In a cellular automaton, a finite pattern is called a sawtooth if its population grows without bound but does not tend to infinity. In other words, a sawtooth is a pattern with population that reaches new heights infinitely often, but also infinitely often drops below some fixed value. Their name comes from the fact that their plot of population versus generation number looks roughly like an ever-increasing

sawtooth wave The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle. A single sawtooth, or an intermittently triggered sawtooth, is called a ...

In rules with small replicators

For instance, in Rule 90, a one-dimensional elementary cellular automaton, the population size starting from a single live cell follows

Gould's sequence Gould's sequence is an integer sequence named after Henry W. Gould that counts how many odd numbers are in each row of Pascal's triangle. It consists only of power of two, powers of two, and begins:. :1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, ...

, which has a

self-similar __NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...

sawtooth pattern. On each step whose number is a power of two, the population crashes from a high of the step number plus one to a low of only two live cells. As the population grows with this pattern, its live cells trace out the rows of a Sierpinski triangle. The sawtooth shape of this pattern can be used to recognize physical processes that behave similarly to Rule 90.. In Rule 90 and in many cellular automata such as Highlife, the sawtooth pattern is based on the existence of a small replicator, which in Rule 90 consists of a single live cell.

In Life

In Conway's Game of Life, replicators are large and difficult to construct. Instead, the first sawtooth in Life was constructed by Dean Hickerson in April 1991 by using a

loaf A loaf ( : loaves) is a (usually) rounded or oblong mass of food, typically and originally of bread. It is common to bake bread in a rectangular bread pan, also called a loaf pan, because some kinds of bread dough tend to collapse and spread ...

tractor beam. For a number of years the least infinitely repeating population of any known sawtooth was 262 ON cells, attained by a sawtooth found by David Bell on July 9, 2005.

Expansion factor

The expansion factor of a sawtooth is the limit of the ratio of successive heights (or equivalently, widths) of the "teeth" in plots of population versus generation number. Some sawtooths do not have an expansion factor under its standard definition because some sawtooths have growth that is not exponentially-spaced.

References

{{Conway's Game of Life Cellular automaton patterns