Sawtooth 201

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Sawtooth 201
Pattern type Sawtooth
Number of cells 201
Bounding box 79 × 55
Expansion factor 47
Discovered by Adam P. Goucher
Year of discovery 2015

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.


The pattern can be seen below. 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.

x = 55, y = 79, rule = B3/S23 49b2o$49b2o2$23b2o5b2o$23b2o5b2o5$43bo5bo$42b3o3b3o$41b2obo3bob2o3$23b o7bo12bo3bo$22bo2bo3bo2bo11bo3bo$26bobo$26bobo$26bobo$22bo2bo3bo2bo$ 23b3o3b3o7$43bo$43bobo$43b2o$30b2o$30b2o2$10b2o$8b2ob2o8bobo$8bo2bo10b o$8bo2bo$9b2o33b2obo3bob2o$32bo11bo2bo3bo2bo$9b2o20bo13b3o3b3o$8bo2bo 19b3o$2o6bo2bo$2o6b2ob2o$10b2o9bobo$21b2o$22bo2$45b2o$45b2o2$20bo$20bo bo$20b2o2$25b2o$25b2o$6b4o15b2o$4b2o4b2o13bo$4b2o5bo12bobo$6b2obobo12b ob2o$11bo$7bo3bo$7bo4bo12b2o$9b3o3bo9b2o$9b2o4bo$15b2o$17bo$17b3o3$20b o$19bob5o$18b2o5bo$18b2o3bo2bo$26bo$20b2obo2bo$23bo2bo$24b2o$24b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ZOOM 8 WIDTH 800 HEIGHT 800 ]]
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RLE: here Plaintext: here

The population is equal to 201 at generations 0, 1840, 88320, 4152880, 195187200, 9173800240, 431168613120, ..., 40 (47n − 1), ... (OEISicon light 11px.pngA257319), giving an expansion factor of 47.


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.


  1. 1.0 1.1 Adam P. Goucher (April 13, 2015). Re: Smaller Sawtooth (discussion thread) at the forums
  2. Tanner Jacobi (April 13, 2015). Re: Smaller sawtooth (discussion thread) at the forums
  3. Dave Greene (April 17, 2015). Re: Smaller sawtooth (discussion thread) at the forums