Day 16, MAT115

Last Time

Next Time

Announcements:
- Sorry, I haven't gotten your Egyptian multiplication homework graded yet.
- According to the website, you have a homework due today. But it involves Egyptian division, and since we just learned how to do that using the unit fraction table, let's put off the homework. Let's make that homework due Thursday after the break.
- Have a good and safe fall break!
Question of the day: are humans symmetric?
Today we'll start with a few more Egyptian divisions: using either the multiplication table backwards, or the unit fraction table, or one other sneaky trick.
- Here's a really ugly one.
  $\frac{17}{25}$
  - = 1/3 + 1/5 + 1/15 + 1/25 + 1/50 + 1/75 + 1/150
    By the Unit Fraction Table,
  - = 1/2 + 1/8 + 1/25 + 1/100 + 1/200
    By the multiplication table
  - = 1/3 + 1/5 + 1/75 + 1/10 + 1/30
    By a method I've not shown you yet!
    And finally,
  - = 1/2 + 1/6 + 1/75
    By a method due to Fibonacci, given in the Liber Abaci
- Here's a relatively easy one: Suppose Fatima had 3 loaves to share between 4 people. How would she do it?
  (Think about what the answer means, in terms of bread.)
- A little trickier:
  1. How would you divide 5 by 7? (use the multiplication table backwards)
  2. How can we use the unit fraction table to get the same answer?
- How would you like to do story problems like this one?!
  Notice the numerals and fractional notation, as we see in our text on pages 13 and 29.
Now let's talk about symmetry:

Symmetry, as wide or as narrow as you define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.
Hermann Weyl (German Mathematician and Physicist; 1885 - 1955)
- Where have we used symmetry already?
  - Yanghui's triangle had an error in it....
  - Pascal's triangle is full of beautiful symmetry, which we even generalized -- and we used symmetry as one of the important preserved quantities.
  - There's a certain symmetry between Egyptian multiplication and division -- one is left-handed, the other right-handed.
  - There's a type of symmetry in the golden rectangle: within a golden rectangle is found a perfect scale copy of itself. The same thing holds true for Fibonacci numbers -- there's a symmetry in the bunny pair table. Each newborn pair is just like every other newborn bunny pair.
  - The Fibonacci spiral has a scale symmetry. As you zoom out on it, it will continue to look the same. A Fibonacci reflection.
  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 problems, and we can use symmetry to make life more interesting. Let's start with the latter:
  - Contradancing
  - Haeckel and the radiolaria: On the Discovery Docket: Proteus
  - Animals quite frequently show some kind of symmetry
- Now to help us break down some of the most important elements of these topics, I offer 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.
  Part of your homework: to do the problems on the four pages of this symmetry handout. (If a problem says something like "on sheet F1-2", don't worry about it.)
- 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?
- Links:
  - Nature by Numbers -- nice video!
Website maintained by Andy Long. Comments appreciated.