Orbits of the Hopalong Map

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This Demonstration plots a number of orbits of the two-dimensional iterative map known as the Hopalong map. The Hopalong map can be represented by the recursion equations: , , with and ranging over the reals and equal to either 0 or +1.


Different orbits are generated using different initial points defined by the "" locators. Initially, 25 orbits are plotted. You can drag the initial points or add or delete new ones (Alt+Click on Windows, Command+Click on Macintosh) inside the plot. Alternatively, up to 50 random initial points can be launched.

The map parameters , , and can be varied manually or a random set can be used. Use the "range" slider as a magnifier to see more detail inside some orbits. Check "full range" to see all points from all orbits.


Contributed by: Erik Mahieu (July 2011)
Open content licensed under CC BY-NC-SA



The Martin map equations were taken from [1] and [2].

The simplified, two-parameter equations for the map were suggested to the author by Barry Martin.

The Martin map is also called the "Hopalong-Attractor".

See also the German website: Huepfer.


[1] M. Trott, The Mathematica GuideBook for Programming, New York: Springer–Verlag, 2004 pp. 347–349.

[2] B. Martin, "Graphic Potential of Recursive Functions," in Computers in Art, Design and Animation (J. Landsdown and R. A. Earnshaw, eds.), New York: Springer–Verlag, 1989 pp. 109–129.

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