Math 130, Spring 2020

Information for students

bCourses Site
DSP students should speak to the instructor as soon as possible, even if you don't have a letter yet.
Guidelines on what to do if you think you may have a conflict between this class and your extracurricular activities. In particular, you must speak to the instructor before the end of the second week of classes.
Academic honesty in mathematics courses: A statement on cheating and plagiarism, courtesy of Michael Hutchings.
How to get an A in this class, courtesy of Kathryn Mann.


The required text for this course is The Four Pillars of Geometry by John Stillwell. You can download a copy of this book for free on campus through the UC library (if that link doesn't work, just search for the book at This book is a wonderful introduction, but a little too easy for us, so there will be lots of required supplementary readings supplied by the instructor. We will also use some excerpts from Hartshorne's Geometry: Euclid and Beyond Euclid. I recommend this to students wishing to go further. It can also be downloaded on campus.

Homework, Readings, etc.

(will be updated throughout the course)

January 21: intro, Euclid's postulates Reading: Stillwell 1.1 Euclid's Elements (for the adventurous, here's a Greek version) January 23: Euclid's constructions; Thales' theorem Reading: Stillwell 1.1–1.4 / Hartshorne Sections 1–2 Random: story about Thales by Plutarch (starts at VI, goes onto page 419) Activity: Euclid: the game Activity: Play with Geogebra January 28: arithmetic with Euclidean constructions, square roots Reading: Stillwell 1.5–1.6 Reading: Critique of superposition, Hartshorne pp 31–34 Worksheet 1: click here Review of Fields: fields (16.1—2) * Homework 1 (due on Tuesday, February 4): click here January 30: parallel postulate, areas, Thales' theorem revisited Reading: Stillwell 2.1–2.6 / Hartshorne Section 22 February 4: equidecomposability Some resources: Hartshorne Sections 22, 24, Wikipedia, interactive demonstrations For fun: Hinged dissections * Homework 2 (due on Tuesday, February 11): click here February 6: constructibility Reading: field extensions Reading: Constructible n-gons and field extensions (from Conjecture and Proof by M. Laczkovich) Reading: Degrees of field extensions (from Algebra (2nd ed.) by M. Artin) Reading: Hartshorne Sections 28–29 Video: Construction of 17-gon (other videos: 1 and 2) February 11: impossible constructions, regular polygons, incidence axioms Reading: see February 6, 12 Reading: Regular polygons (from Galois Theory by Ian Stewart) * Homework 3 (due on Tuesday, February 18): click here February 12: Hilbert's incidence and betweenness axioms Reading: Hartshorne Sections 6–7 Notes on the real projective plane: click here February 18: intro to project; betweenness axioms Reading: Hartshorne Section 7 * Project: Description, Grading Scheme, Topic Suggestions * Homework 4 (due on Tuesday, February 25): click here February 20: congruence axioms, other axioms Reading: Hartshorne Sections 8–11 February 25: starting projective geometry Reading: Stillwell 5.1–5.4 Reading: How to Win the Lottery with Projective Geometry (from How Not To Be Wrong, by Jordan Ellenberg) February 27: projections, fractional linear transformations, invariants Reading: Stillwell 5.5–5.9 March 3: Midterm (in class) Material: everything up to the end of Feb 20 class, all homework questions but not: critiques of Euclid; superposition; equidecomposibility; Galois theory No cheat sheets or notes may be used; you don't need a compass/straightedge, although you may bring them if you want You will be given a list of Hilbert's axioms (I1-3), (B1-4), (C1-6), (P), (E), projective plane axioms You do not need to cite Euclid's axioms by number (you can assume unique lines); any construction will be "from the axioms" unless stated I will not ask you to reprove anything we did in class, but you should understand the techniques that go into the proofs, and how to use them You might need to give definitions of terms from class March 5: cross-ratio, other projective planes Reading: Stillwell 5.7–5.9 * Homework 5 (due on Thursday, March 12): click here March 10: no class! March 12: projective Pappus, Desargues; planar ternary rings Video: on bCourses; 8:15 orders of projective planes, 17:00 projective Pappus, 26:20 projective Desargues, 38:00 planar ternary rings, 49:55 (projective) plane from a planar ternary ring, 56:30 examples, 1:02:15 from projective plane to planar ternary ring, 1:18:13 Pickert-Hall theorem, Wedderburn's little theorem, final remarks Reading: Stillwell 6 March 17: geometry via transformation groups, quaternions Video: on bCourses Reading: Stillwell 7.1–7.3, 7.6 * Homework 6 (due on Tuesday, March 31): click here March 19: more quaternions, group of rotations of the sphere Video: on bCourses Reading: Stillwell 7.6, 7.8 March 31: Möbius transformations; hyperbolic lines; angles; disc model Video: on bCourses Reading: Stillwell 8.1–8.5 * Homework 7 (due on Tuesday, April 7): click here April 2: hyperbolic distance; area of spherical triangle Reading: Stillwell 8.5–8.7 April 7: area of hyperbolic triangles; hyperbolic circle; notes on geodesics * Homework 8 (due on Tuesday, April 14): click here April 9: geometry on surfaces There are a lot of readings, but you don't need to read them all or in detail. It's a new topic, so I want to give you lots of choices of reading material. Reading: Here (ignore the "3–manifold" parts) and here Reading: Chapter 6 of Introduction to Topological Manifolds by John Lee (just to get an idea; there's lots of topology you may not know) Reading: John Conway's ZIP proof of the classification of compact surfaces April 13: classification of surfaces Reading: see April 9 * Homework: no more homework! Just work on your project. April 20: geometry on surfaces Reading: see April 9 * Homework: no more homework! Just work on your project.