Hopalong Fractals

This applet iterates equations of the form

x(n+1) = y(n) - sign(x(n)) * sqrt(abs(b*x(n)-c))
y(n+1) = 1- x(n)

where b, and c are parameters and sign(x(n)) returns 1
if the argument is positive, -1 otherwise.

The Controls
Run Starts iterating and drawing.
New Colors This button generates a new pseudorandom color-LUT
DotSize provides for selection of the dot size to be used for painting. Range is from 1..5.
Randomize b, Randomize c These controls can be used to manipulate the parameters c and b. When clicked, new random values for the parameters b resp. c, are generated
Range b, Range c These edit fields can be used to set the range of the random values for parameters b and c. The input must be in -100 ... 100. Example: If Range is set to 5, then the resulting random numbers will range from -5 ... +5
Offset b, Offset c These edit fields allow to define an offset which is added to the randomly generated parameters b and c. The input must be in -500 ... 500.
Example : If Range is 5 and Offset is 50 then the resulting values for b resp. c will range from 45 ... 55
Scale This selector defines a scaling factor for the generated pictures.
Type This selector allows to chose from the 6 possible formulas used for iteration
1. x(n+1) = y(n) - sign(x(n)) * sqrt(abs((b * x(n)) - c ))
2. x(n+1) = y(n) + sign(x(n)) * sqrt(abs((b * x(n)) - c ))
3. x(n+1) = y(n) + sign(x(n)) * sqrt(abs((b * x(n) * x(n)) + c ))
4. x(n+1) = y(n) - sign(x(n)) * (abs((b * x(n)) - c ))
5. x(n+1) = y(n) - (sign(x(n)) * cos(abs((b * x(n)) - c )))
6. x(n+1) = y(n) - (sign(x(n)) * log(abs((b * x(n)) - c )))

y(n) = 1.0 - x(n) ( for all types )


If no dots are painted, select dotsize > 1 !!

After typing in values into the editfields, don't forget to hit the ENTER key !!!

Some of the modes need b resp. c values in a certain range. In this case, no dots are painted and nothing seems to happen. Simply define new parameters.

These fractal types were found by Barry Martin, Aston University, Birmingham,
England and has been published in the Scientific American's column
"Computer Recreations" by A. K. Dewdney.


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