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##### Records

Minimum-population sawtooth

##### Records

David Bell has reduced the minimum repeating population of any known sawtooth from 262 to 260 cells. The sawtooth, similar to his 262-cell variant (from 2005), uses a 4-engine Cordership and stationary p256 gun. The space between the dynamic and static ends of the sawtooth is alternately filled with gliders and emptied.

The population plot confirms the sawtooth behaviour of the pattern, showing that it is an *exponential* sawtooth. This is not the only possible envelope for a sawtooth, as Dean Hickerson built a *parabolic* sawtooth, so named because the convex hull of the population graph is a parabola.

David's sawtooth, despite having a very low population, still may not be optimal. There are two figure-8 oscillators between the p256 gun and four-engine Cordership, which could potentially be removed. Apparently, he searched for a sawtooth in which the interior of the gun performs this task, but to no avail.

We leave it as a challenge to the reader to create a sawtooth with a smaller minimum repeating population.

UPDATE: Thanks to Matthias Merzenich for noticing that I had confused David Bell and Paul Tooke; the article has now been amended.

There is a huge variety of sawtooth patterns, summarised in the table below:

Name | Discoverer | Year | Return rate | Maximal population |

Sawtooth 1212 | Dean Hickerson | 1991 | Exponential | O(t) |

Cord puller | Dean Hickerson | 1991 | Exponential | O(t) |

Parabolic sawtooth | Dean Hickerson | 1991 | Quadratic | O(sqrt(t)) |

Sawtooth 633 | Dean Hickerson | 1992 | Exponential | O(t) |

Hacksaw | Dean Hickerson | 1992 | Exponential | O(t) |

Sawtooth 562 | Hickerson & Coe | 1992 | Exponential | O(t) |

Sawtooth 1846 | Dean Hickerson | 1992 | Exponential | O(t) |

Sawtooth 362 | David Bell | 1992 | Exponential | O(t) |

Externally-timed sawtooth | David Bell | 1992 | Exponential | O(t) |

Moving sawtooth | David Bell | 2005 | Exponential | O(t) |

Sawtooth 260 | David Bell | 2010 | Exponential | O(t) |

O(sqrt(log(t)))-diameter | Adam P. Goucher | 2010 | Exponential | O(log(t)) |