Math 445 Syllabus and Topic Outline

Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Week 8 Week 9 Week 10

  1. Monday, 1/05/98

    Topic: Similarity Transformations and Dilations (1)
    King will be absent on Wed on Friday and Curjel will substitute. For this reason the topic of dilations will be split between Monday 1/5, the 487 lab on Wed 1/7, and class on Monday, 1/12. The written assignment on this material will be in Assignment 3. It is strongly recommended that you look over this assignment in advance so that you can answer some of the problems during the Wed. lab. Assignment 1 is a preview exercise for Wed. It is a review of material from 444 (and also from 126).
    Reading
    • Sibley, pp. 163-4,
    • King, Eye for Similarity (handout),
    • Bix, pp. 246-261 (handout)
    Handouts
    • Assignment 1: Problems for Curjel on parametrization of lines
    • Syllabus and Topic Outline (part of this Web page)
    • King, Eye for Similarity Transformations

  2. Wednesday 1/07/98

    Topic: Coordinates, Points and Lines in 3-space (1)
    Guest Instructor Caspar Curjel. Note that this is a hands-on in-class activity. Some of the HW assignments will be done or at least begun in class.
    Reading
    Curjel Handout
    Assignment 1 due.
    Problems for Curjel on parametrization of lines.
    Handouts
    • Assignment 2: Problems for Curjel on parametrization of lines

  3. Math 487 Lab #1 Wednesday 1/07/98

    Topic: Dilations
    Sam Adli will preside. A lab sheet will outline some explorations based on the reading Eye for Similarity Transformations. Some of this can be used for homework problems.
    Email List
    Everyone should be on the email list. If you are getting email from the list this week you are on; otherwise, you need to suscribe. Ask Sam or a lab assistent for help.
    Handouts
    • Lab Sheet #1
    • Bix, pp. 246-261
    • Problem 1 from 444 Final
    • Key Facts on Dilations
    • Homework Problems #3 (this is also on the web)

  4. Friday, 1/09/98

    Topic: Coordinates, Points and Lines in 3-space (2)
    Guest Instructor Caspar Curjel. This is a hands-on in-class activity.
    Reading
    Curjel Handout
    Assignment 2 Due.
    Problems for Curjel on parametrization of lines.

Back to Top. --- Back to Math 445 Home Page.

  1. Monday, 1/12/98

    Topic: Similarity Transformations and Dilations (2)
    Class exercise: worksheet on vector and matrix formula for dilations.
    Reading
    Same as in Monday, 1/5.
    Quiz 1 Today
    Basics on Similitudes, also special problem: Problem #1 from 444 Final.
    Handouts
    • Worksheet on Vectors and Dilations
    • Quiz 1

  2. Wednesday 1/14/98

    Topic: Secants of circles and Power of a Point
    This is based on facts we have learned about circles, but the use of these ideas leads to some sophisticated and powerful concepts such as power of a point and the radical axis. It is important to sort out the definitions and the relationships here and how the ideas are applied to carry out constructions.
    Reading
    Sved, Chapter 1
    Assignment 3 Due
    Problems on Dilations. Two errata in printed version of 1(f) and 2(a) corrected in email list and in this web version.

  3. Math 487 Lab #2 Wednesday 1/14/98

    Topic: Orthogonal Circles and Inversion
    This lab will be based on Chapter 9 of Geometry Through the Circle, so be sure to bring your copy of the book.
    Handouts

  4. Friday, 1/16/98

    Class activity on radical axis
    Given the axis, construct orthogonal circles. How to construct the axis.
    Reading
    Sved, Chapter 2
    Handouts
    • Worksheets on the radical axis.

Back to Top. --- Back to Math 445 Home Page.

Monday, 1/19/98. Holiday: Martin Luther King Day


  1. Wednesday 1/21/98

    Topic
    Inversion. Definition. Images of lines and circles under inversion.
    Reading
    Sved, Chapter 2. Also same topics in Ogilvy.
    Assignment 4 Due.
    Problems on circles and radical axes.

  2. Math 487 Lab #3 Wednesday 1/21/98

    Topic: Inversion as a Transformation
    This lab will be based on Chapter 10 of Geometry Through the Circle, so be sure to bring your copy of the book.
    Handouts

  3. Friday, 1/23/98

    Topic
    Inversion and sterographic projection
    Reading
    Sved, first part of chapter 2.
    Quiz 2
    Orthogonal circles, power of point, radical axis. See study problems on assignment sheet #4.
Back to Top. --- Back to Math 445 Home Page.

  1. Monday, 1/26/98

    Topic
    More Inversion
    Reading
    Ogilvy, Chapter 3

  2. Wednesday, 1/28/98

    Topic
    Apollonian Circles and Harmonic Division
    Reading
    • Sved, Chapter 3. (for lab)
    • Ogilvy, Chapters 1 and 2.
    Assignment 5 Due.
    Problems on inversion.

  3. Math 487 Lab #4 Wednesday 1/28/98

    Topic:
    Dr. Whatif's Euclidean Geometry in a Disk and other models. This lab will be based on Chapter 3 of Sved.
    Handouts

  4. Friday, 1/28/98

    Topic
    Solving Geometry Problems using Inversion
    Reading
    Ogilvy, Chapter 4
    Handouts
    • Blue handout - figures for inversion applications
    Quiz 3
    Inversion, images under inversion. Study questions appended to Assignment 5.
Back to Top. --- Back to Math 445 Home Page.

  1. Monday, 2/2/98

    Topic
    Introduction to Non-Euclidean Geometry
    Reading
    Sved, Chapter 4

  2. Wednesday, 2/4/98

    Topic
    More Non-Euclidean Geometry with an introduction to models for Non-Euclidean Geometry
    Reading
    • Sved, Chapter 4.
    • Sved, Chapter 5. (for lab)
    Assignment 6 Due.
    Figures from DWEG geometry. Problems on Non-Euclidean Geometry.

  3. Math 487 Lab #5 Wednesday 2/4/98

    Topic:
    Poincare Disk Model for non-Euclidean Geometry -- the basic constructions. This lab will be based on Chapter 5 of Sved.
    Handouts

  4. Friday,2/6/98

    Topic
    Harmonic tetrads using only a straightedge.
    Reading
    Ogilvy, pp. 978-104
    Handouts
    • A page with the line AB and a point C on it for drawing a "tent".
    Quiz 4
    Overall orthogonal circles -- Apollonian circles and harmonic division and how this fits with orthogonal circles, inversion, and geometric models. Study questions appended to Assignment 6.
Back to Top. --- Back to Math 445 Home Page.

  1. Monday, 2/9/98

    Topic
    Harmonic sets and triangles; cross-ratio and length
    Reading
    Ogilvy, end of Chapter 3 and handout
    Handouts
    • Three sides on harmonic sets via Menelaus and Ceva, chords and sines of angles, and cross-ratio.

  2. Wednesday, 2/11/98

    Topic
    A bit more about the Hyperbolic Plane and the Poincare model.
    Reading
    • Review Sved, Ogilvy and Eye for Similarity Transformations
    Assignment 7 Due.
    Poincare model etc.

  3. Math 487 Lab #6 Wednesday 2/11/98

    Topic:
    Wrap-up on Poincare and circles.
    Handouts
    • Lab Sheet #6

  4. Friday, 2/13/98

    Midterm Test
    Comprehensive test.
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Monday, 2/16/98. Holiday: Presidents Day


  1. Wednesday, 2/18/98

    Topic
    Introduction to Spherical Geometry. Great circles as "lines". Antipodal points. Angles between circles. Orthogonal circles and poles.
    Extra credit in-class quiz
    A problem from the Midterm (with more time) for extra credit.
    Reading
    • Lenart Sphere lab handout
    • Sved, chapter 2, pp. 29-32
    • Sved, pp. 73-74
    • Sved, pp.78-79, #9
    No Assignment Due.

  2. Math 487 Lab #7 Wednesday 2/18/98

    Topic:
    Spherical geometry, including Girard's area formula
    • With Lenart Spheres
    • With Sketchpad (using stereographic projection)
    Handouts

  3. Friday, 2/20/98

    Topic: More construction with Lenart Spheres
    . Concurrence of perpendicular bisectors (described orally) and concurrence of angle bisectors (an exterior bisector of one triangle is the interior bisector of another). =
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  1. Monday, 2/23/98

    Topic:
    Spheres and Platonic Solids. How tessellations by polygons on the sphere are related to the Platonic solids.
    • Start with octants (a spherical octagon).
    • Centers of octants give the cube, the dual of the octant.
    • Four of the eight cube vertices give a tetrahedron.
    • The cube side length is the diagonal length of the pentagons for the dodecahedron.
    • The icosahedron is the dual of the dodecahedron.

  2. Wednesday, 2/25/98

    Topic
    Introduction to Tessellations. Guest instructors: Mark Haberl and Lanae Starkey.
    • Regular tessellations
    • Tessellations and Wallpaper groups
    • Tessellations by quadrilaterals (for more on this, see the lab).
    Assignment 8 Due.
    Spherical geometry.
    Project Description Due.
    Some people got this late. Do it ASAP.

  3. Math 487 Lab #8 Wednesday 2/18/98

    Topic:
    Tessellations
    • Quadrilaterals and Triangles
    • Building new tessellations from regular tessellations
    Handouts

  4. Friday,2/27/98

    Topic
    More tessellations and symmetry
    Quiz 5
    Spherical geometry
    • On the sphere: great circles and antipodal points, how to construct orthogonal circles, measure distances as angles, etc.
    • In the plane: how to construct the antipodal point in the plane; how to construct the sorts of figures in the homework.
    Back to Top. --- Back to Math 445 Home Page.

    1. Monday, 3/2/98

      Topic:
      Representations of infinity. Jon Helman and Karen Wilson, guest instructors. Many Escher handouts.

    2. Wednesday, 3/4/98

      Topic:
      Fractals
      Assignment 9 Due:
      Tessellations