Day 15, MAT115

• tetrahedron is self dual
• hexahedron and octahedron are dual
• dodecahedron and icosahedron are dual

Amoeboid protozoa Circogonia icosahedra:

and even saw their duals in nature (e.g. Molybdenum Dichloride).

• Key
• Median: 6.25 (ouch!)
• 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!

• We started off with regular polygons in two dimensions, and discovered that there were similarly super-symmetric objects in three-dimensions.

  People have made use of these solids for many unusual purposes: for example, someone constructed this Platonic grape arbor:

  

  And I read a story in the Guardian, about the science behind Salvador Dali's works (such as The Sacrament of the Last Supper):

  

  In a gorgeous video, Prof. Carlo Sequin begins by showing how to start from regular polygons -- two-dimensional regular polytopes -- and build all the Platonic solids -- three-dimensional polytopes; but he eventually shows how to build all the regular polytopes in all higher dimensions!).

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

• A little further along in the video, Prof. Sequin describes all the symmetries of the four-dimensional version of the octahedron.

  Can you guess how many symmetries there are for the octahedron? The question is, in how many ways can you rotate 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)?

• Finally, towards the end of the video, Prof. Sequin uses duality to obtain the third class of polytopes.

  Back to regular polygons for the moment:

  What is the dual of
  • an equilateral triangle?
  • a square?
  • a pentagon?
  What can we say in general?

• translations (e.g. wallpapers)
• symmetry of scale

• Wallpaper is often made up of a single repeated pattern, sometimes reflected, and then translated over the entire wall periodically to make an attractive impression.

  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:

  

  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)

• Today's Question of the Day:
  What is the most beautiful rectangle?
  • 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.

• 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)