We present what we argue is the generic generalization of Conway's "Game of Life" to a continuous domain. We describe the theoretical model and the explicit implementation on a computer.
Generalization of Conway’s ”Game of Life” to a
continuous domain - SmoothLife
Stephan Rafler
N¨urnberg, Germany
frlndmr@web.de
November 27, 2024
Abstract
We present what we argue is the generic generalization of Conway’s
”Game of Life” to a continuous domain. We describe the theoretical model
and the explicit implementation on a computer.
1
Introduction
There have been many generalizations of Conway’s ”Game of Life” (GoL) since
its invention in 1970 [1]. Almost all attributes of the GoL can be altered: the
number of states, the grid, the number of neighbors, the rules. One feature
of the original GoL is the glider, a stable structure that moves diagonally on
the underlying square grid. There are also ”spaceships”, similar structures that
move horizontally or vertically.
Attempts to construct gliders (as we will call all such structures in the fol-
lowing), that move neither diagonally nor straight, have led to huge man-made
constructions in the original GoL. An other possibility to achieve this has been
investigated by Evans [2], namely the enlargement of the neighborhood. It has
been called ”Larger than Life” (LtL). Instead of 8 neighbors the neighborhood
is now best described by a radius r, and a cell having (2r + 1)2 −1 neighbors.
The rules can be arbitrarily complex, but for the start it is sensible to consider
only such rules that can be described by two intervals. They are called ”birth”
and ”death” intervals and are determined by two values each. These values can
be given explicitly as the number of neighbors or by a filling, a real number
between 0 and 1. In the first case, the radius has to be given, too, in the last
case, this can be omitted.
The natural extension of Evans’ model is to let the radius of the neighbor-
hood tend to infinity and call this the continuum limit. The cell itself becomes
an infinitesimal point in this case. This has been done by Pivato [3] and inves-
tigated mathematically. He has called this model ”RealLife” and has given a
set of ”still lives”, structures that do not evolve with time.
1
arXiv:1111.1567v2 [nlin.CG] 7 Dec 2011
2
SmoothLife
We take a slightly different approach and let the cell not be infinitesimal but of
a finite size. Let the form of the cell be a circle (disk) in the following, although
it could be any other closed set. Then, the ”dead or alive” state of the cell is
not determined by the function value at a point ⃗x, but by the filling of the circle
around that point. Similarly, the filling of the neighborhood is considered. Let
the neighborhood be ring shaped, then with f(⃗x, t) our state function at time t
we can determine the filling of the cell or ”inner filling” m by the integral
m = 1
M
Z
|⃗u|
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Reference
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