An iteration due to ~~artist~~ mathematician [1] Barry Martin produces intricate doiley-like patterns by iterating a simple mathematical function. I ran across this via [2].

The images produced are sensitive to small changes in the starting parameters *x* and *y*, as well as to the parameters *a*, *b*, and *c*.

Here are three examples:

And here’s the Python code that was used to make these plots.

import matplotlib.pyplot as plt from numpy import sign, empty def make_plot(x, y, a, b, c, filename, N=20000): xs = empty(N) ys = empty(N) for n in range(N): x, y = y - sign(x)*abs(b*x - c)**0.5, a - x xs[n] = x ys[n] = y plt.scatter(xs, ys, c='k', marker='.', s = 1, alpha=0.5) plt.axes().set_aspect(1) plt.axis('off') plt.savefig(filename) plt.close() make_plot(5, 5, 30.5, 2.5, 2.5, "doiley1.png") make_plot(5, 6, 30, 3, 3, "doiley2.png") make_plot(3, 4, 5, 6, 7, "doiley3.png")

[1] Tom Crilly identified Martin as an artist in [2] and I repeated this error. Barry Martin informed me of the error in the comments below.

[2] Desert Island Theorems: My Magnificent Seven by Tony Crilly. The Mathematical Gazette, Mar., 2001, Vol. 85, No. 502 (Mar., 2001), pp. 2-12

If you’re interested in Hopalong attractors like this, see: https://examples.pyviz.org/attractors

Those are reminiscent of a Poincare section of a KAM torus. I suspect they’re the same phenomenon at some level. The KAM torus arises generically by a nonlinear perturbation of a periodic motion. In this case the underlying periodic motion is found when a=b=c=0.

If you google Barry Martin you will find that the Hopalong iteration is not due to ‘The Artist’ Barry Martin but to me the mathematician Barry Martin. I must have created a lot of custom for the artist, but I don’t mind that! Tony Crilly has long since apologized for the error. My discovery was pure serendipity when I was researching the Henon attractor. I retired in 1998 am now 84 and still besotted by maths. Hopalong continues to provide surprises.