The idea is to have iterative fractals of continuous functions (see also IFS or iterated function system). In the limit, it's hard to tell whether such a fractal would be continuous by the end of things, but at the beginning it looks possible. Below is an example of a non-continuous function (Koch curve):
Base (iteration 0):
_____________________
Generator (iteration 1):
/\
/ \
_______/ \________
Iteration 2:
__/\__
\ /
___/\___/ \___/\___
Okay, so it's pretty ugly. I hope the general idea is clear however. My idea is to take some continuous ("nice") function and iterate it in a similar manner; examples of "nice" functions might include sin, a binomial with rotational symmetry about the origin, or other functions with such rotational symmetry.
Here's one function I'm thinking of (sin(pi*(x+1))):
Base (iteration 0):
----------------------------------------------------------------------------------------------------
Generator (iteration 1):
,,oO```Oo,,
,oO` `Oo,
,/` `\,
,/` `\,
,/` `\,
/` `\
,/ \,
/` `\
,/ \,
/` `\
,/ \,
,/` `\,
`\ /`
\, ,/
`\ /`
\, ,/
`\ /`
\, ,/
`\, ,/`
`\, ,/`
`\, ,/`
`Oo, ,oO`
``Oo,,,oO``
Iteration 2:
,.ooOO''OOo.,,
/'` `''OOo.,
/ `'\,
|` `\,
|, \
\ '|
|, ,|
\ /
\ |'
\, /
`\ ,/ ,,.ooOO''````'\,
\, /` ,,.ooOO'`` |
----------`\---------,/----------------------,,.ooOO'``----------------------/`--------\,-----------
| ,,.ooOO'`` ,/ `\
`\.,,,,..ooOO'`` /` \,
,/ `\
,| \
/ \
|` `|
|, \
\ `|
\, ,|
`\., /
`'OOo.., ,./
``'OOo..oOOO'`
The generalized form of iterations (as I see them) are as follows:
Variables:
t (for Theta. The equations below are generally considered parametric. The range for t will probably be -q to q where q is an "interesting" solution of the generative function.)
x (the infamous variable from algebra. I'll be using x for graphing as well.)
y (the companion variable for x. Ditto the graphing bit.)
p (not quite sure where I'm getting the label from. This is the number of subsections to divide the prior iteration into when iterating.)
Functions (grouped according to iteration):
Base (iteration 0):
f0(t)=t
g0(t)=0
Generator (iteration 1):
f1(t)=t
g1(t)=(pick an "interesting" function)
Iterations (iteration a):
Let F(t) be the derivative of fa-1(t).
Let G(t) be the derivative of ga-1(t).
Let s be the integral from -q/(p^(a-1)) to q/(p^(a-1)) of sqrt(F(t)+G(t)).
fa(t)=fa-1(t)-sqrt(G(t)/(1+G(t)))*ga-1((p^(a-1))*s*t)/((p^(a-1))*s)
ga(t)=ga-1(t)+sqrt(1/(1+G(t)))*ga-1((p^(a-1))*s*t)/((p^(a-1))*s)
The difficulty I keep having is the integral ("s" above). That is a basic length integral for parametric equations instead of standalone equations. Specifically, sin(x) is nasty in this integral. In graphing the fractal, let x=fa(t), and y=ga(t).
It appears that sin(x) would be quite a fascinating function to use here, but for that length integral. x^3-x is also interesting, but also difficult. |
