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2011 January 30

Minimum-population sawtooth

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.

David Bell's 260-cell sawtooth.

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:

NameDiscovererYearReturn rateMaximal population
Sawtooth 1212Dean Hickerson1991ExponentialO(t)
Cord pullerDean Hickerson1991ExponentialO(t)
Parabolic sawtoothDean Hickerson1991QuadraticO(sqrt(t))
Sawtooth 633Dean Hickerson1992ExponentialO(t)
HacksawDean Hickerson1992ExponentialO(t)
Sawtooth 562Hickerson & Coe1992ExponentialO(t)
Sawtooth 1846Dean Hickerson1992ExponentialO(t)
Sawtooth 362David Bell1992ExponentialO(t)
Externally-timed sawtoothDavid Bell1992ExponentialO(t)
Moving sawtoothDavid Bell2005ExponentialO(t)
Sawtooth 260David Bell2010ExponentialO(t)
O(sqrt(log(t)))-diameterAdam P. Goucher2010ExponentialO(log(t))