Sawtooth 201

Sawtooth 201 is a diagonal sawtooth discovered on April 13th, 2015,[1] and was the smallest known sawtooth in terms of its bounding box until the appearance of Sawtooth 195 on October 31st of the same year. It was also smallest in terms of its minimum repeating population of 201, until the appearance of Sawtooth 181 on April 28th.

It functions by letting two glider streams of period 46 retract a block, created by collision with a spark from a 58P5H1V1, one cell at a time. The retracted block is deleted via interaction with a blocker, and the streams are allowed to return to the now-farther-away 58P5H1V1 to create another block.

Population

The population is equal to 201 at generations 0, 1840, 88320, 4152880, 195187200, 9173800240, 431168613120, ..., 40 (47n - 1), ... ( A257319 ), giving an expansion factor of 47.

History

The original design by Tanner Jacobi used a twin bees shuttle to delete the retracted block, resulting in a population of 213.[2] Adam P. Goucher noticed that a blocker would suffice on the same day, reducing the population to 201.[1] Dave Greene optimised the bounding box of the pattern by moving the blocker and spaceship inwards by two full-diagonals and rephasing the blocker by 4 generations; this also reduced the expansion factor from 474 to 47.[3]

Due to a mismatch between the period 46 guns and the period 5 spaceship in the original pattern, full retraction only resulted in the minimum population every fourth cycle. The expansion factor of this pattern is asymptotic to 47 if each of the four sub-tooths per cycle is counted as a separate peak; otherwise the expansion factor is 474 = 4,879,681. For the original version of Sawtooth 201, the minimum population recurs at T=0, T=234224640, T=1142941759764480, etc. For Sawtooth 213, the spaceship starts a few cells closer, which makes a big difference to the speed of the cycles, but not to the expansion factor: T=0, T=11232640, T=547659593220480, T=2672404111505817299520, and so on.

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gollark: Your alleged validity is a mere lyriclyal construct.
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gollark: Apiaristic bee memes.
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References
  1. Adam P. Goucher (April 13, 2015). "Re: Smaller Sawtooth". ConwayLife.com forums. Retrieved on July 3, 2016.
  2. Tanner Jacobi (April 13, 2015). Re: Smaller sawtooth (discussion thread) at the ConwayLife.com forums
  3. Dave Greene (April 17, 2015). Re: Smaller sawtooth (discussion thread) at the ConwayLife.com forums
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