Patterns for Simple Cellular Automata in a Universe of Dense-Packed Spheres

R ules for describing certain simple cellular aut omata in a uni ver se of dense-packed sp heres are presented an d explored . Speclfically investigated are oscilla tors and other stable forms for a rule int roduced in [2] as a. candidate for 2. game of Life. T he "signature" of a form is defined , and t he signature s are given for all known smaUer forms for the above ru le . Rates of growth a rc di scu ssed and computer implementation techn iqu es are mentioned. 1. Int roduction In a three-dimensional un iverse of close-packed spheres, wh ich is a lso called a hexahedral tesselation [4J, each sphere, or cell, has 12 immediate neighbors. T he spat ia l relationsh ip between neighbo rs is best illust rated by placing a cell at the cente r of a cuboct ahedron. Th e 12 corne rs th en show the locati on of neighbor cells (see figure l -AL which are arranged in fou r intersecti ng planes, each containing a hexagonal ring of six neighbors touching th e cent ra l cell. For examp le, in figure I-A, the four planes are defined by (1,2,3,4,5,6) , (1,7,8,4,12,10), (2,7,9,5,12,11), and (3,8,9,6,10,11). F igure 1-8 shows the neighbors of a central cell as spheres; here, the cente r cell is surrounded by its neighbors and hence is ba rely visible. Although one can easily envision space when it has been packed full (see figu re I -C), a disp lay of just a few cells arranged in some configuration can be confusing. To help alleviate th is problem, all illustrations of configurat ions are shown from th e same perspect ive: namely, the viewer is always looking st ra ight down , in a dir ecti on perpendicular to plane (1,2,3,4,5,6) . 1. 1 Some no tation Let n denote a universe of dense-packed spheres; R3 will be used to denote our occas ional reference to a. th ree-dimensional Cartesian uni verse. A cell can have two states, which we will ca ll living and dead. T he next gene rat ion sta te of a cell is de termined by the current state of the cell and its 12 immediate @ 1987 Complex Systems Publicat ions, Inc.