MAT115: Math for Liberal Arts

Also please begin reading the Mathemalchemy comic book (in particular the part about the chipmonks), which will give you an important mathematical overview of the installation as well as detail on the prime factorizations and the sieve of Eratosthenes. These readings (and video) will be the focus of Thursday's quiz.

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

• 64
• 210
• 229

Check out Vi Hart teaching us how to Binary hand-dance. See if you can dance along! Then you might continue on, and see how she uses binary trees to prepare her Thanksgiving feast (vegetarians and vegans: serious meat warning....).

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

• Imagine that you're playing Fraudini's trick with a friend: for each of the following numbers, indicate which cards contain that number by writing each number as a sum of powers of 2 (you might use a tree to create the factorization):
  1. 53
  2. 49
  3. 31

• For the following use the method of "primitive counting" described in class:
  1. Turn the following into the appropriate string of 1s and 0s by drawing the appropriate ternary tree:
     1. 32
     2. 63
     3. 97

  2. Turn the following strings of 1s and 0s into the appropriate number of sheep. Drawing the ternary tree is desired!
     1. 1,0,1,0,1,0
     2. 1,0,1,0,1,0,1
     3. 1,0,1,1,0,0,0,1

• 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?

This one-page reading gives a third version of the triangle, from the Arab world, and a little background on the influence of Indian culture on our number systems.

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

• 21
• 171
• 360
• 400
• 3600
• 47331

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 (but only match to sample is working). Look for them on the left side-bar.)

• Creating The Never-Ending Bloom (the artistry of John Edmark)
• The Art of John Edmark | Talk by Paul Dancstep | Exploratorium: Fibonacci spiral checkerboards (and other symmetries of scale)

• Geometry (a local copy)

  This one introduces you to another generalization of the Platonic solids -- to the Archimedian solids, and then to the more general Johnson solids, of which there are 92.

• Symmetry (a local copy)

For Thursday: explore the 17 wallpapers patterns, and try to find ways to distinguish them.

This site can help out a lot! Distinguishing the 17 wallpaper groups. In particular, take a look at the glide reflection symmetry.

Email the resulting image to me (taking a screenshot of it may be easiest).

• play her Mobius Music Box. (If you liked that one, there's more here!)
• tell the story of Wind and Mr. Ug
• play with her food (fruit by the feet)

• Unknot ("Wedding Band"; untwisted paper)
• (Pair) of mirrored right-handed and left-handed Mobius Bands
• Hopf Link (cut a twice-twisted band in two; or cut a Mobius band into "thirds", to create a short mobius band linked with a twice-twisted band in a Hopf Link)
• Trefoil knot (cut a thrice-twisted band "in two")
• "Solomon's knot" (cut a four-twisted band in two, to make a link)
• Cinquefoil knot (cut a five-twisted band "in two")

We'll have another gallery of those, and more prizes.... Mixed media collections are encouraged (you can make them out of fruit roll-ups if you wish...).

• This was assigned because it relates to knots and links, but it also relates to graphs. Vi is doing double duty for us here! (I hope that you also watched her video about the Mobius Music Box).
• Notice her graphs in the first part of the video, including the graph of an octahedron!
• By making a closed-curve squiggle graph, you're creating a knot. (A knot is a single piece of string, in which the two cut ends are ultimately tied back together to create a single continuous piece of string -- which may be "knotted"!)
• "Borromean snakes": "No two snakes are actually linked to each other" (1:22).
• Just before the Borromean snakes there is a trefoil snake (1:14).
• "What kind of knots are you drawing and can you classify them?"

• 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.

Here's another version of that, with a transcript.

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

Several of our paradoxes are covered in this Numberphile's video, Infinity Paradoxes.