MDDM: Fractals: Going back a couplatwotree pages

Judy Panetta judy at firemist.com
Fri Dec 7 19:22:46 CST 2001


Digressing a bit, but this small point was brought to my attention the other
day...FWIW

"...circl'd by the hellish Cusps of Peaks unnatural, - frozen in midthrust,
jagged at every scale." MD pg.134

"Hmph," says my physics, computer science, cum artist friend, "'jagged at
every scale,' humph...fractals."


fractal: any of various extremely irregular curves or shapes for which any
suitably chosen part is similar in shape to a given larger or smaller part
when magnified or reduced to the same size.

In the mood for some elementary functions? C'mon, it hasn't been that long
since we took calculus...an' doncha jus love da math:

A fractal is simply a graph of a different type of function. If you're ready
to get into the thick mathematics, including the third element of fractal
geometry, imaginary numbers. Here's a fractal function: f(n) = f(n) * f(n) +
c or f(n)^2 + c. This is known as the recursion law. This specific equation
will form a fractal known as the Julian set. In this equation, c = a complex
number(contains an imaginary number). It can be of any value and the result
will be a different Julian set. n stands for the coordinates of the point.
Where coordinates are (x, y), in fractal geometry, it would be represented
as x + iy. In other words, x is the constant, and y is the imaginary number.
In fractal geometry, the x-axis represents the reals, and the y-axis
represents the imaginaries. Back to the fractal function, we use the new
coordinates (x + iy) for n. Instead of the result of the function being a
line, it represents only one point. The point is located at n, the
coordinates. Take the point (2 + 1i). For our c value, we'll use (1 + 1i).
Remember, the c value can be any complex number. Now we'll plug it into the
equation.

f(n) = f(2 + 1i) =

(2 + 1i)(2 + 1i) + (1 + 1i) =

2*2 + 2i + 2i + i^2 + 1 + 1i =

5 + 5i + -1 = .............remember i^2 = -1

4 + 5i

These are our new coordinates. Remember, if you run a set of coordinates
through a function, the result is a new set of coordinates. 4 + 5i is the
new set of coordinates. The work shown above represents one iteration. We
continue to run each new set of coordinates through the function until we
can prove that the point will a) leave the graph (example: on a ten by ten
graph, the new coordinates are (-234, 97)) or b) never leave the graph (the
rule is after 200 iterations, if the point is still on the graph, it will
never leave.) This is how a color is selected. If the point leaves after one
iteration, it is assigned a color. Every point after, that leaves the graph
after one iteration, is that same color. All points that leave after two
iterations will be assigned a different color, and so on. Every point that
never leaves the screen is assigned one color, usually black. After doing
this process for each and every point of the graph, the result could look
something like this Julian set.

http://www.kcsd.k12.pa.us/~projects/fractal/what2.html





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