Fundamentals of Geometry Math 3181 January 10 th, 2013 Instructor: Dr. Franz Rothe Office: Fretwel 345D Phone: 687 4908 Office hours: TR 4 p.m. - 5 p.m. and W 11 a.m. - 1 p.m. and by appointment Best book: Manuscript: Robin Hartshorne Geometry: Euclid and Beyond second printing Springer 2000 ISBN 0-387-98650-2 Franz Rothe Several Topics from Geometry http://math.uncc.edu/ frothe/3181all.pdf Examinations: 2-3 tests Final Exam M, May 6, 2 p.m. - 4.30 p.m. in FRIDAY 383 or: Tu May 7, 2 p.m. - 4.30 p.m. in FRIDAY 381 2-3 Homework sheets some weekly quizzes Attendance and participation Grading Letter grades are only given for the entire course. They are assigned based on the part of the total possible credit (points) you have gained. I intend to use the following scheme: 1000 points - 880 points A 879 points - 760 points B 759 points - 640 points C 639 points - 520 points D otherwise F This table is only intended to be a guideline. It assumes the total of 1000 points. The grading will be adapted to the total actually occurring at the end of the course. Tools You need to have a straightedge, compass, protractor and colored pencils for this class. As simple as these tools are, they are really essential. Please load on your computer the drawing system http://www.geogebra.org/cms/ It is really useful. The files with your drawings can be exported easily in different formats, for example as.png (portable network graphics). I request in the homework several drawings made with this computer system. 1
During the semester, please print out the relevant parts of my manuscript and make them available to you as hardcopy. You need to have available from the manuscript for all lectures p. 1-31 (table of content, references, brief Euclid, list of Hilbert axioms and so on) Course Topics: Most topics are included in the manuscript, or very good available in Hartshorne s book Euclid and Beyond. It is the best geometry book I know and rather cheap to get on amazon.com. Too, you can try the website abebooks.com. The part of my table of contents given below list the very important topics. You are expected to get fairly acquainted with these parts by some more leisurely reading. Of course, I need to make some selections. This semester, I intent to elaborate many details on area in neutral and Euclidean geometry. Attendance The general attendance rules of UNCC do apply. I expect you to be in the class regularly. I am willing to disregard 2 absences. The real problem begin with absences more than that. In case a list of attendance is passed, I allow myself to subtract 5 points for unexcused absense. Please make sure that you actually get the list of attendance that I pass around. If your name is missing, please add it to the list. Whatever the reason for your absence, you have to get all material you missed. Absence is no excuse for missing an exam, not knowing or not being prepared. Discipline Be aware that withdrawal from a course after the deadline to drop is nearly impossible. The instructor has no authority to assign a W grade after the drop date. Therefore make up your mind early in the first week of the semester, which course to take and check whether you can manage your workload. Please check relevant drop dates. It is necessary to create an atmosphere of politeness and concentration. I intend to speak loud enough, but within natural limits. English is not my native language. If you a have a question, please speak up, such that everybody can hear you. All cheating in the tests is clearly forbidden. There are things that are not tolerable, as for example regular use of cell phone in class. Not allowed are talking in class, coming late or leaving early, using cell phones during class time (at least regularly, there may be exceptions to people with sick relative, babies at home, and so on.) Further reminders Since nobody can control what material is available to you during tests, I announce this guideline, which I believe is fair: Your answers have to be not only formally correct, but even more important: they have to deal with the problem in question. I consider to deduct points, if you answer not only the present but some other problems without making clear what you do. Your work has to have the quality to be understandable by yourself or some friend also some time, say a year later. 2
Final The course has a final prepared by the instructor. I shall adapt the exact content of the final to the selection of the material that I have actually stressed during the course. As you can see from the grading scheme, it is essential to get a decent grade on that final. To this end, you need to have a working knowledge of the entire course material. Otherwise you can easily loose a letter grade, or more, because of the common final. Do not fool yourself thinking that having heard about a procedure means actually mastering it. We shall learn to appreciate careful definitions and correct statement of questions, methods, theorems and results. We have to develop the mathematical abilities required for successful problem-solving which include: recognizing similar problems in different disguises, adjusting the notation to the present case, preparing appropriate drawings, breaking up a problem in several steps, going, if appropriate, to a more general or a more special case, transferring a known method to a different situation, giving answers in precise formulation. Goals of the Course 1. To deliver a sound foundation of geometry, keeping up with the standard of rigor established by Hilbert in his ground breaking work Foundations of Geometry of 1899. 2. To learn Euclidean geometry thoroughly, especially congruence, similarity, area, constructions. 3. To discuss the theory of segment field, proportions and area in Euclidean geometry, as done by Hilbert in detail. 4. To discuss the history of the parallel postulate, the attempts to prove it, and finally the discovery of its independence. 5. To understand different axiomatic systems of geometry, the relations of their parts, the importance of models on one side, and general theorems on the other side. 3
I Neutral Geometry 15 1 Hilbert s Axioms of Geometry 16 1.1 Logic.................................... 16 1.2 David Hilbert s axiomatization of Euclidean geometry......... 18 1.2.1 Introduction from Hilbert s Foundations of Geometry....... 18 1.2.2 Hilbert s axioms............................ 19 1.3 Frege s Critique and Hilbert s answer................... 22 1.4 About the consistency proof for geometry................ 23 1.5 General remark about models in mathematics.............. 24 1.6 What is completeness?........................... 25 2 Incidence Geometry 28 2.1 Elementary general facts.......................... 28 2.2 Finite incidence geometries........................ 32 2.3 Affine incidence planes........................... 38 2.4 Projective incidence planes........................ 40 2.5 Introduction of coordinates........................ 42 2.6 The Fano Plane............................... 47 2.7 Finite affine and projective incidence planes............... 52 2.8 Finite coordinate planes.......................... 54 2.9 The Prime Power Conjecture about Non-Desarguesian Planes..... 55 3 The Axioms of Order and Their Consequences 62 3.1 Order of points on a line.......................... 62 3.2 Bernays Lemma.............................. 70 3.3 Plane separation.............................. 72 3.5 Interior and exterior of a triangle..................... 79 3.6 Left and right, orientation......................... 83 3.7 The restricted Jordan Curve Theorem.................. 85 5 Congruence of Segments, Angles and Triangles 101 5.1 Congruence of segments.......................... 101 5.2 Some elementary triangle congruences.................. 106 5.3 Congruence of angles............................ 112 5.4 SSS congruence............................... 119 5.5 The equivalence relation of angle congruence............... 122 5.6 Constructions with Hilbert tools..................... 129 5.7 The exterior angle theorem and its consequences............ 138 5.8 SSA congruence.............................. 158 6 Measurement and Continuity 167 6.1 The Archimedean axiom.......................... 167 7 Legendre s Theorems 183 7.1 The First Legendre Theorem....................... 183 7.2 The Second Legendre Theorem...................... 185 7.3 The alternative of two geometries..................... 190 4
7.4 What is the natural geometry?...................... 195 8 Neutral Geometry of Circles and Continuity 200 8.1 Immediate consequences of neutral geometry.............. 200 8.2 The tangent is the limiting position of a secant............. 205 8.3 Mutual placement of two circles...................... 209 8.4 Continuity principles for circles...................... 211 10 Towards a Natural Axiomatization of Geometry 249 10.1 The Uniformity Theorem......................... 249 10.2 A hierarchy of planes............................ 254 10.3 Wallis axiom................................ 255 10.4 Proclus Theorem.............................. 257 10.5 More about Aristole s axiom........................ 260 11 Area in neutral geometry 263 11.1 Equidecomposable and equicomplementable figures........... 263 11.2 The winding number............................ 266 11.3 Area of rectilinear figures......................... 267 11.4 A standard equicomplementable form for a figure............ 272 11.5 The role of the Archimedean axiom.................... 275 12 A Simplified Axiomatic system of Geometry my own Suggestion 278 12.1 A simplified axiomatization of geometry................. 278 12.2 Fundamental constructions with Euclidean tools............. 281 12.3 Euclidean tools are at least as strong as Hilbert tools.......... 284 II Euclidean Geometry 293 1 Some Euclidean Geometry of Circles 294 1.1 Thales Theorem.............................. 294 1.3 Construction of tangents to a circle.................... 308 1.4 A bid of philosophy............................ 309 1.6 Angles in a circle.............................. 316 1.7 On the nature of the tangent....................... 325 2 Simple Euclidean Geometry 329 2.1 Five constructions of the parallel..................... 329 2.2 Dividing a segment into any number of congruent parts......... 333 2.3 Some triangle constructions........................ 337 3 Pappus, Desargues and Pascal s Theorems 340 3.1 Pappus Theorem.............................. 342 3.2 Desargues Theorem............................ 346 3.3 Pascal s Theorem.............................. 350 4 Arithmetic of Segments Hilbert s Road from Geometry to Algebra 354 4.1 Fields.................................... 355 4.2 Construction of the field of segment arithmetic............. 355 5
4.2.1 Commutativity............................ 357 4.2.2 The distributive law......................... 359 4.2.3 The associative law.......................... 360 4.2.4 A direct proof of commutativity of segment arithmetic...... 361 4.2.5 Alternative proof of associativity.................. 363 5 Similar Triangles 365 5.1 Basic properties from Euclid........................ 365 5.2 Some exercises............................... 369 5.3 Secants in a circle............................. 378 5.4 Trigonometry................................ 385 5.6 Oriented angles............................... 392 5.7 Unwound angles.............................. 393 5.8 Extension to unwound angles....................... 394 6 Area in Euclidean Geometry 400 6.1 Equidecomposable and equicomplementable figures........... 400 6.2 The Theorem of Pythagoras and related results............. 402 6.4 Euclidean area............................... 414 6.5 The role of the Archimedean axiom.................... 417 6.6 Some uniqueness results for justification................. 419 6.7 About the volume of polyhedra...................... 423 6.8 The parallelogram equation........................ 424 6.9 Heron s formula for the area of a triangle................. 426 7 Coordinate Planes Descartes Road from Algebra to Geometry 429 7.1 A Hierarchy of Cartesian planes...................... 429 7.2 About Archimedean fields......................... 432 7.3 Congruence in a Pythagorean plane.................... 433 7.4 Transfer of an angle............................ 435 7.5 Enough rigid motions exist in a Pythagorean plane........... 438 7.6 Euclidean fields and intersection properties of circles.......... 442 Selected Important References [5] H.E. Dudeney, The Canterbury Puzzles, and other curios problems, Nelson, London, 1929. [8] Marvin J. Greenberg, Euclidean and Non-Euclidean Geometry, fourth ed., W.H. Freeman and Company, New York, 2008. [9] Marvin J. Greenberg, Old and new results in the foundations of elementary plane Euclidean and Non-Euclidean geometries, The American Mathematical Monthly 117 (2010), 198-219. [10] Robin Hartshorne, Geometry: Euclid and Beyond, second printing, Springer, 2002. [12] David Hilbert, Foundations of Geometry, 2nd English ed., Open Court, La Salle, 1971. 6
[17] Stefan Mykytiuk and Abe Shenitzer, Four significant axiomatic systems and some of the issues associated with them, The American Mathematical Monthly 102 (1995), 62. [18] David Park, The Grand Contraption the world in myth, number, and chance, second printing, Princeton University Press, 2005. [23] John Stillwell, The Four Pillars of Geometry, Springer, 2005. 7