International Journal of Mathematics and Mathematical Sciences
Volume 8 (1985), Issue 3, Pages 579-587
Weak gardens of Eden for -dimensional tessellation automata
Mathematics Department, University of Central Florida, Orlando, Florida, USA
Received 16 July 1984
Copyright © 1985 Michael D. Taylor. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
If is the parallel map associated with a -dimensional tessellation automaton, then we say a configuration is a weak Garden of Eden for if has no pre-image under other than a shift of itself. Let the set of weak Gardens of Eden for and the set of Gardens of Eden (i.e., the set of configurations not in the range of ). Typically members of satisfy an equation of the form where is the shift defined by . Subject to a mild restriction on , the equation always has a solution , and all such solutions are periodic. We present a few other properties of weak Gardens of Eden and a characterization of for a class of parallel maps we call -characteristic transformations in the case where there are at least three cell states.