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2011 May 07

Restricted Patterns
Quadratic population growth from one row of cells

quadratic growth diagram
Quadratic growth pattern of width 1.
Stephen Silver, 20 April 2011.
Uses a breeder by Nick Gotts.
Following up on an open problem he originally posed in 1998, Stephen Silver has constructed a minimal-height Life pattern that exhibits quadratic population growth -- a switch-engine breeder based on Nick Gotts' 26-cell quadratic-growth pattern, evolved from an initial pattern that's just a single cell in height. The other dimension could probably be optimized considerably, though -- the pattern is just slightly over a million cells in length (!), and takes a million ticks to evolve into the final breeder form.

At right is a diagram shows what the full pattern looks like, with a sample section of the generating line of cells expanded to explain the mechanism used to construct the breeder. Line sections are arranged to produce exactly-timed two-glider salvos, which collide to produce LWSSes, which in turn collide to build the breeder. A multi-step reaction at the X axis produces the second glider in each pair with an exactly-timed delay relative to the first one.

1xN breeder after 2M ticks
The width-1 breeder after two million ticks, showing the first six switch engines
heading NW and SW, plus other stable and traveling detritus left over from
the construction process.
The breeder is based on Nick Gotts' 26-cell quadratic-growth pattern. It is incrementally constructed by colliding LWSS streams travelling parallel to the baseline.

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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.

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