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2009 March 24

Unusual Growth Rates
Pi In A Cloudy Sky

Dean Hickerson's 'Life Computes Pi' patternUpdate:Until recently, Dean Hickerson's Life pages have been available only in Web-archive form, with no images available.

Updated versions of these pages have now been moved to a new home on Tomas Rokicki's website, and the images and text files are all available again.

The image at right is from an intriguing family of patterns constructed in mid-2006. The family includes 'Life Computes Pi' and a number of 'Clouds' variants. There's really no substitute for watching these evolve in real time in a high-speed Life simulator, but a few surprising pictures of later stages of their evolution are shown below.

The pattern to the right is the starting configuration for 'Life Computes Pi', which consists of four breeders creating lines of guns that recursively stifle each other's output. The gliders appear to be spiraling outward, but in fact each set of four guns affects only itself, and any finite area around the center of the pattern will eventually repeat an earlier state.

As the number of ticks (t) increases, the population of the entire pattern approximates (pi-2)/720 t^2. At four million ticks, when the images below were captured, this works out to a value of pi correct to two places after the decimal point... so this is not quite the most efficient way to calculate pi.

'Clouds' variant of 'Life Computes Pi' pattern at 4 million ticksThe image to the right shows the large-scale shape generated by this family of objects after several million generations. The variant shown here is known as 'Clouds', because a complex feedback effect between the quadrants creates ever-larger rough-edged clouds of gliders as the pattern grows in size.

In 'Clouds', gliders in intersecting streams crash to form a block instead of mutually annihilating; the block then deletes the next glider that hits it from either one of the streams.2009-03-23-Life-clouds-wide.PNG

The selected area (green rectangle) is shown in greater detail in the next image, and a further magnification is shown after that. Individual gliders would not be visible for several more factor-of-two zooms below the level of the last image

2009-03-23-Life-clouds.png 2009-03-23-Life-clouds-close.PNG

Very small changes to the initial conditions can cause large differences later on. This pattern is identical to 'Clouds' except for the addition of a block in the southeast quadrant, which deletes a single glider. The early evolution of the pattern is not much affected, but by the time half a million generations have gone by, the rough-edged clouds of gliders have completely disappeared.