting great circle arcs form a spherical triangle (see figure); while a spherical triangle must be distorted to fit on another sphere with a different radius, the difference is only one of scale. In differential geometry, spherical geometry is described as the geometry of a surface with constant positive curvature. There are many ways of projecting a portion of a sphere, such as the surface of the Earth, onto a plane. These are known as maps or charts and they must necessarily distort distances and either area or angles. Cartographers’ need for various qualities in map projections gave an early impetus to the study of spherical geometry.  Riemann sphere Elliptic geometry is the term used to indicate an axiomatic formalization of spherical geometry in which each pair of antipodal points is treated as a single point. An intrinsic analytic view of spherical geometry was developed in the 19th century by the German mathematician Bernhard Riemann; usually called the Riemann sphere (see figure), it is studied in university courses on complex analysis. Some texts call this (and therefore spherical geometry) Riemannian geometry, but this term more correctly applies to a part of differential geometry that gives a way of intrinsically describing any surface. 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 the mid-19th century it was shown that hyperbolic surfaces must have constant negative curvature. However, this still left open the question of whether any surface with hyperbolic geometry actually exists. hyperbolic planeHyperbolic plane, designed and crocheted by Daina Taimina. In 1868 the Italian mathematician Eugenio Beltrami described a surface, called the pseudosphere, that has constant negative curvature. However, the pseudosphere is not a complete model for hyperbolic geometry, because intrinsically straight lines on the pseudosphere may intersect themselves and cannot be continued past the bounding circle (neither of which is true in hyperbolic geometry). In 1901 the German mathematician David Hilbert proved that it is impossible to define a complete hyperbolic surface using real analytic functions (essentially, functions that can be expressed in terms of ordinary formulas). In those days, a surface always meant one defined by real analytic functions, and so the search was abandoned. 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. Such a surface, as shown in the figure, can also be crocheted. models of hyperbolic geometryIn the Klein-Beltrami model for the hyperbolic plane, the shortest paths, or geodesics, are chords (several examples, labeled k, l, m, n, are shown). In the Poincaré disk model, geodesics are portions of circles that intersect the boundary of the disk at right angles; and in the Poincaré upper half-plane model, geodesics are semicircles with their centres on the boundary. In the 19th century, mathematicians developed three models of hyperbolic geometry that can now be interpreted as projections (or maps) of the hyperbolic surface. Although these models all suffer from some distortion—similar to the way that flat maps distort the spherical Earth—they are useful individually and in combination as aides to understand hyperbolic geometry. In 1869–71 Beltrami and the German mathematician Felix Klein developed the first complete model of hyperbolic geometry (and first called the geometry “hyperbolic”). In the Klein-Beltrami model (shown in the figure, top left), the hyperbolic surface is mapped to the interior of a circle, with geodesics in the hyperbolic surface corresponding to chords in the circle. Thus, the Klein-Beltrami model preserves “straightness” but at the cost of distorting angles. About 1880 the French mathematician Henri Poincaré developed two more models. In the Poincaré disk model (see figure, top right), the hyperbolic surface is mapped to the interior of a circular disk, with hyperbolic geodesics mapping to circular arcs (or diameters) in the disk that meet the bounding circle at right angles. In the Poincaré upper half-plane model (see figure, bottom), the hyperbolic surface is mapped onto the half-plane above the x-axis, with hyperbolic geodesics mapped to semicircles (or vertical rays) that meet the x-axis at right angles. Both Poincaré models distort distances while preserving angles as measured by tangent lines.  CSTC COLLEGE OF SCIENCES TECHNOLOGY AND COMMUNICATION, INC. CSTC College Bldg. Gen. Luna St. Maharlika Hi-way, Pob. 3, Arellano Sub. Sariaya Province of Quezon R4A Registrar’s Office: 042 3290850 / 042 7192818 CSTC IT Center: 042 7192805 Atimonan Contact Number: 042 7171420 III. Application(Performance Task 60%) Multiple Choice.Choose the correct answer. 1. IV. Assessment(Written Works-20%)  CSTC COLLEGE OF SCIENCES TECHNOLOGY AND COMMUNICATION, INC. CSTC College Bldg. Gen. Luna St. Maharlika Hi-way, Pob. 3, Arellano Sub. Sariaya Province of Quezon R4A Registrar’s Office: 042 3290850 / 042 7192818 CSTC IT Center: 042 7192805 Atimonan Contact Number: 042 7171420 V. Reflection(Performance Task -60%) For you, what is the significance of studying Non-Euclidean Geometry? VI. References Prepared by: Arlene Joy S. Arco Instructor Reviewed by: VICTOR M. DISILIO, MAEd History of Mathematics Teacher