Mathematics 100 A
MATHEMATICS IN THE CONTEMPORARY WORLD
Calvin College, Fall 2006

Instructor Gerard A. Venema Office: North Hall 287 Office hours: 10:30 - 11:20 a.m. M. W. F. (Other times by appointment) Phone: 616-526-6402 Email: venema@calvin.edu Homepage: http://www.calvin.edu/~venema

Textbook The Heart of Mathematics An invitation to effective thinking by Edward Burger and Michael Starbird The textbook has its own website:   www.heartofmath.com.

Course Description

This is a course in mathematics designed to meet the needs of students in the humanities and liberal arts. The course will focus on some of the major ideas of mathematics and the impact those ideas have had on our culture. The course has three major goals:

1. To attain a better understanding of some rich mathematical ideas.
2. To build thinking skills that can be used to analyze problems in other areas of life.
3. To develop a new perspective on mathematics and the way it is used in the world.

I hope you will come to see that mathematics is a human activity that requires both creativity and imagination. My goal in teaching the course is to help you learn to appreciate mathematics and to discover the power of mathematical thinking.

The course will likely be quite different from mathematics courses you had in high school. There the emphasis was on technique and computational skills. I will ask you to think and analyze rather than to work routine exercises. The course will not help you "brush up on your basic math skills" but it will play an important role in your total liberal arts education.

(The photo shows author Mike Starbird teaching his students about the fourth dimension.)

Prerequisites

The only prerequisites for the course are an open mind and a willingness to put aside preconceived prejudices about mathematics.

Syllabus

Homework

Homework will be assigned each day. Written exercises are to be turned in on the due-date specified. You are encouraged to work with other members of the class on the homework assignments, but you should write up your own individual solution to each exercise. You should watch the course website for new homework assignments.

An important part of each homework assignment is reading the appropriate sections of the textbook. Do not neglect that part of the assignment. Read slowly and think carefully about the ideas presented. The usual rule of thumb is that you should expect to spend two hours working outside of class for each hour in class.

Test and Final Exam Schedule

• Test 1. Friday, September 29     [Topics]     [Practice congruence problems]
• Test 2. Friday, October 27     [Topics]
• Test 3. Friday, December 1     [Topics]
• Final Exam. 1:30 p.m., Tuesday, December 12

Grades

Course grades will be based on scores on homework, the three tests, and the final exam. The homework will count as much as one test; the final exam will count as much as two tests.

Some Course-Related Links

• Fibonacci Spiral generator
• The Great Internet Mersenne Prime Search
• The Westminster Catechism
• "Hotel Infinity" lyrics
• Social Satire in Flatland
• Flatland   (the book)
• Flatland   (the movie)
• The 3D Platonic Solids
• Platonic Solids in All Dimensions
• Robert Lang's mathematical origami
• Instructions for making an origami hyperbolic paraboloid
• Illustrations of regular tessellations in the three 2D geometries
• Platonic spheres
• Banchoff recalls Dali
• The mathematician and the artist
• Banchoff with folding hypercube
• Banchoff at Calvin College
• The Acme Klein Bottle Co.
• Nature article on dodechedral universe
• More on dodecahedral universe
• National Geographic article on dodecahedral universe
• Fields Medals awarded 2006
John Conway's game of life
• The Koch snowflake fractal
• Devaney's Dynamical Systems page

Homework Assignments

Homework Assignment 1 (Due Friday, September 8)

1. Read "Welcome" (pages ix -- xii). This section sets the tone for what is to come; it explains what the book is about and how you should read it.
2. Read "Surfing the book" (pages xiii -- xxvi). This section gives a preview of what is contained in the various chapters of the book.
3. Read §1.4, page 25. The section is less than one page long, but the "Lessons for Life" are important.
4. Try out Mindscapes 1 and 3, page 26, on three friends (preferably not mathematics or science majors). Write a short report on their answers. Do their answers agree with your own?
5. Solve Mindscape 2, page 26. Explain your reasoning in one concise paragraph.

Homework Assignment 2 (Due Monday, September 11)

• Work Mindscapes 4, 5, and 7 on pages 26 and 27.

Homework Assignment 3 (Due Wednesday, September 13)

1. Read §2.1.
2. Work Mindscapes I.2, I.5, I.6, I.7, and II.4, pages 44ff.
3. Bring at least 1 million of something (anything) to class Wednesday. Be prepared to explain how you know that you have 1 million of them.

Homework Assignment 4 (Due Friday, September 15)

1. Read §2.2.
2. Work Mindscapes I.1, I.2, I.3, I.12, and II.5, pages 56ff. [Note: Some copies of the textbook have a typographical error in the statement of Mindscape I.3. You should look for a pattern in (F_n+1)² - (F_n-1)² rather than (F_n+1)² - (F_n)².]

Homework Assignment 5 (Due Monday, September 18)

1. Work Mindscapes I.5, I.6 on page 58 and II.4 on page 61.
2. Read §2.3.
3. Work Mindscapes I.7, I.8, and I.9 on page 78.

Homework Assignment 6 (Due Wednesday, September 20)

• Work Mindscapes I.3, I.4, I.11, I.12, I.19, and II.10 on pages 78. [Hint: For II.10, you can use your sieve rather than the hint in the book.]

Homework Assignment 7 (Due Friday, September 22)

1. Read §2.4.
2. Work Mindscapes I.1, I.2, I.4, I.6, I.8, and I.9 on pages 90 - 91.

Homework Assignment 8 (Due Monday, September 25)

• Handout on Euclidean algorithm.

Homework Assignment 9 (Due Monday, October 2)

1. Read §2.5.
2. Work Mindscapes I.1, I.3, I.4, I.5, I.8, I.9 on pages 109 and 110.

Homework Assignment 10 (Due Friday, October 4)

1. Read §2.6 and §2.7.
2. Work Mindscapes I.1, I.2, I.3, and I.5 on pages 120-121.
3. What would go wrong if you tried to use your argument from I.5 to prove that √4 is irrational?
4. Read I Kings 7:23. What numerical value does the passage assign to π? Is this consistent with what the textbook says on page 119? Explain.
5. Work Mindscapes I.1, I.2, I.3, I.4, I.5, and I.15 on pages 134-135.

Homework Assignment 11 (Due Monday, October 9)

1. Read §3.1.
2. Work Mindscapes # I.4, I.8, II.2, II.4, II.5 on pages 143-145.

Homework Assignment 12 (Due Wednesday, October 11)

1. Read §3.2
2. Work Mindscapes I.3, I.4, I.6, I.7, I.8, I.11 on pages 158-159.

Homework Assignment 13 (Due Friday, October 13)

1. Work Mindscapes I.12, I.13, II.6, II.7 on pages 159-161.
2. Read §3.3.
3. Work Mindscapes I.7, I.9 on page 171.

Homework Assignment 14 (Due Monday, October 16)

1. Read §3.4.
2. Work Mindscapes I.1, I.5, I.6, I.8, I.10, II.1, II.4 on pages 187-190.
3. Read pages 1 - 13 and 30 - 33 of "Infinity and the Absolute."

Homework Assignment 15 (Due Wednesday, October 18)

1. Read §3.5
2. Work Mindscapes I.1, I.2, I.4, I.5, I.8, I.9 on pages 203-205.

Homework Assignment 16 (Due Monday, October 23)

1. Read §4.3
2. Work Mindscapes I.2, I.4, I.6, I.7, I.9, and II.2 on pages 245-247.

Homework Assignment 17 (Due Monday, October 30)

1. Read Social Satire in Flatland
2. Read Flatland
3. Answer Flatland questions

Homework Assignment 18 (Due Wednesday, November 1)

1. Read §4.5
2. Work Mindscapes I.3, I.6, I.7, I.9, II.2, and II.4 on pages 284 - 286.
   [Hint for I.7: Click here for a picture of the soccer ball. Note that you can rotate the soccer ball with your mouse.]

Homework Assignment 19 (Due Monday, November 6)

1. Read §4.6
2. Work Mindscapes II.1, II.2, II.3, and II.4 on page 303.
   [Hints: You may want to try out your drawings for 1 and 2 on a round ball---like a basketball. Click here to see my drawing of a triangle on the sphere. You will need to find a protractor to use for 3 and 4.]

Homework Assignment 20 (Due Wednesday, November 8)

1. Read pages 249 - 252 in textbook.
2. Work Mindscapes I.5, I.6, II.1, and II.2 on pages 263 and 266.

Homework Assignment 21 (Due Friday, November 10)

• Work the homework assignment in the handout on Regular Tessellations

Homework Assignment 22 (Due Monday, November 13)

1. Read §4.7
2. Work Mindscapes I.1, I.2, I.3, I.7, and II.1 on pages 320 - 322

Homework Assignment 23 (Due Wednesday, November 15)

1. Read page 327 and §5.1
2. Work Mindscapes I.2, I.4, and I.7 on pages 338 - 339

Homework Assignment 24 (Due Friday, November 17)

1. Read §5.3
2. Work Mindscapes I.2, I.3, I.4, I.7, I.8, and I.9 on pages 365 - 366

Homework Assignment 25 (Due Monday, November 20)

1. Read §5.2
2. Work Mindscapes I.3, I.4, I.5, I.8, I.13, and I.14 on pages 351 - 353

Homework Assignment 26 (Due Wednesday, November 22)

• Work Mindscapes I.9, I.10, I.11, I.16, and I.17 on pages 352 - 353

Homework Assignment 27 (Due Monday, November 27)

• Worksheet on "The Proof" video

Homework Assignment 28 (Due Wednesday, November 29)

1. Read page 399 and §6.1
2. Work Mindscapes I.2, I.3, and I.4 on page 407

Homework Assignment 29 (Due Monday, December 4)

1. Read §6.2
2. Work Mindscapes I.7 - I.14 and I.16 - I.18 on pages 420 - 421

Homework Assignment 30 (Due Wednesday, December 6)

1. Read §§6.3 & 6.6
2. Work Mindscapes I.8, I.9, I.15, and I.16 on pages 449 - 451
3. Work Mindscapes I.6 and I.10 on page 509