Day 24, MAT115

• Today we begin one of my favorite topics: Fractals!

  You may have never heard of fractals, but you've encountered them.

• Here are some videos on today's
  1. Gallery of Fractals
  2. Making Fractals

1. A gallery of knots
2. Distinguishing knots, using Reidemeister moves and Tri-colorability

This is fun stuff, folks -- I hope that you're going to enjoy this!

• Other rectangles turn golden: you've seen fractals in this class already, and maybe heard me mention them. I might have mentioned them in the context of the Fibonacci numbers, and in the context of the Fibonacci and Golden spirals.

  In the latter two cases, you recall that by constructing spirals with squares in certain ways, we ultimately achieve a golden rectangle, which contains a perfect copy of itself (only smaller): the side lengths are in the ratio 1 to $\phi$, where \[ \phi=\frac{1+\sqrt{5}}{2} \]

  • Now: what kind of rectangle will you get if you repeat the spiral process with a non-Fibonacci rectangle to start (rather than a 1x1 square)? For example, you might start with a 3x1 rectangle, and then append the biggest square you can each time.

    Turns out, that if you do it over and over and over again, you'll still obtain rectangles that becomes more and more golden!

    But that's because we were adding a square each time. Let's try another process....

  • The A0 Paper Fractal

    "A paper" is something you've no doubt encountered before: it's the long sheets we occasionally use (usually "A4" paper), types of paper far more common in Europe.

    "A paper" is constructed so that, if folded in half (do we say the long way or the short way?), you get a sheet which has exactly the same shape -- that is, the ratio of its side lengths is the same as the original sheet.

    

    Let's see what the dimensions of A paper must be so that this requirement is fulfilled....

    Call the long dimension of the A1 paper above $x$, and the short dimension 1.

    • Then the ratio $x:1$ is the same as the ratio $1:\frac{x}{2}$.
    • Solve for $x$: \[ \frac{x}{1}=\frac{1}{\frac{x}{2}} \] and we've got another quadratic equation on our hands! Fortunately we recall how to "work our quads" (and this one happens to be particularly easy): \[ x^2=2 \] so $x=\sqrt{2} \approx 1.4142$.

    Let's check: here are the "official" paper sizes (in mm):

    

    Pick a height, and divide by the width, and what do you get? Approximately 1.4142.... E.g. A1: $841/594 \approx 1.4158$.

  • Why not an A0 paper spiral? Let's try that.... This is the spiral, created using non-squares ("A-rectangles") each time:
    

    We could go on forever! There's "the world within the world". This is a fractal.

  • Fractals
    • The concept of fractals is the most modern mathematics that we study in this course. We've worked our way up from the Babylonians all the way to the 21st century.

      Here is our "intuitive" definition of a Fractal:

      A world within a world!

      By which I mean that there's a perfect copy of the fractal contained within itself -- and perhaps infinitely many!

      For example, in the A0 Fractal, the whole rectangle (A0 paper) contains smaller (but perfect) copies of itself -- A1, A2, A3, .... papers.

      This key notion is more formally called "self-similarity": "a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts)."

    • Some other examples you have already seen during our course:
      • Dual Platonics: e.g. a tetrahedron within a tetrahedron within a tetrahedron.... Maybe cooler are the other pairs, where we alternate solids (cube then octa, or icoca then dodeca), then repeat the process:
        

      • The tree diagram of the rabbit problem that gave rise to the Fibonacci numbers:
        

      • I shared my "Flirting with Death Spiral" a couple of weeks ago:
        

      • Often these arose out of the idea that we carry out a process, then "do it again, do it again"....

    • Here are some examples you might have seen before:
      • Here's a suitable mascot for this unit: Bessie the fractal cow:
        

      • World's Most Famous Fractal (Benoit Mandelbrot, Novel Mathematician, Dies at 85: 10/14/2010):

    • So what is a fractal? A couple of things we'll generally see:
      1. It's a process or graph that possesses "self-similarity".

      2. It often possesses a simple and recursive (or iterative) definition -- that is, we do something, and then we "do it again!"

    • Let's check out a beautiful gallery of natural fractals.

      It turns out that nature loves fractals, just like it loves Fibonacci numbers.

      I found this image in my February issue of Science magazine:

      

    • Building deterministic fractals with sticks
      • Consider the famous "Koch" fractal:
        1. Start with a straight stick.
        2. Replace the middle third of the stick with a little "tent" -- two third-length sticks.
        3. Do it again and again!
        

        This fractal becomes infinitely long, but in a confined space! Very strange.... but this type of strange behavior is typical of fractals.

        Let's make a variation of that. Use your graph paper, and a stick along the edge that's 27 units long. Best to use a pencil with a good eraser!

        Instead of triangular tents, we'll make square tents.

      • Here are some more lovely examples (including the Koch fractal, but check out the beautiful bushes!).

      • The Sierpinski triangle is an area fractal.
        • Triangular paper is good for Sierpinski....
        • Triangular paper is good for Sierpinski....
        • Let's create one by subtraction.
        • After we're finished removing all that stuff, how much area is left in the triangle?

    • Now let's introduce a little bit of randomness: Sierpinski's triangle again

      The Chaos game - generating fractals using random movement!

      • Once again, simple rules lead to complex objects. This time, however, we don't proceed methodically, but rather haphazardly - and yet we still produce the complex images!
        • with 10 points
        • with 100 points
        • with 500 points
        • with 1000 points
        • with an infinite number of points!

        From randomness comes order; from simple rules comes complicated objects! Then all hell broke loose....

• L-Systems in PostScript, from which I generated this image:
  

• I first encountered the L-Systems fractals at britton.disted.camosun.bc.ca/fractals_arcytech/lsystems.html, but that's a dead link. However, I copied the files, and have a local copy here. However, web browsers these days don't seem to like the applet tag in javascript, so this doesn't work anymore in a standard web browser...:(