# Sawtooth

A sawtooth (plural: **sawtooths**^{[note 1]}) is a finite pattern whose population grows without bound but does not tend to infinity. In other words, it 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 graph.

The first sawtooth was constructed by Dean Hickerson in April 1991 by using a loaf tractor beam (a technique that was also used in the construction of sawtooth 633). The least infinitely repeating population of any known sawtooth is 177, attained by Sawtooth 177; the smallest bounding box of any known sawtooth is 62 × 56, attained by a variant of the same pattern, Sawtooth 195.^{[1]}

## 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 they have growth that is not exactly exponentially-spaced (see parabolic sawtooth and sawtooth 1163).

## See also

- Caber tosser
- Sawtooths (category)

## Notes

- ↑ The term "sawteeth" is not in common use.

## References

- ↑ Maia Karpovich (October 31, 2015). Re: Smaller sawtooth (discussion thread) at the ConwayLife.com forums

## External links

- Sawtooth at Wikipedia
- Sawtooth at the Life Lexicon