Fractals

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Fractals

The Geometry of Nature

Roll Over Euclid & Tell Old Plato the News...

A fractal is a geometrical object that repeats itself at ever smaller scales. A fractal is generally a fragmented shape that can be split into parts, each of which approximates a reduced-size copy of the whole. This property is called self-similarity. The term fractal was coined by Beno�t Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes infinite iteration. Iteration refers to processes that repeat indefinitely. A fractal has the following features:

Infinitely zoomable.
Indescribable in traditional Euclidean geometry.
Self-similarity repeating itself at ever smaller scales.
Fractional dimension.
Recursive definition.
Simulates natural objects like clouds, mountain ranges, lightning bolts, coastlines, and snow flakes.

Fractal Dimensions

You are traveling through another dimension — a dimension not only of sight and sound but of mind. A journey into a wondrous land whose boundaries are that of imagination.

—Rod Serling

Here is a rather trivial question with an obvious answer: How many lines will it take to make a line twice as long as the red line? Answer: 2 of them. Notice that a line has one-dimension and that 2¹ = 2. Likewise, to make a line 3 times as long will require 3 copies of the original line. 3¹ = 3.
How many squares will it take to make a square twice the size of the small square? Answer: 2² = 4 of them. How many squares will it take to triple the size of any square? Answer: 3² = 9 of them. See the pattern? You have seen it before in your study of squares.
How many cubes will it take to make a square twice the size of the small cube? Answer: 2³ = 8 of them. How many cubes will it take to triple the size of any cube? Answer: 3³ = 27 of them. See the pattern? You have seen it before in your study of cubes.

This is the rule:
If you know how much you want to scale an object,
and you want to know how many copies you need,
raise the scale you want the object to the dimension of the object.

Scale^Dimension = Copies

This rule applies also to shrinking an object. If you want to shrink an object to half its size, raise ½ to the object's dimension. For example, to shrink a square by one-half, notice that
½² = ¼. This means that ¼ of a square scales down the square by ½.

Let's apply this idea to fractals. With fractals, we know the scale we want, and how many copies we need; but we do not know the dimension. Using the formula above, how do we solve for dimension d? Answer: use the logarithm function on your calculator. A logarithm, in case you forgot, is simply another name for exponent.

log Scale^Dimension = Dimension x log Scale = log Copies

which is to say

Dimension = log Copies / log Scale

Classical Fracitals

Here are examples of fractals from the Wikipedia and elsewhere. Observe their scaling factor and count the number of copies required to enlarge them or shrink them.

Euclid's Triangle

7 Stages in the Line with its Middle Thirds Missing

Square Dust

Sierpinski

Holey Trinity

carpet

Cell Phone Antenna or a Moth-Eaten Carpet?

sponge

Sponge Bob

snowflake

Snow Flake

stool

Toad Stool

Sausage

Link Sausage

$fractalbox$
Fractal Box

Analyze the fractals above and complete the table below. In the SCALE column, enter the scaling factor needed to make a larger copy. In the COPIES column, enter how many copies you need to make a larger copy. If you do not understand these instructions, ask someone. Enter integers in the first two columns. In the DIMENSION column, enter the dimension of the fractal. This number requires 4 place precision:

Dimension = log Copies / log Scale

10 Point Table — 1 Point Per Fractal (after the first 3)

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Last Modified 12/4/08 1:12 PM