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I hope that this has you imagining "Russian Dolls", which are those which stack in such a way that there is a perfect copy of a doll inside each doll, only smaller (self-similarity at a smaller scale):
Today we're going to examine fractals in a couple of different contexts, and then talk about how to make a few.
In the latter two cases, you recall that by constructing spirals with squares in certain ways, we ultimately achieve (or tend to) a golden rectangle, which contains a perfect copy of itself (only smaller): the side lengths are in the ratio $\phi$, where \[ \phi=\frac{1+\sqrt{5}}{2} \]
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....
"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.
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: $841/594 \approx 1.4158$.
We could go on forever! There's "the world within the world". This is a fractal.
(By the way, the spiral image above has dimensions 424x300 -- the ratio? 1.4133333333333333....)
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)."
You'll be making one of very own Spiral Fractal as half of this week's quiz grade. So get on it! Let me illustrate....
It turns out that nature loves fractals, just like it loves Fibonacci numbers.
I found this image in a recent issue of Nature:
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 short edge, that's 88 units long. Best to use a pencil with a good eraser! We'll want a stick that's 81 squares long (because we're dividing into thirds).
Instead of triangular tents, we'll make square tents.
The Chaos game - generating fractals using random movement!
One of the most interesting fractals arises from what Michael Barnsley has dubbed ``The Chaos Game'' [Barnsley]. The chaos game is played as follows. First pick three points at the vertices of a triangle (any triangle works---right, equilateral, isosceles, whatever). Color one of the vertices red, the second blue, and the third green.
Next, take a die and color two of the faces red, two blue, and two green. Now start with any point in the triangle. This point is the seed for the game. (Actually, the seed can be anywhere in the plane, even miles away from the triangle.) Then roll the die. Depending on what color comes up, move the seed half the distance to the appropriately colored vertex. That is, if red comes up, move the point half the distance to the red vertex. Now erase the original point and begin again, using the result of the previous roll as the seed for the next. That is, roll the die again and move the new point half the distance to the appropriately colored vertex, and then erase the starting point.
From randomness comes order; from simple rules comes complicated objects! Then all hell broke loose.