Day 27, MAT115

This session will be captured on Zoom, if I remember to turn it on, and record it.

Our page of zooms and the play-by-plays.

Plus an article from the New York Times, which describes why geometry is our special human superpower....

There are also some throwback questions, to make sure that you're getting ready for our next exam (not this Friday, but the following Friday, 4/8).

An Euler example:

Keep going:

What's the fewest number of doors you could close (or add!) that would make possible

1. an Euler path?
2. an Euler circuit?

(By the way: don't call it Fibonacci Nim, because that gives away a secret toward the strategy; just call it "Nim", or "Fraudini Nim"!)

This work features one of the topics we'll be studying down the road -- the Platonic solids. But it is also famously regarded as a painting in which the artist made conscious use of the golden ratio, which we cover today. The dimensions of this painting (the rectangular frame) create a (nearly) "golden rectangle" -- the most beautiful rectangle, according to some of the ancient Greeks.

• We'll use the hexagonal paper to create a version of the Pascal triangle.
• Now let's find those Fibonaccis....
  

Ordinarily we build the Fibonacci spiral by building bigger and bigger rectangles. The shapes of the rectangles change as we go along, in such a way that the ratio of side lengths are Fibonacci numbers. Let's look at the sequence of the ratios....

So let's recap the spiral building process, with a focus on those side ratios.

• Start with a piece of graph paper (the wide way), and darken the square in the 10,10 spot (ten from the left, ten down from the top). You might want several colors, if you've got them -- might as well make this pretty!
• Rule: attach the largest square you can: first left, then down, and then continue in a counter-clockwise fashion.
• As you add each square you create a new rectangle. Compute the ratio of side lengths (larger over smaller).
• Do it again, do it again, do it again! (You could do it forever -- but we run out of paper.)
• What do you notice about the dimensions of the rectangle? It's the ratio of those dimensions that interests us most -- the ratio of dimensions (larger to smaller) dictates the shape of the rectangle. Two rectangles that have the same ratio have the same shape.

  1x1	1/1=
  2x1	2/1=
  3x2	3/2=
  5x3	5/3=
  8x5	8/5=
  13x8	13/8=
  21x13	21/13=
  34x21	34/21=

• The thing we are approaching is the so-called golden rectangle, whose side-lengths are in the ratio called "the golden ratio". But we will derive the golden rectangle in a different way -- the way that the Greeks did it! When we're done, we'll have also defined the golden mean, aka the golden ratio.

• Nature loves Fibonacci numbers. Many items in nature spiral, and, if you see spirals, look for Fibonacci numbers:
  • Begin with a rectangle.
  • Remove the largest square possible from the rectangle, to leave a rectangle.
  • If the rectangle that results has exactly the same shape as the original rectangle, then that rectangle is golden:
    

  What this means is that the golden rectangle preserves its shape under this operation, of being reduced in size by having the largest square removed. That's not true of the Fibonacci rectangles -- their shapes are changing, as we can see by the changing ratios. Ultimately, at the "center", is a square!

  However if we look back at the Fibonacci spiral sequence of rectangles as they're growing, we see that they're tending toward a "golden rectangle".

• If you did your reading (and of course you did!), you know that the ratios of successive Fibonacci numbers tend toward a certain number, which is called the "golden mean" or "golden ratio": \[ \varphi=\frac{1+\sqrt{5}}{2} (\approx 1.618033988749895) \]

  Let's show that this golden ratio is, in fact, the side-length ratio of the golden rectangle described above. And our secret weapon will be your old friend, that old favorite, the quadratic formula!

  While we might think about the Fibonacci spirals as being created by attaching squares, the golden rectangle is created by removing squares. We just do things backwards....

  We can build beautiful rectangles by pasting squares together, or we can define beautiful rectangles by taking them away....

(sidelengths: 466x288, with ratio 1.6180555555555556!)

You'll be making these, too, soon, using your own photos. Let's do one in class now (if we're lucky this will work!)

Here's a video clip of Fred Astaire and Ginger Rogers. They say Fred Astaire was pretty good (well, "the best dancer ever"); but Ginger Rogers did everything Fred did, only backwards (and in heels...).

That's like this golden rectangle process, and like Egyptian division: backwards and in heels.