Sawtooth (cellular automaton)

In a cellular automaton, a finite pattern is called a sawtooth if its population grows without bound but does not tend to infinity.

[1] Their name comes from the fact that their plot of population versus generation number looks roughly like an ever-increasing sawtooth wave.

For instance, in Rule 90, a one-dimensional elementary cellular automaton, the population size starting from a single live cell follows Gould's sequence, which has a self-similar sawtooth pattern.

As the population grows with this pattern, its live cells trace out the rows of a Sierpinski triangle.

[4] 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.

The population growth in Rule 90 starting from a single live cell, measured by Gould's sequence .
The number of alive cells plotted versus the number of elapsed generations for the first sawtooth discovered in the Game of Life
Example of a sawtooth pattern living through several drops below the maximum live cell count. Click on the image to see the cell pattern.