Parabolic sawtooth
Parabolic sawtooth  
View static image  
Pattern type  Sawtooth  

Number of cells  889  
Bounding box  126×114  
Expansion factor  n/a  
Discovered by  Dean Hickerson  
Year of discovery  1991  
 

Parabolic sawtooth is a diagonal sawtooth that was discovered by Dean Hickerson on June 26, 1991. It is of special note because unlike most other sawtooths, its graph of population versus generation number is a sawtooth graph with parabolic (as opposed to linear) envelope and its population returns to 1208 in amounts of time that are quadratically spaced (as opposed to exponentially spaced, like most sawtooths). It can be reduced by using the Simkin glider gun instead of the Gosper glider gunbased p120 guns.
The pattern works by repeating the following operation for each n ≥ 0:
 A 4glider salvo is sent southeast toward a block A, arriving in generation 20n^{2} + 144n + a[n mod 3], where a[0]=a[2]=131 and a[1]=91.
 Block A is pushed 1 unit southeast and another block, B, is created upstream from block A. Every 108 generations, 2 gliders hit B and pull it 3 units northwest. Eventually block B gets deleted by a glider, at generation 20n^{2} + 180 n + b[n mod 3], where b[0]=193, b[1]=223, and b[2]=227.
 Another 4glider salvo is sent toward block A.
The population is minimal around the time block B is deleted. The minimum repeating population that appears is 1208 in generations 180n^{2} + 540n + 210 and 180n^{2} + 660n + 450. The population is maximal around the time block B is created: there are about n/30 2glider salvos on their way toward block B around generation t = 20n^{2} + 144n, so the population is about n/3 ~ sqrt(t/180) at that time.
Videos
External links
 Parabolic sawtooth at the Life Pattern Catalog