A Spherical XOR Gate Implemented in the Game of Life
Are there uniquely spherical cellular automata machines? Might there be computational processes that come about more naturally on spheres than they would in the plane? This chapter describes an exploration of geodesic grids as environments for cellular automata (CA) and specifically addresses the movements of Game of Life (GoL) gliders whose interactions are affected by the positive curvature of spheres. 2D CA are typically arranged on regular planar grids with periodic boundary conditions — equivalent to the topology of a torus. This chapter instead considers the dynamics of CA on spheres. The unavoidable discontinuities that arise from mapping a 2D grid onto the sphere are accepted as integral components of the environment. A novel XOR gate built on GoL is demonstrated, utilizing the double-crossing of glider paths following geodesic great circles.
KeywordsCellular Automaton Cellular Automaton Great Circle Platonic Solid Spherical Curvature
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- 1.Berlekamp, E.R., Conway, J.H., Guy, R.K.: What Is Life? Winning Ways for Your Mathematical Plays, vol. 2: Games in Particular, Chap. 25. Academic Press, London (1982) Google Scholar
- 2.Grunbaum, B., Shephard, G.C.: Tilings and Patterns. Freeman, New York (1987) (balanced tilings explained in pp. 129–134) Google Scholar
- 3.Kiester, R.A., Sahr, K.: Planar and Spherical Hierarchical, Multi-resolution Cellular Automata. Computers, Environment and Urban Systems. Elsevier, Amsterdam (2008) Google Scholar
- 4.Langon, C.: Life at the Edge of Chaos. Artificial Life. Addison–Wesley, Reading (1992) Google Scholar
- 6.Rendell, P.: Turing universality of the Game of Life. In: Adamatzky, A. (ed.) Collision-Based Computing. Springer, Berlin (2002) Google Scholar
- 7.Rennard, J.P.: Implementation of logical functions in the Game of Life. In: Adamatzky, A. (ed.) Collision-Based Computing. Springer, Berlin (2002) Google Scholar
- 8.Yukita, S.: Dynamics of cellular automata on groups. IEICE Trans. Inf. Syst. E82-D (10), 1316–1323 (1999) Google Scholar
- 9.Yukita, S.: Cellular automata in non-euclidean spaces. In: Proceedings of the 7th WSEAS International Conference on Mathematical Methods and Computational Techniques in Electrical Engineering, pp. 200–207. World Scientific and Engineering Academy and Society, Stevens Point (2005) Google Scholar