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Conservation laws in physics are numerical invariants of the dynamics of a system. This article concerns conservation laws in a fictitious universe of a cellular automaton. We give an overview of the subject, with particular attention to problems of combinatorial flavor. 1 Kepler’s Laws and Selective Observation Much of science arises from looking at nature through a highly selective glass, which eliminates all the irrelevant details and singles out a particular feature to be studied. The strength of this approach flourishes when the filtered feature has a nice description of its own, which is independent of the eliminated parts. A conservation law is the simplest of such descriptions. It asserts that a certain quantity associated to a system remains constant throughout the evolution of the system. Perhaps the first example of a conservation law found in physics is the intriguing discovery by the German astronomer Johannes Kepler (1571–1630) [27] of the laws governing the motion of the planets. Kepler knew from his large collection of data, gathered from astronomical observations, that each planet follows, not a perfectly circular orbit, but an elliptic one, with sun on one of the focal points. The speed of the planet is not uniform either. Whenever it is farther from the sun, the planet moves slowly, while once it comes closer to the sun it circles around ∗Research supported by the Academy of Finland grant 211967. Figure 1: Kepler’s selective observation: dS ′ = dS it faster. Kepler was able to put this quantitatively, by realizing that the axis connecting the planet to the sun sweeps out equal areas within equal time segments (Figure 1). In other word, all through its orbit, the area-sweeping rate dS/dt of each planet remains constant (cf. [1]). 1 In this article we discuss such laws in cellular automata. A cellular automaton (CA for short) is an abstract structure, consisting of a d-dimensional checkerboard (d = 1, 2, 3, . . .). Each cell of the board has a state chosen from a finite set of states. The state of each cell changes with time, according to a uniform, deterministic rule, which takes into account the previous state of the cell itself and those in its neighborhood. The changes, however, happen synchronously, and in discrete time steps. One of the simplest CA exhibiting a non-trivial conservation law is the Traffic CA, which resembles cars moving on a highway. This is a one-dimensional CA, consisting of an infinite number of cells arranged next to each other on a line. Each cell has two possible states: (interpreted as a “car”) or (“empty space”). At each step, a car moves one cell forward, if and only if, its front cell is empty. Figure 2 shows a typical space-time diagram of the evolution of the Traffic CA. Not surprisingly, the number of cars on the highway is preserved by the evolution of the CA. As a two-dimensional example, consider the following discrete model of an excitable medium due to Greenberg and Hastings [16]. The CA runs on a twodimensional board. Each cell is either “at rest” (state ), or “excited” (state ), or is in a “refractory phase” (state ). A cell which is at rest remains so unless it is “stimulated” by one or more of its four neighbors (i.e., if at least one of the neighbors is excited). An excited cell undergoes a 1-step refractory phase, before going back to rest. Typically, a configuration of the infinite board contains a number of “singularities” with waves continuously swirling around them. See Figure 3 1This was later coined the law of conservation of angular momentum. Figure 2: A typical space-time diagram of the Traffic CA. Time evolves downward. The highway is directed toward left.