Day 14, MAT115

• I'm sorry that I've not gotten your homework's graded yet. I've got a key here, in case you want to make sure that you're doing them right, however.

• Materials for today:
  1. Here are a couple of videos:

     1. a video covering some practice on Egyptian Division

     2. a video covering today's new topic of symmetry
  2. Here are a couple of docs:
     1. Examples using multiplication backwards
     2. Example using the Unit Fraction Table

1. That Egyptians, although they were "base 10" people, probably also gravitated toward the powers of 2 (they liked doubling and halving).
2. That "fair divisions" -- making portions look fair to their enslaved populations -- may have had important ramifications socially. The Family Circus comic makes this plain: Dolly things that her sandwich, cut into two pieces, is somehow less than PJ's sandwich, because it's been cut into four pieces.....

   

• We started last time with some simple examples, and we should do a warmup, that sounds like a hard one (but it's easy, because it divides evenly -- I want to show that it's basically just multiplication backwards): what's 14941/223

  1	223
  2
  4
  8
  16
  32
  64

  On the right we build the "missing part" of the quotient -- 14941 -- and on the left we build up our answer, using the powers of 2 corresponding to the "useful" doubles of 223.

  The divisor is the thing that gets doubled. In this case we get lucky, and it divides evenly.

• Let's try this one: 31 divide by 13? This is about as tricky as it gets! We'll do it in two ways: as "multiplication backwards", and using the Unit Fraction Table: Egyptians had this fascination with fraction that had a numerator of 1. The only fraction (other than the the unit fractions) they used was 2/3. Nonetheless, just FYI, 2/3 can be written as \[ \frac{2}{3} = \frac{1}{2} + \frac{1}{6} \]

1	13
2

Hermann Weyl (German Mathematician; 1885 - 1955)

• Where have we used symmetry already?
  • Yanghui's triangle had an error in it....
  • There's a certain symmetry between Egyptian multiplication and division.
  • There's a type of symmetry in the golden rectangle: within a golden rectangle is found a perfect scale copy of itself.

  Where else does symmetry appear, and how do we use it?

  Are humans symmetric? Turns out that we're purported to have a "good side" and an "evil side"!

• Symmetry can be used in two different ways:
  • to simplify, and
  • to complexify!
  We can use symmetry to solve some problems, and we can use symmetry to make things more interesting. Let's start with the latter:
  • Contradancing
  • Haeckel and the radiolaria: On the Discovery Docket: Proteus

• Now to help us break down some of the most important elements of these topics, I want to distribute this worthy handout, or four pages of it at any rate.

  It covers two kinds of symmetry that are very important: rotational and reflective.

  So let's take a look at the definitions.

  Your homework: to do the problems on the first three pages of this symmetry handout (see the assignments page).

• In the plane of 2-Dimensions,
  • What is the most symmetric rectangle?
  • What are regular polygons?
  • How many regular polygons are there?
  • What kinds of symmetry do they possess?
  • Which regular polygons can be used exclusively to "tile" the plane, like a bathroom floor?

• Unit Fraction Table