Day 15, MAT115

• Key
• Median: 8
• Let's talk about a few things that folks did well at, as seen on the key.
• Some of you brought your mirrors (used your reflective phone screens, etc.). Good thinking!

• The table:

  	# of Vertices	Edges	Faces	faces at each vertex	sides at each face
  Tetrahedron	4	6	4	3	3
  Cube	8	12	6	3	4
  Octahedron	6	12	8	4	3
  Dodecahedron	20	30	12	3	5
  Icosahedron	12	30	20	5	3

• Each Platonic solid has a "twin" -- called its dual -- which we can discover from the table.

Dualing examples:

• Pastel drawing of a compound of Molybdenum Dichloride:
• Superconductivity:

• The cube you've undoubtedly done before.
• The tetrahedron can be drawn to be a "distorted" Mercedes symbol
• Here are all the solids. Practice drawing them! Where's the duality?

• In the following graphic, you see another way to represent these objects in two-dimensions, by unfolding them:

People have made use of these solids for many unusual purposes: for example, someone you know constructed this Platonic grape arbor on his farm in Canada:

Salvador Dali's works included a fair amount of math (such as The Sacrament of the Last Supper):

We'll watch about 7 minutes of this, but I encourage you to watch the rest when you can:

Can you guess https://en.wikipedia.org/wiki/Octahedral_symmetry? The question is, in how many ways can you rotate or reflect an octahedron to get what appears to be the same octahedron (which which, in fact, involved mixing it up by moving vertices, edges, and faces around)?

Back to regular polygons for the moment:

• an equilateral triangle?
• a square?
• a pentagon?

In how many different ways (from the standpoint of distinctly different symmetries) can one create such a pattern?

Suppose you have a design, of a mouse say, such as the prototypes of the mice of Mathemalchemy:

Nine of the possible wallpaperings are shown on the bakery wall of Mathemalchemy:

And throughout the exhibit you can see various other "wallpaperings", in different media.

The seventeen distinctly different wallpaper groups: (the following images are taken from this Wikipedia page, Wallpaper group)

• We've made some Fibonacci spirals -- but today we want to make Golden spirals.

• As we move toward the so-called "golden mean" and "golden rectangle", we'll start with a nice Fibonacci spiral (we've done this already: see this site for more on these spirals).

  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.
    • 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 Fibonacci 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....

    But each rectangle remaining is a perfect copy of the original! In contrast to our Fibonacci spiral, in which case each was successively more beautiful, each of these is a perfect copy -- smaller, yes, but perfectly the same. This is an example of symmetry of scale. And while we can, of course, tile the floor with these, we're going to need some tiles of vastly different sizes (some getting incredibly small).

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

• Now what can we do with these ideas?

  Here is a somewhat silly example -- A Fibonacci spiral fractal comic I made recently, which I have entitled "Flirting with Death Spiral". Infinite fun! I'll be having you make one of these yourself, so you'll want to think of a good photo to use....

  

  (sidelengths: 466x288, with ratio 1.6180555555555556!)

  By the way, Salvador Dali's image is sized 334x208: side ratio, 1.6057692307692308

• One last example of tiling with the exact same shape of scaling sizes is provided in a beautiful video which we may watch today, if we have time.