Mathematics 618 (Foundations of Geometry) Spring 2018
Under construction, all content subject to change

• Syllabus (.pdf)
• Midterm 1 will be in class on Wednesday March 28.
• Solutions for Midterm 1 (.pdf)
• Midterm 2 will be in class on Wednesday May 16.
• I plan to have Midterm 2 graded by 9.00am on May 17. I will reply to any email (around that time) giving a score and current grade.
• Answers for Midterm 2 (.pdf)
• Sample Second Midterm Exam with numerical answers (.pdf)
• Another sample 2nd midterm (.pdf)
• Another sample 2nd midterm (.pdf)
• Sample Second Midterm Exam with full solutions (.pdf)
• The final exam will be on Wednesday May 23 at 4pm.
• Sample Final Exam with answers (.pdf)
• Sample Final Exam (.pdf)

  Planned class topics (under construction, subject to change):

  • 01-31-18 Introduction 1.1. Euclid's Geometry, Absolute Geometry 1.2.
    • Preliminary Reading: Sections 1.1, 1.2 pages 1-15.
    • Homework: Exercises 2, 7, 8, 9, 16, 31 from pages 16 and 17.
  • 01-31-18 The 5th postulate 1.4.
    • Preliminary Reading: Section 1.4 pages 21-24.
    • Homework: Exercises 1, 2 from page 24.

    Main topics from 01-31-18. The five postulates. Euclid's Theorems 1, 4, 5, 16, 27,29. Absolute Geometry. Historical discussion of attempts to prove the 5th postulate and the discovery of hyperbolic geometry.

  • 02-07-18 Geodesics on the Sphere 11.1. Great circles.
    • Read Before class: Section 11.1 pages 143-156.
  • 02-07-18 Dot and cross products of vectors 11.2. Vector triple products.
    • Review before class: Dot and cross products of vectors.
    • Read before class: Section 11.2.

    Main topics from 02-07-18. Dot and Cross products of vectors. Spherical geometry. Great circles. Geodesics.

  • 02-14-18 Spherical Triangles. Spherical laws of sines and cosines. Spherical Trigonometry 11.2, 11.4
    • Bring to class a mandarin orange, tangerine or similar fruit.
    • Read Before class: Sections 11.2, 11.4.
    • Read before class: Everything anyone would ever need to know about spherical trig.
    • Homework: Exercises 1, 3, 4 page 150.
    • Homework: Exercises 10,11 pages 149-150.
  • 02-14-18 Spherical Trigonometry 11.2, 11.4
    • Homework: Practice problems on Spherical Trig (.pdf)

    Main topics from 02-14-18. Spherical laws of sines and ccsines. Poles and polar triangles. Polar law of cosines. Spherical Pythagorean Theorem. Examples of solution of spherical triangles. Examples of navigation problems. Napier's pentagram.

  • 02-21-18 Section 11.3: Spherical area. Catch up and review Chapters 1 and 11. Discussion of homework problems.
    • Read Before class: Chapters 1 and 11
    • Bring any questions about homework to class.

    The main part of the class is review of Chapters 1 and 11: I will answer any review questions and solve any requested homework problems. Make sure to come prepared with questions about Chapters 1 and 11. If we run out of questions, I will start material from Chapter 3.

  • 02-28-18 Inversion 3.1.
    • Read Before class: Section 3.1 pages 41-49.
    • Homework: Exercises 3, 4, 6 from pages 48-49.
    • Geometers Sketchpad tools for inversive geometry
    • An alternative set of Geometers Sketchpad tools for inversive geometry

    Main topics from 02-28-18. Inversion, images of lines and circles. Inversions negate directed angles. Orthogonal circles and inversion.

  • 03-07-18 College closed due to snow. No class. Midterm 1 will be postponed by a week.
  • 03-14-18 Midterm 1 will not be in this class. It has been postponed by a week. Applications of inversion 3.2.
    • Read Before class: Section 3.2 pages 49-51.
    • Homework: Exercises 1, 2, 3, 4, 5 from page 51.
    • Review Problems on Inversive Geometry, only problems 1,2,3,11,12,13,17,22 apply to this year's course.

    Catch up and review of Chapter 3. Discussion of homework problems.

    • Read Before class: Chapter 3 pages 41-51.
    • Bring any questions about homework to class.

    Main topics from 03-14-18. Applications of inversion. Review of Chapter 3, discussion of homework problems.

  • 03-21-18 College closed. Class canceled due to snow. Sorry. Midterm 1 will have to be postponed until next Wednesday. I will update class plans for April and May later.

  • 03-28-18 Review of any topics previously covered.
  • 03-28-18 Midterm 1.

    All later content will be rescheduled as follows because of lost snow days. The two midterm exams are each extended by 75 minutes and all later classes will run as long as necessary to cover scheduled material. I have reorganized this material to fit into the number of classes available, but classes will have to be a little longer than normal to be able to cover all of this material. However, I will do my best to be efficient in class so that extra class minutes are as few as possible.

  • 04-18-18 Hypebolic geometry. Introduction. The half plane module. Measurement of length. Geodesics. Measurement of angles. Isometries. Hyperbolic reflections. Chapter 4.
    • Read Before class: Chapter 4 pages 53-61.
    • Homework: Exercises 1, 2, 3 from page 56.
    • Homework: Exercises 1, 2, 3, 4, 5 from page 59.
    • Homework Ch 4: Exercises 1, 3, 5, 8, 9, 10, 12, 13, 14 from pages 61-62.

  • 04-25-18 The disk model of hyperbolic geometry. Circles in hyperbolic geometry. Euclid's postulates and the hyperbolic plane. Hyperbolic area. Chapters 5 and 7.
    • Read Before class: Section 5.1 pages 67-75.
    • Read Before class: Section 7.1 pages 89-93.
    • Read Before class: Section 7.2 pages 93-96.
    • Homework: Exercises 1,2,3,4,7 pages 74-75.
    • Homework: Exercises 1,2,3, 4 page 93.
    • Homework: Exercises 1,2,3,4,7 page 97.
    • Review Problems on Hyperbolic Geometry.

  • 05-02-18 Hyperbolic trigonometry 8.1, 8.2.
    • Read Before class: Sections 8.1, 8.2 pages 99-104.
    • Homework: Exercises 1,2,3,4,5 page 104.
    • Homework: Exercises 5,6,7,8,11 page 107.
    • Additional Review Problems on Hyperbolic Trigonometry for midterm 2 and the final.

    Main topics from 05-02-18. Hyperbloic trig functions: sinh, cosh, tanh, sech, csch, coth. Hyperbolic trigonometry of right triangles. Napier's Pentagon for right triangles hyperbolic trig.

  • 05-09-18 Classification Theorems 2.4. Rigid Motions and absolute geometry 2.5.
    • Read Before class: Sections 2.4, 2.5 pages 38-40.
    • Homework: Exercise 5 from page 40.
  • 05-09-18 Rigid motions 2.1. Translation, Rotation, Reflection 2.2. Glide Reflection 2.3.
    • Read Before class: Sections 1.4, 2.1, 2.2 pages 21-34.
    • Read Before class: Section 2.3 pages 35-37
    • Homework: Exercises 1, 2, 3 from page 27.
    • Homework: Exercises 1, 2, 3, 4, 6 from page 34.
    • Homework: Exercises 1, 2, 4, 5 from page 37.
    • Practice problems on Euclidean Transformations (.pdf)

  • 05-16-18 Review if required. Midterm 2.