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2005 August 10

Transcendental Objects
New Sawtooth Patterns

David Bell has constructed two new smaller sawtooth patterns. The first one undercut the size of the previous smallest known sawtooth by 6 cells. Sawtooth pattern with minimum repeating population of 269

4-engine Cordership with minimum population of 134 To improve on this new record, Bell constructed a 4-engine Cordership with a minimum population of 134, compared with the 3-engine Cordership's minimum population of 149:

4-engine Cordership with minimum population of 136 There is at least one other variant with a larger population, but which still beats the 3-engine Cordership. This has a minimum population of 136:

The problem with the 3-engine Cordership is that the 2-engine "wing" component swings wildly in population from small to large, and in the 3-engine Cordership the two (overlapping) wing components are locked together in a non-optimal relative phase, so that whenever one half has a small population, the other half has a large population.

By adding another switch engine in the middle to create two independent wing components, the relative phases can be adjusted in an optimal manner to reduce the population, even taking the extra switch engine into account.

Note: these new Corderships have a smaller minimum population, but the 3-engine Cordership still has a smaller average population. The average population of the 3-engine Cordership is 193.125, whereas the average population of the 134-cell-minimum Cordership is 217.854.

Sawtooth pattern with minimum repeating population of 262 David Bell constructed a sawtooth with a minimum repeating population of 262 based on the 136-cell-minimum Cordership:

Moving sawtooth with repeating minimum population of 1239 Bell has also created a revised moving-sawtooth pattern using the small c/3 period 27 rakes discovered since they were constructed. Further optimization is probably still possible.

The pattern works in the same way as Bell's previous moving sawtooths, by using the output of c/3 rakes to ignite the blinkers from a c/2 blinker puffer, with the number of blinkers to be consumed growing on each cycle.

Another possible sawtooth design using 'lineship' technology, which Bell has not been able to complete as yet, is presented here.