Schrandt-Ulam Cellular Automata
In 1967, Schrandt and Ulam  proposed a curious growth rule.[more]
A cube of the next generation is formed if:
(1) It is contiguous to one and only one cube of the current generation.
(2) It touches no other previously occupied cube except if the cube should be its "grandparent".
(3) Of this generation satisfying the previous condition, all those that would touch each other are eliminated. There is an exception for those cubes that have the same parent; these are allowed to touch.
This Demonstration shows this growth rule. Since the code is slow, precalculated values are used after 30 steps.[less]
 R. G. Schrandt and S. M. Ulam, "On Recursively Defined Geometrical Objects and Patterns of Growth," Technical report LA-3762, Los Alamos, NM: Los Alamos Scientific Laboratory, 1967. permalink.lanl.gov/object/tr?what=info:lanl-repo/lareport/LA-03762.