Math 130, Fall 2015

Information for students

  • Syllabus

  • 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 M. Hutchings.

  • Policy on absences for tests and midterms.

  • How to get an A in this class

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 (link) (if that link doesn't work, 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 (link)

Week-by-week list of readings and activities

(will be updated throughout the course)
  • Sept. 27th: A review of Hartshorne's Geometry: Euclid and Beyond Euclid by D. Henderson.

Week 2: Euclid's constructions with straightedge and compass.
  • Stillwell, chapter 1 (and a little of chapter 2).

  • Activity: Euclid: the game

  • online version of Euclid's elements, with comments. You were given the definitions and postulates as a handout.

Week 3: Parallels, and the theory of area
  • Stillwell, chapter 2

  • Short reading: Commentary on Euclid's method of superposition, from Hartshorne. (despite the weird page numbering, these pages are in order!)

  • 111 ways to prove the pythagorean theorem

  • Handout: Proposition 35 and 38 from the online version of Euclid's elements

Week 4: (non)-constructible figures. "Algebra is the answer"
  • Euclid's construction of the regular pentagon, from Hartshorne. Compare with your construction from HW2.

  • Constructible n-gons, followed by a quick introduction to field extensions. From a wonderful book "Conjecture and Proof" by M. Laczkovich.

  • (Optional) videos of Prof. Eisenbud and Gauss' 17-gon: video 1 video 2

  • (Optional) Viete's construction of the 7-gon

Week 5: Hilbert's axioms.
  • Reading from Hartshorne: sections 6-8

Week 6: Hilbert's axioms, continued.
  • Reading from Hartshorne: sections 8-10

Week 7: Midterm exam on Tuesday, Intorduction to projective geometry on Thursday. Discussion of independent project.
  • Reading to be done by the beginning of Week 8: How to win the lottery with projective geometry
    an excerpt from ``How not to be wrong" by Jordan Ellenberg.

  • Midterm solutions
    Midterm problem statements (with point values)

Week 8: Projective geometry
  • Selection from Ellenberg, see above

  • Reading from Stillwell, chapter 5.

  • Just for fun: anamorphic drawing.
    And a very sophisticated understanding of projective transformations in OK GO's music video The writing's on the wall.

Week 9: More projective geometry
  • Selections from Stillwell, chapter 6

Week 10: Transformation groups, plane and spherical geometry.
  • Stillwell, chapter 7

  • Quaternions and rotations
    More than you wanted to know, but you might be especially interested in the practical advantages of using quaternions over ordinary matrices.

Week 11: Introduction to hyperbolic geometry
  • More on quaternions and 4-dimensional geometry: Hypernom the game. Explained here

  • Stillwell, chapter 8 up to 8.4

  • Mobius transformations: video by Douglas Arnold and Jonathan Rogness, with explanation here
    And an interactive applet by Terry Tao.

Weeks 12-13: More hyperbolic geometry; practice for presentations
  • Reading from Stillwell, chapter 8.

  • Many tilings of hyperbolic space by Jos Leys

  • movies of isometries of hyperbolic space by Goodman-Strauss

  • Here is a template that you can use to make this Escher tessellation on a wrinkly-paper model of hyperbolic space!

  • Notes on area of hyperbolic triangles from the end of the last class.

Week 14-15: Student presentations on independent projects.
  • Schedule

Week ∞ (not part of the course, but on the horizon).
      Some geometry books that you might like to read in the future
  • The shape of space by J. Weeks. A wonderful introduction to geometry and topology and the question "what is the geometry of our universe?" There is a short (and less mathematical) movie inspired by the book here .

  • Three-Dimensional Geometry and Topology by W. Thurston. This book begins with an introduction to the hyperbolic plane, and then goes much further...


Homework

Weekly homework assignments will be posted here.
  • Problem set 1 due Tuesday, September 8
    selected solutions

  • Problem set 2 due Tuesday, September 15
    selected solutions

  • Problem set 3 due Tuesday, September 22
    selected solutions
    • See above for links to the readings.
    (also here are the videos mentioned in the problem set: video 1 video 2
    Note: the last problem should say "where k and n are relatively prime". This has been added to the statement!

  • and... Just for fun (not to hand in) challenge problem on equidecomposability. Due never, but tell me if you solve part b!

  • Problem set 4 due Tuesday, September 29
    selected solutions

  • No homework due Tuesday, October 6 (you have a midterm!)

  • Problem set 5 due Tuesday, October 13
    selected solutions

  • Independent project assignment
    Some suggestions for topics

  • Problem set 6 due Tuesday, October 20
    selected solutions

  • Problem set 7 due Tuesday, October 27
    selected solutions

  • Problem set 8 due Tuesday, November 3
    selected solutions

  • Problem set 9 due Tuesday, November 10
    selected solutions

  • Problem set 10 due Tuesday, November 24.
    Note: there was a typo in Question 3.c), which has now been corrected.
    The formula given there is a special case of the Gauss-Bonnet theorem which says that angle defect (in our case, 2pi - exerior angle sum) is equal to the area of a polygon multiplied by the curvature of the space.
    Hyperbolic space has constant negative curvature, in our calculaions we're using curvature -1.

    triangle paper
    selected solutions to HW 10

  • Resources for your independent project:
    General advice on how to get started writing
    Guidlines for peer review of a paper
    Grading rubric for oral presentation and written report

  • Presentation schedule

  • Review for the final exam
    Your final exam takes place on Wed, December 16, 3pm-6pm, in the usual classroom. I will hold office hours on Monday and Tuesday, times to be announced by e-mail.

Worksheets

Worksheets that were given in class
  • Worksheet 1 from September 3

  • Worksheet 2 from September 17

  • Worksheet 3 from September 29

  • Worksheet on anamorphic (perspective based) writing from October 13

  • Worksheet 4 (Pappus theorem) from October 22

  • Worksheet 5 (Rotations) from October 29

  • Worksheet 6 (Tiling the plane with reflections) from November 12