From: "R. William Gosper" <rwg@osots.com>
http://gosper.org/jaggies.png is a 4x blowup of a detail of a Golly (Game of Life) screen, where each pixel is 2^24 x 2^24 cells. The diagonal line marked by arrows is a row of equally spaced gliders, (dx,dy) = (14412108,16685960). The gap at the green arrow is a legitimately missing glider, but every other dot on that line represents one glider. I.e., Golly darkens a pixel iff there's anything in the 2^24 square it represents. Note that dy>dx. Puzzle: How can the nonmonotonicity at the black arrow be anything but a display bug? (Does anybody remember BRMs and DDAs?)
Both dx and dy are < 2^24. So, going up-right, the next glider can either be on the same row of pixels, or one row up; on the same column, or one column to the right. That means it might be up, up-right, right, or in the same pixel. The arrow points to a place where it goes right (but not up). The places where you don't go up are relatively rare (once every ~184 gliders) because 16685960 is close to 16777216=2^24. Row number and column number as functions of glider number are related to the continued fractions for dy/2^24 and dx/2^24. They will look like fractals until you get to the LCM. The jaggies are like the beating of those two fractals against each other, and resolve to a pattern when you get to the LCM of all three numbers. That is, when the number of gliders in a string is equal to the LCM. For a similarly incoherent explanation of this same integer-division jaggy phenomenon (only with irrational numbers) see the section called "Why the Fractal Patterns?" on this page: http://www.tiac.net/~sw/2005/03/Mandala/index.html Off Google: "BRM -. Binary Rate Multiplier,. and DDA - Digital. Differential. Analyzer" Don't know if a DDA analyzes the way I just did. related to Bresenham's algorithm. Yeah but Bresenham is just an optimized way of drawing the same jaggies. --Steve