Iterations II - Hopalong

Here we have a look at the Martin-Attractors, also known as Hopalongs, a sort of orbit-fractals. They are images of a simple two-dimensional iteration system. The name Hopalong is derived from the fact, that such an image is built of points hopping along on an elliptical path starting from one point in the center. Hopalong orbits were discovered by Barry Martin from the Aston University, Birmingham, England. A.K. Dewdney presented the Hopalongs in the magazine "Scientific American" (1986) and in Germany they became famous because they have been mentioned in the magazine "Spektrum der Wissenschaft".

The algorithm

An image is calculated using three parameters a, b and c. Dewdney's article contains the following programme:

INPUT a, b, c
x = 0
y = 0
PLOT(x, y)
FOR i = 1 TO num
xx = y - SIGN(x) * [ABS(b*x - c)]^0.5
yy = a - x
x = xx
y = yy

ABS is the absolute value function. SIGN(x) is the same as x/ABS(x). If x>0 then SIGN(x) = 1, if x<0 then SIGN(x) = −1 and if x = 0 then the result of SIGN(x) is zero, too.

After a certain number of points the color is changed. Normally The image doesn't depend on the first point. These attractors have the butterfly effect. That means if you change the parameters a bit, you'll get a total new image with very little similarity to the first one.

Hopalong Program

Screenshot of the Hopalong Program

The program

A modern version of the program is available for a free download. The data files for the gallery images are included.


Parts of the Iterations sections were created in cooperation with Henning Kopp.


Math&Art Gallery


Galerie Cosch


© 2007 Ulrich Schwebinghaus