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Turns out, that if you do it over and over and over again, you'll get a rectangle that gets more and more golden.
Let's try: we'll use a piece of grid paper (looking short and squat), and a 3x1 rectangle (start it running the tall-way at the 8th column, 12th row.
Now, are the ratios looking golden?
"A paper" is something you've no doubt encountered before: it's the long sheets we occasionally use (usually "A4" paper), which is 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 , and the short dimension 1.
In both cases there's "the world within the world". Both are fractals.
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)."
It turns out that nature loves fractals, just like it loves Fibonacci numbers.
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!
Intstead of triangular tents, we'll make square tents.
The Chaos game - generating fractals using random movement!
Here's a better version of that, that has more general options.
From randomness comes order; from simple rules comes complicated objects! Then all hell broke loose....