1Son of Dr. Teri Perl, formerly a lecturer in our Department.6 February 2002MATH 350 GEOMETRY JT SMITHOutline 1 SPRING 20021. This is the first of many lecture outlines that you’ll follow through the course.They correspond not to its logical subdivisions, nor to specific class meetings, butto my preparation sessions. You’ll need to become adept at using them online anddownloading them.2. They are Adobe Acrobat *.pdf files. To read them you must have installed thefree Acrobat Reader, which your Internet browser will use to display them. Thesefiles contain blue underlined links to various others. On my website you’ll findinstructions for obtaining the reader software. (Follow the link.) The requiredsoftware should be installed on University computers.3. Go over the syllabus in detail.4. Introductiona. Read the text from the beginning through chapter 1. Some of this iscovered in class informally.b. It’s augmented by the paper Mashers mathematical that I authored withDan Wheeler (to be published soon in Math horizons).c. Wheeler’s Hilbert masher has the same symmetry as one of the others.Which?d. Follow this link to figure 1.1.3. Demonstrate with a meter stick that themural is inaccurate. Follow this link to art deco in Napier.e. I’m trying to locate a book by David Hockney, recently reviewed by JedPerl1 in The new republic, that seems to have information on mechani-cal/optical devices that Renaissance artists may have used for perspectivedrawing. With Field 1997, that might lead to a fascinating term project.f. Euclid [ca. 300 B.C.] 1945. The Optics of Euclid, translated by Harry EdwinBurton, Journal of the Optical Society of America 35:357– 372. This is theearliest surviving writing on perspective. It’s readable. I have a copy,should you want to read it and our Library can’t retrieve it from storage.Springer recently published another translation, interleaved with theArabic source.g. Follow this link to the American Mathematical Society’s Mathematicalsubject classification, and demonstrate it.5. Geometric examplesa. You must start your search for a term project early in the course, beforemuch of the detailed mathematics.b. Many fine term project topics involve examples of geometric design.c. Moreover, you’ll appreciate the theory most if you’re familiar with manyexamples.d. Therefore, I’ll present visual and tangible examples in class every day, ifpossible.Page 2 MATH 350 OUTLINE 1e. I particularly want you to gain visual familiarity with the various kinds ofsymmetry that we’ll classify late in the course.6. Linear algebraa. Later in the course we’ll analyze 2D and 3D motions and other transforma-tions with analytic methods—sophisticated uses of systems of linearequations.b. Probably all the needed properties of linear systems are covered in pre-calculus algebra courses. But their large-scale use demands the efficientterminology and notation of matrix algebra.c. Some of you may not have thought seriously about linear systems for someyears. Moreover, matrix algebra is not officially a prerequisite to thecourse. Some of you have learned it in other courses.d. Therefore I’ve provided in appendix C a quick presentation of the necessarymaterial. Read it.e. The appendix actually does this for n dimensional systems in general,whereas we often use only two dimensions. Therefore I’ve posted on theweb a 2D version of the same material. If matrix algebra is new to you, youshould read that document before appendix C.f. During the first part of the course, I’ll devote a little class time to thismaterial. The midterm examination will have at least one problem on it.g. As a first installment, consider vector addition and scalar multiplication.7. Foundationsa. Chapter 2 of the text, on the axiomatic foundation of Euclidean geometry,is in a way its most sophisticated part. I’m not going to lecture on it, butwill simply use that method in presenting an overview of the subject. Asit unfolds, I’ll comment on the choice of undefined concepts and axioms,and the arrangement of the theorems.b. The goal of this part of the course is not to impart facility with the axiom-atic method nor skill in proving elementary geometry theorems, but torefamiliarize everyone with the content of elementary geometry, and toprovide the basis for solving some fairly difficult problems.c. Read chapter 2 so that you’ll recognize references I make to it. There arealso some suggestions for term paper topics.8. The Birkhoff–Moise axiom systema. This outline covers chapter 3 from the beginning through section 3.1, andchapter 4 from the beginning through section 4.1.b. Incidence axiomsi. Undefined concepts: point, line, planeii. The underlying, or prior, theory includes set theory.iii. Incidence axioms (note: for us axiom = postulate)iv. Some theoremsMATH 350 OUTLINE 1 Page 3c. Glimpses of historyi. Euclid 1st printed ed. 1482, Latin, definitionsHeath 1925 ed. definitionspostulatesii. Legendre 1834 “modernized” Euclidlater 1800s translation concepts p. 13postulates p. 18iii. Stone–Mallory 1937 ed. (my schoolbook?) postulatesiv. Hilbert 1902 English ed. conceptsaxiomsbiography and portraitv. Birkhoff 1931 paper conceptsbiography and portraitBirkhoff & Beatley 1940 text based on thispaper is not easy to showvi. Moise & Downs 1964 SMSG text conceptsd. Incidence geometry, also known as descriptive geometry, is a whole fieldof mathematics, although a minor one. I suspect that there are interestingPhD level problems remaining in higher dimensional incidence geometry.9.Assignment 1a. Section 4.1, exercise 1 or 2 or 3.b. Section 4.1, exercise 4 or 5 or