Math 130, Spring 2020

Information for students

Syllabus
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.

Textbook

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 lib.berkeley.edu). 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 (2^nd 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.

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