MAT115R: Math for Liberal Arts

Also begin watching the Mathemalchemy documentary (you might just want to skim through it at the moment to get the big idea of it). I'll point out some of the highlights as we go along....

Read Rock Groups

Try computing the prime factorizations for some numbers (and draw their trees): or are they prime?

• 64
• 210
• 229

• Make sure that you can draw all of the Complete Graphs up to \(K_8\), using colors to illustrate the divisibility of the number.
• Please read Early Concepts of Number and Counting for next time.

• "...a messenger could be sent to another tribe with the message that they wish to trade 20 baskets of food for 15 pearl necklaces, say. This could have been done by simply indicating the point on his body that corresponded to the correct number of objects." Use the "Body counting" guide on page 3 to answer the following:
  • Which parts of the body would the messenger point to, for 20 and 15?
  • And if it were 33 baskets of food for 11 pearl necklaces?
  • Would you trade left foot little toe baskets of food for right knee pearl necklaces?
  • Do you notice any rhyme or reason to the ordering of the body parts?

• Draw a cartoon of the Baker and the woman buying bread, comparing their tally sticks.
• Draw a knotted string like that of "The Example 3643", but for the number 1294.
• How do the South Africans mentioned count to numbers above 100?

1. How do we write the following numbers of sheep by partition (using the ternary trees?):
   • 9 sheep
   • 31 sheep
   • 54 sheep

2. Can you go backwards? How many sheep is meant by the following:
   • 1,0,1,1
   • 1,1,0,1,1,0,0
   • 1,0,0,1,0,1,1,0,1
3. Try the binary card trick on your friends and family. Are you successful in reading minds? (Don't give away your secrets!)

• Write the first sixteen natural numbers as unique binary strings.
• Given the binary string, tell what number it is:
  • 1011
  • 1101100
  • 100101101

• Check out Vi Hart teaching us how to Binary hand-dance. See if you can dance along!
• Word of the day: binary! Try out the Great Fraudini's trick on an unwitting victim; make sure that you can do it.

  If your victim were thinking of the following numbers, on which cards do the numbers appear, and why?

  • 63
  • 27
  • 42
  Can you draw the tree and write the number using the primitive counting trick?

• Background History of "Pascal"'s triangle
• Properties of "Pascal"'s triangle

If you want to solve the following problems, we can use particular rows of Pascal's triangle:

• If you toss a fair coin 7 times, what are the chances that it comes up

  0 heads	1 head	2 heads	3 heads	4 heads	5 heads	6 heads	7 heads

• If there are three friends in a Facebook "group" (or graph), call them A, B, and C, draw all possible configurations of the Facebook by adding in arcs as "friendships" (and relate your results to a row in Pascal's triangle).
• If you have 8 friends, but can only take 5 in your car, how many different carloads are possible?

1. To compute the number of Facebooks possible for five people, you would have to calculate the number of arcs possible (10), and then go to the 10 row of the table for Pascal's triangle. Either compute the 10th row (not a bad exercise), or look it up, and then answer these questions:
   • How many ways can you create a Facebook with 2 friendships between the people?
   • How many ways can you create a Facebook with 4 friendships between the people?
   • How many ways can you create a Facebook with 6 friendships between the people?
   • How many ways can you create a Facebook with 0 friendships between the people?

2. Use the combinations formulas from class to find the following:

   \[ \begin{array}{c} C^3_{2} = \cr C^5_3 = \cr C^7_6 = \cr C^7_4 = \cr C^{10}_6 = \cr \end{array} \]

1. Demonstrate Egyptian multiplication by multiplying the following (write out the table, and check your work):
   1. 13*34
   2. 23*79
   3. 81*123
   4. 255*256

1. Demonstrate Egyptian division by dividing:
   1. 9/4
   2. 13/8
   3. 30/5
   4. 105/7
   Try these using the same sort of "doubling/halving" table that we use for multiplication.

Please read the following from Georges Ifrah's "The Universal History of Numbers":

• Babylonian's Positional System
• Mayan's Positional System

• 21
• 171
• 360
• 400
• 3600
• 47331

You should finish the translation of the Mayan calendar, on your own, so that you know how to write and interpret Mayan numbers.

Sean noticed that the first four numbers at the top were just multiples of 177. But that eventually fails. In which column?

1. Please read this short description of the Fibonacci numbers.

2. As usual, Vi Hart makes math fun! Doodling in Math: Spirals, Fibonacci, and Being a Plant [Part 1 of 3]; here is Doodling in Math Class: Spirals, Fibonacci, and Being a Plant [2 of 3] and Part 3.

3. Write the following numbers as a sum of non-consecutive Fibonacci numbers:
   1. 155
   2. 24
   3. 99
   4. 128

4. If you were playing me in Fibonacci Nim, two questions: do you go first or second, and, if first, what's your first move?
   1. 47
   2. 24
   3. 34
   4. 19

You might try some of the "art" features.

Email me this, and we'll hold an art show of spirals, with "get out of quiz free" prizes.... (Due Wednesday, 3/5)

1. Read this symmetry handout (pages 1-6, 8).

   Carry out the following exercises on that page:

   1. A1, A4 (p. 1)
   2. B1, B3, B4 (p. 2)
   3. C1 (p. 3)

2. Also read this article (from The New York Times, March 22, 2022), as we move into geometry: Is Geometry a Language That Only Humans Know? (here's the source -- the New York Times). Neuroscientists are exploring whether shapes like squares rectangles and and our ability to them recognize are part of what makes our species special.

   (There are two fun games described in this article, to test your instinctual geometric abilities. Look for them on the left side-bar.)

You will be allowed to use these as a cheat sheet for the next exam.

• Find examples of company logos which involve Platonic solids (a Google search, e.g. cube logos will get you started)

• Draw 2-dimensional projections of each of the Platonic solids. That is, a realistic view of a platonic solid on 2-dimensional paper. Try your hardest to do this well!

• Count vertices, faces, and edges for a soccer ball, and verify that it satisfies Euler's formula.

• Construct floor plans of houses that have Euler paths through them (remember that doors are "bridges" and rooms are "land masses", in terms of the Konigsberg bridges problem).

• Enjoy this bizarre Vi Hart video of a Sierpinski triangle made of Candy Corn!
• Attempt these exercises

Can you identify some fractals in nature, or in human society? Can you create some fractals of your own, using sticks, rectangles, triangles, etc.? Try using the on-line gasket maker to make some fractals, and try out the L-system fractals. What can you create?

You've already created a fractal of your own, if you tried my Fibonacci Spiral Fractal maker!

• Mobius Music Box
• Pachelbel's Music Box Canon in D (not Mobius, but having seen the first you might enjoy this!)
• Wind and Mr. Ug: a Mobius tale

• Snakes and Graphs
• Borromean Onion Rings

Check out this careful introduction to using Reidemeister moves: Knot Theory - What's Knot to Love?

• Decide whether the Borromean rings are tricolorable or not.
• Decide whether the two five knots are tricolorable or not.
• Identify the knots or links in this "story", which I call A Knotty Tale. You may need to apply the Reidemeister moves to convince yourself that a picture of a knot is really the unknot, say. You can also try tricoloring pictures, to eliminate some candidates, or bolster the claims of others.

For Monday: please provide a one paragraph description of your project and a one paragraph description of your logo; I want to make sure that everyone has something interesting to share on the following Monday and Tuesday, April 28 and 29.

For a much more thorough introduction to both Reidemeister moves and tricolorability, check out Tricolorability of Knots, by Kayla Jacobs.

A review of knots: see if you can follow along as I make a transition between the circle and that thing that looked like a four knot (but is truly an unknot). Is it not tricolorable in each rendition?

1. Watch What is Zeno's Dichotomy Paradox? - Colm Kelleher
2. Please read The Hilbert Hotel.
3. This on-line article summarizes our work in the Hilbert Hotel.

Then watch the Numberphiles episode which covers the material from today's lecture.

Make sure to email me your logo at least several hours before Monday's class!

For those struggling with Cantor's diagonalizability argument, listen in to see what Vi Hart has to say about it!