Math 44, Mathematics in Art, Culture, and Society: A Liberal Arts Math Class
Spring 2019 Home Page

Syllabus ("Green Sheet")

Homework:
Chapter 1 to be added in and turned in with homework from chapter 2: do any five of the following problems which begin on page 33: #3, 6, 8, 12, 13, 16, 17,18. Note that there are hints following the problems!

Ch. 2.1: I: 4,6,9,10-12,14,15,18,21
Ch. 2.2: 1-7,16,18,26
Ch. 2.3: 1-3,6,7,13,14,19,20,22,37,39
Ch. 2.4: 3,4,5,7,9,19,28,29,30,32
We will not cover chapters 2.5-2.7, but I will discuss briefly.
We will not cover chapter 3, but I will discuss it briefly.

Ch. 4.1 #2,11,12,13,15,18,19
Ch. 4.2 #5,7,21
Ch. 4.3 # 1,2,4,
Ch. 4.4 # 1-5, 18
Ch. 4.5: 4,11,12

Ch. 5.1: 4-6, 12,14,16,22,25,26,36
Ch. 5.2: Do any three of 8,9,10,13,14. Also do 18,19,22, any one of 26-28, 37
Ch. 5.3: 2,4,5,7-10
Ch. 5.4: 1-5,13,16,19,20

Chapter 6.1: 1,2,3,5,7,8
Chapter 6.2: 3,5,13,23
Chapter 6.3: 1-5, 15, 16
Chapter 6.5: #10, 12-19

Math 44 Final Exam Study Guide

1. One-dimensional cellular automata, like those we did in class.
2. Given a two-dimensional design, find the symmetries in it.
3. A problem involving the formulas F – E + V = 2  and DV = 2E, where F is the number of faces, E is the number of edges, V is the number of vertices, and D is the degree of each vertex.
4. Given a linear repeating pattern, find how it continues at a distant point, like problems we went over in the last class.
5. What makes a fractal a fractal, and what are some examples in nature or the world?
6. Decide whether a figure is a knot or the unknot.
7. Eulerize a graph by adding the fewest number of double back edges.
8. Decide whether a figure is a certain link or not (see the chapter on knots!)
9. Question on some of the recent mathematicians we studied.
10. The art gallery theorem, and the applications of modular arithmetic.
11. Find if a graph has a Hamiltonian Cycle or an Euler Circuit.
12. Calculate the fractal dimension – a calculator might help.
13. A logic problem like the one we did on Arc, Barc, and Carc.
14. A problem on the Fibonacci sequence, F(1)= 1, F(2)= 1, F(3)= 2, F(4)= 3, F(5)= 5, F(6)= 8,…
15. Find the symmetries in a frieze pattern, like the ones we worked on in class. (Those symmetries are either rotation, reflection, translation, or glide.)
16. A pigeonhole principle problem.
17. A take-away game problem, like those we did in class.
18. Given the degrees of the vertices of a graph, draw the graph.
19. Complete a figure so that it has certain specified symmetries, like those we worked on in class.
20. Given a polyomino, find a tiling of the plane with it and identify the symmetries present.
Bonus: answer a question about your reaction to the Math 44 class.

Class 22, Thu., Jun. 20, 2019
Final exam is Thursday, June 27, 4-6 PM (no class Tuesday).
Final exam is cumulative and has many multiple choice questions (bring a half-page scantron).
It is open book, open notes, old exams OK, caculator but not communication capable device OK.
Bring Ch. 6 homework to the exam, I will check at start of exam and give back to you - all students need to bring ch. 6.
I will have your take-home exam graded and can hand back on Tuesday, June 25 when I give final for my other class from 1:45 to 3:45.
Problems in sample tests below that we did not study or discuss are not included; no essay question.

Class 21, Tue., Jun. 18, 2019
We went over answers to take-home exam, went over homework for chapter 6 (due during final), and reviewed for final.

Here's a final exam sample problem list from a previous quarter. Here's another list of sample problems. 
And here are some solutions to the problems in the lists above. 

Class 20, Thu., Jun. 13, 2019
We went over calculation of fractal dimension, and also learned about iterating functions that represent population levels, and can produce mathematical models like that known as the butterfly effect. Please bring that two page handout to class on Tuesday completed and ready to turn in.
Also due on Tuesday is the take-home exam.
Please be working on chapter 6 homework, which we will discuss on Tuesday.

Class 19, Tue., Jun. 11, 2019
I gave back homework and activity sheets.
I also gave back the take-home exams and asked you to redo many of the problems. I went over details about many of them.
Your reworked take-home exam is due next Tuesday, June 18.

One of the items we went over had to do with modular arithmetic.
The Hidden Role of Modular Arithmetic reviews modular arithmetic and relates it to some other problems from chapter 1.
You should print this out and make sure you understand it!

We also set a due date of this Thursday, June 13 for the handouts on cellular automata, cylinder games, and torus games.
We started work on finding the dimensions of a number of fractals. Work on those and bring back to class on Thursday.

Chapter 6 homework is above.

Class 18, Thu., Jun. 6, 2019
We learned about how the Mandelbrot set is derived from a simple quadratic (2nd power) complex number function.
Here's an even more colorful zoomed version of the Mandelbrot set.
Here is a description of chaos theory and dynamical systems.
We worked more on the cylinder/torus problems - bring to class Tuesday to turn in.
We worked on another one-dimensional cellular automata - bring to class Tuesday to turn in.
We began learning about chaotic systems.

Chapter 5 homework will be due next Tuesday, June 11.

Class 17, Tue., Jun. 4, 2019
We leaned about frieze symmetries - there are seven kinds of linear frieze patterns.
We also worked on problems on the surface of a cylinder and a torus.
Then we went over a one-dimensional cellular automata that produces the design of a Sierpinski triangle.
We also learned how to calculate fractal dimension = [log (# new copies)]/[log (magnification factor)]

Chapter 5 homework will be due next Tuesday, June 11.

Class 16, Thu., May 30
We went over a take-away game example.
Students in the class made progress on problem 6, though no one came up with an explanation why what you think you discovered is true - that would be interesting!

We learned about complete graphs: K(n) is the complete graph on n vertices, with an edge from every vertext to every other vertex.
K(m,n) is the "complete bipartite graph on m and n vertices" with two sets of vertices, one set with m vertices, the other with n vertices, and an edge between each vertex in the m-set and each vertext in the n-set. We saw that K(3,3) (also known as the utility graph) cannot be drawn on the plane without edges crossing. Same with K(5), the pentagram, which we made with a string loop last week.
We also went over material on surfaces such as the torus, and saw how to draw K(3,3) and K(5) on a torus without edges crossing. I suggested you figure out how to draw K(6) on the torus without edges crossing.

Class 15, Tue., May 28
We went over more about Euler Circuits (a series of edges traveling on every edge exactly once and ending at starting vertex),
and Hamiltonian cycles (a series of edges passing through each vertex exactly once and ending at starting vertex).
We also learned how to "Eulerize" a graph by adding "double-back" edges in order to create a graph with all even degree vertices that then has an Euler circuit.

We also began chapter 5 material on surfaces and knots, and made some simple knots with our arms!

Class 14, Thu., May 23
We worked some on take home exam, also went over some graph theory.

Class 13, Tue., May 21
Class gave reports.

Class 12, Thu., May 16, 2019
We learned about Euler's Theorem, that the number of faces F, edges E, and vertices V of a polyhedron satisfy the equation V – E + F = 2.

We will not cover chapters 4.6 and 4.7. We started on chapter 5 and learned about graphs, and found all eleven graphs with four vertices (that have no loops or multiple edges).

We learned about Euler Circuits and Hamiltonian Cycles.

We also looked at the "Small World Phenomenon," and "six degrees of separation,"
Here are some more links:
Six degrees of Kevin Bacon.
Oracle of Bacon, where you can look up actor's links to other actors.
Kevin Bacon's charitable SixDegrees.org.
The Erdos Number project.
We also even learned about even Erdos Bacon Sabbath numbers!

How to make polyhedra with our fingers, see these two short videos:
Handmade Math
Polyhedra at Your Fingertips

Papers due on Tuesday by 6:15 PM!
Chapter 4 homework due next Thursday, May 23.

Class 11, Tue., May 14, 2019
We went over several types of symmetry practiced finding tilings or tessellations of pentominoes and found some symmetries in the tilings.
Please read this handout showing you how to find the symmetries in a tiling.

We learned how to make a finger geometry tetrahedron and cube.
Here are two videos on the finger polyhedra:
Scott Kim talking about and demonstrating the four-finger tetrahdron. 
The four-hand tetrahedron, and others.
We also built all five platonic solids and several other "Archmedean" or "semi-regular" polyhedra.

Chapter 4 homework will be due a week from Thursday.
Let me know if you have questions about the paper due next Tuesday.
Here are some titles and subjects of students' papers in Math 44 and Math 46 from previous quarters.

Class 10, Thu., May 9, 2019
We went over three symmetry handouts:90 degree rotational puzzle, Rotational symmetry handout, Reflection symmetry.
Please complete all three and turn in on Tuesday, May 14.

Symmetry leads us to the field of "tilings" or "tessellations." There is an enormous amount of recent mathematics and art  on this subject. Take a look at David Eppstein's enormous collection of links on tilings. He calls his collection of such links the Geometry Junkyard; it's very entertaining, in a mathematical and arts sort of way!

We also found - and named - all twelve pentominoes; here's a page on pentominoes.
Here are the names we came up with for the pentominoes.
Here's a handout showing you how to find the symmetries in a tiling (from a previous quarter - you did not have the assignment to find a tiling with a pentomino for homework, we worked on it in class instead!) Here are hundreds of links to some amazing tiling sites. Pentominoes each have five identical unit squares, linked complete edge to edge. For example, we might try to solve the same problem for the "hexomino:" there are 34 hexominoes, each having six squares. We did not find all the 6-ominoes, or hexominoes, but here is a diagram indicating briefly how all 34 hexominoes tile the plane. These are not shown in complete detail.

We also saw a modular arithmetic trick using mod 9: write down a 5 digit number, rearrange the digits and change one of the non-zero digits in the smaller of your two numbers (the original and the rearranged number) to zero, then subtract the smaller number from the larger. The result of the subtraction will have "digital root, " meaning the sum of the digits, mod 9, equal to the digit that was changed to zero.

Since we haven't finished going over chapter 4, your homework for chapter 4 will not be due this coming Tuesday.

Class 9, Tue., May 7, 2019
We heard the mathematician reports today, all very interesting, thank you!

Chapter 4 homework is above. It will be due next Tuesday, May 14.

I handed out a 90 degree rotational puzzle for you to work on. Please work on it and bring to class Thursday, we will finish it in class. (If you missed class, print it!)

Third Essay Assignment.
You have a short paper on a subject related to the mathematics of the course that catches your interest due Tue, May 21. You will turn the paper in via Turnitin.com.

Here's the description of the essay:
Report on an article or chapter from a popular book about mathematics. The report will be one to two pages long, typewritten, (it must be at least 600 words), and will cover the mathematics from one to several chapters of a book from the following list; other books or sources may also be used. You must use published material, not just web sites, unless you get permission from the instructor, and you MUST cite your sources. A short oral report to the class will also be required. You may do a powerpoint if you wish.
You should include in what you write and talk about:
(1) why you chose this topic, 
(2) what you learned, and
(3) what you think about the subject in question.
(4) What you might like to find out about the subject in the future. 

You may not do a report on the Fibonacci numbers or the golden ratio!

Examples of books with mathematical content:
The Mathematical Tourist and Islands of Truth, by Ivars Peterson.
Any of the books of Martin Gardner on mathematics (over 15 titles).
Game, Set, Math and Does God Play Dice by Ian Stewart, or other titles on math by Stewart.
The Mathematical Experience by Davis and Hersh.
A Number For Your Thoughts and Numbers At Work and At Play by Stephen P. Richards.
Tilings and Patterns by Grunbaum and Shepard.
Mathematical Snapshots by Steinhaus.
Mathematics: The New Golden Age by Keith Devlin, or other titles by Devlin.
The Emperor's New Mind by Roger Penrose.
The Mathematics of Games by John Beasley.
Archimedes' Revenge by Paul Hoffman
What is Happening in the Mathematical Sciences, ed. by Barry Cipra, Vols 1-5 (on reserve in campus library)

Examples of books with cultural content:
Ethnomathematics by Marcia Ascher.
You can also consult this Multicultural Mathematics Bibliography. Many of the references are in our library, and the bibliography contains call numbers for those that are in the library.
A number of Martin Gardner's books are in the De Anza library.

Class 8, Thu., May 2, 2019
We went over the answers to exam 1 during the first hour.
During the second hour we learned how to put two rhythms together to create a "polyrhythm" with length the least common multiple of the number of beats in the two rhythms.
For example, we had already seen how a 4 beat and 6 beat rhythm combine to produce a 12 beat rhythm.
We then learned how to draw a star polygon to show the rhythmic structure of the two combined rhythms, using a circular modular or clock arithmetic diagram (section 2.4 in the text)..

We also learned about the four types of planar symmetry by physically acting them out:
Translation, as in p --> p
Reflection, as in p --> q
180 degree rotation, as in p --> d
Glide, as in p --> b
We will be studying these in more detail in the coming weeks, as they are the basis for all symmetries in the plane.

Class 7, Tue., Apr. 30, 2019
First exam was today.

We went over lots of review problems on modular arithmetic, the UPC code, and the take-away game, for example.

Your mathematical biography is due Tuesday. If you were not in class last Thursday, you have been assigned someone to report on, see:
Mathematician bios assignment: background on various contemporary mathematicians.
The two books Mathematical People and More Mathematical People are on two-hour reserve in the De Anza Library.

On Thursday we will go over material in chapter 4.3 and 4.4 on the golden rectangle and symmetry.

Class 6, Thu., Apr. 25, 2019
Your first exam is Tuesday. It is open book, open notes, calculator allowed but not communication capable devices.
You will turn in your chapter 1 and 2 homework (see list above) at the end of the exam, which will take the second hour of class.
Bring a scantron half sheet for the exam.
See the study guides in the previous entry on this page below.

Today we went over UPC and ISBN codes, as examples of the uses of modular arithmetic in business and industry.
We went over the Pythagorean theorem (Ch. 4.1) and the Art Gallery Theorem (Ch. 4.2).
We went over the tea and milk problem from chapter 1 and saw how it could be made into a card trick with black and white cards or down-facing and up-facing cards.
We saw a parity card trick with the 20 high ranking cards in the deck - here is the site with the 4 rows of 5 trick that I did in class.

We also saw part of the film about Martin Gardner.
How to make polyhedra with our fingers, see these two short videos:
Handmade Math
Polyhedra at Your Fingertips

Here is someone doing that "tie a knot without letting go of the ends" trick!

Mathematician bios assignment: links for your report on various contemporary mathematicians.
If you were absent today, I've assigned you someone to report on, see assignment and links above.

Class 5, Tue., Apr. 23, 2019
Here is a study guide for the first exam from a previous quarter. 
Here is another practice exam from a previous quarter, more like what you will have, since it's a multiple choice/Scantron exam.
Please bring the half page long scantron for your exam 1. This will be open book, open notes, calculator allowed, one hour.

We went over the handout problems on the pigeon hole principle, the pattern diagram, and the water pouring problem.
We went over more about modular arithmetic.
We went over the 15 game, in which two players alternately remove numbers from the list (1,2,3,4,5,6,7,8,9} until one of them has three numbers with sum 15, which wins. We saw that this game is equivalent to playing tic-tac-toe on the 3 by 3 magic square. And we saw the connection between the magic square and remainders, mod 3.

We went over material on prime numbers, for example why there are an infinite number of primes. We also learned about several unsolved problems:
Are there an infinite number of twin primes, pairs of primes that differ by two?
Is the Goldbach conjecture true: is every even number greater than 2 the sum of two primes?
Are there an infinite number of Mersenne primes, those that are one less than a power of 2, like 3, 7, 31, and 127? Each of these leads to a perfect number, which is the sum of its factors other than the number itself.

You should be working on chapter 1 and 2 homework; you should also be reading the textbook! I mentioned in class that prime numbers are used to encode credit card numbers online, and and modular arithmetic is used to create codes like the UPC code, the ISBN code, etc., which are the subject of homework problems.

Class 4, Thu., Apr. 18, 2019
On Tuesday, April 23, turn in the Fibonacci assignment.
Chapters 1 and 2 homework to be turned in on Tuesday, Apr. 30.

We went over the 5,7, water pouring problem and saw that all numbers of gallons 1,2,3,4,5,and 6 are obtainable. We also saw that it seems likely that if two water containers hold m and n gallons, where n is the larger number, then every integer from 1 to n that is a multiple of the least common divisor (the GCD) of m and n is possible to be measured by pouring water back and forth.

We briefly went over the cops and comedians problem using a grid and coordinates from (0,0) to (3,3), where (m,n) means there are m comedians and n police officers on the right side of the river.

We did the "Where's Fido" logic problem. Here's a handout on the Fido puzzle.

We also examined Pascal's triangle, which arose in finding the number of paths between two points in a rectangular grid, but also as the coefficients in the raising of a binomial like (a+b) to a positive integer power.

We briefly went over the derivation of the golden section phi = 1.618... using "continued fractions;" this is done in detail in the textbook also. Please do read the textbook!
Here's more info about continued fractions and how they relate to the Fibonacci numbers. 
Here's a long intro to continued fractions.

I showed you a way to transform the 15-game into tic-tac-toe on a 3 by 3 magic square. In this game two opponents take turns removing a number from the set {1,2,3,4,5,6,7,8,9} with the goal of taking three numbers with sum 15.

We began studying modular arithmetic as a way to better understand numerical patterns.
Here is a short handout on modular arithmetic;
Here are some videos and links to sites about modular arithmetic, though also read the textbook chapter 2.4:
https://www.youtube.com/watch?v=Eg6CTCu8iio
https://www.youtube.com/watch?v=5OjZWSdxlU0
https://artofproblemsolving.com/wiki/index.php/Modular_arithmetic/Introduction
https://medium.com/i-math/intro-to-modular-arithmetic-34ad9d4537d1

Class 3, Tue., Apr. 16, 2019

We learned how to count Fibonacci numbers on pine cones.
On Tuesday, April 23, turn in the Fibonacci assignment.

Here are Vi Hart's Fibonacci number videos. The second video shows up at the end of the first, same with the third appearing after the second!
Here is a great site about Fibonacci numbers.
Here is a site about Pingala's possible use of the Fibonacci Numbers in ancient India. 
Look up Rachel Hall who credits Indian mathematician/musician with the Fibonacci numbers - see her article "Math for Poets and Drummers" listed at her site.

We also spent a good deal of time going over the pigeonhole principle, the pattern problem, and the 5 and7 gallon container problem.
The water pouring problem: you are given a 5 gallon and 7 gallon container and need to decide if you can use them along with an unlimited supply of water to measure exactly 1, 2, 3, 4, and/or 6 gallons?
Please turn that handout in on Thursday of this week, Apr. 18. Here is the handout page with the dots for use in the billiard solution method.

Please work on the Chapter 1 homework (see above), which we will discuss on Thursday.

Class 2, Thu., Apr. 11, 2019

Your math autobiography is due Tue., Apr. 16. If you have trouble getting into Turnitin.com, let me know. I added everyone's email to the site so you should be able to upload your paper easily.

You should have read through chapter one, have started working on the homework, and begun reading chapter 2 - we started the material in chapter 2.1 in class today.

We learned about the pigeonhole principle. In the pigeonhole principle "magic trick," I asked you to choose seven numbers from the list 1,2,3,...,12. The properties each of your lists had were:
(1) A pair of your numbers had a sum of 13.
(2) A pair of your numbers had a difference of 6.
(3) A pair of your numbers had a difference of 3.
(4) A pair of your numbers had a difference of 1.
(5) A pair of your numbers had the property that one divided the other equally.
We saw how the pigeonhole principle explained why properties 1, 2, 3, and 4 are true. For Tuesday, can you use the pigeonhole principle to explain property 5? Remember, it's all in how you label to (six) pigeonholes! Property 5 is more difficult to explain; in this case the six pigeonholes have different numbers of numbers assigned to them!

We also worked on the "Starbucks" pigeonhole problem: Five Starbucks are located in a 2 mile by 2 mile square area. Why are at least a pair of them less then 1.5 miles apart?
We answered the question by dividing the area into four 1 mile by 1 mile squares, and noticing that by the pigeonhole principle, a pair must fall in one unit square.
The greatest distance apart in the unit square, by the Pythagorean theorem, is the square root of two, which is 1.4.... miles, less than 1.5 miles.

We saw what happened when we play a six beat rhythm at the same time as a four beat rhythm, creating a polyrhythm of length 12 beats.
Twelve is the Least Common Multiple of 4 and 6, or LCM(4,6) = 12.

We worked on the "Police and Comedians" problem from chapter 1, and I suggested a way to examine the problem usng coordinates.

Speaking of coordinates, we also worked on the chapter one problem on using containers of size 6 and 10 gallons to get exactly 8 gallons of water.

All of these puzzles in chapter 1 are famous ones in what is known as "recreational mathematics." The water measuring problem, according to puzzle history expert David Singmaster and others, goes back to a book called "Annales Stadenses" compiled by the monk Abbot Albert in the year 1240 CE in Germany. The billiard ball solution method is apparently due to M.C.K. Tweedie in 1939. If given 3 containers holding x, y, and zgallons of water, there is no known simple method for determining which amounts can be measured if z happens to be smaller than x + y. For example, if you have three containers holding 6, 10, and 15 gallons, and an unlimited supply of water, which amounts 1 through 14 is it possible to measure exactly?

Here's a site where you can get free software for implementing the billiard ball approach to the water pouring problem. You have to download the "Mathematica Player" (also free) first.

Class 1, Tue., Apr. 9, 2019
Today we played the pattern game and the 1 & 2 take-away game, and also learned to make our first names into a clapping slapping rhythm, as well as a marching rhythm. Please practice!

In the take-away game, in which each player removes 1 or 2 counters on each move, the last player to move winning. 
We saw that the losing postions are all multiples of 3.
What happens if a player may remove 1, 2, or 3 counters on each move?

Chapter 1 homework to be added in with homework from chapter 2 when you turn that in, is to do any five of the following problems which begin on page 33: #3, 6, 8, 12, 13, 16, 17,18. Note that there are hints following the problems!

Here is the pattern game we played in class. Print this out and include in your portfolio.

Here are some of the vocabulary words we have used or will use soon during class related to the pattern game. Try to use each one in a sentence, to make sure you understand them: 
multiple: 12 is a "multiple" of 3 and of 4. 3 and 4 are "factors" of 12. Is 13 a multiple of 1? Is 0 a multiple of 13?
horizontal (row): parallel to the horizon. Often means we are thinking about right and left.
vertical (column): up and down
odd numbers : 1,3,5,7,... These numbers are "congruent" to 1, mod 2. Is -1 an odd number? (We'll learn about number "congruence" soon!)
even numbers : 0,2,4,6,8, .... Is 0 even? These numbers are "congruent" to 0, mod 2. Is -6 and even number?
alternate or alternation: a pattern in which two "sub-patterns" are each displayed in every other section of the pattern.