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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.

2009 March 17

Engineered Objects
Working 2c/3 Signal Elbow

2c/3 signal elbow:  Calcyman, 2 June 2008
Working stable 2c/3 signal elbow
Last June, Calcyman noticed the surprising fact that the standard wire termination for normal 2c/3 signals works just as well for the double-length signals put out by a previously known 90-degree 2c/3 signal elbow.

Dean Hickerson's original block-deleting 2c/3 termination almost certainly wasn't designed with this in mind, but it happens to absorb a double-length signal in exactly the same way as a standard signal -- the final stable state is the same in either case. This means that communication speeds approaching 2c/3 can be implemented over long distances in any direction, not just diagonally.

In the accompanying diagram, the input Herschel signal is circled in red. The output signal can be any of a number of optional glider outputs in the Herschel circuit at the bottom.

Two elbows in a row will not work (there's no known way to turn a double-length 2c/3 signal). But in the absence of layout constraints, a single elbow is sufficient to send a 2c/3 signal anywhere in the universe.

2009 March 16

New Oscillators

A summary of new oscillators found in the last few months.

New P10 First is a Period 10 Oscillator found by Nicolay Beluchenko. Like the Period 6 Unix, this oscillator can be chained together in a variety of ways.

New P21 New P7 Here is also are several of Period 7 Oscillators that he found. Noam Elkies showed how two of the smaller ones could be combined with a Bi-Block to produce a Period 21 oscillator.

New P51 pair New P51 This Period 51 Oscillator found by Beluchenko is also the first oscillator of that period found which is not a combination of smaller period oscillators. These oscillators can also share their common blocks when properly phased.

New P47 This Period 47 Oscillator found by Beluchenko consists of several t-tetromino pairs that interact with each other.

New P47 This Period 30 Oscillator found by Jason Summers consists a pair of t-tetrominos which repeatedly bounce back after being pushed together.

New P47 A Period 3 Oscillator with an isolated spark found by Beluchenko.

2009 March 14

Engineered Objects
Smaller "Highway Robber"

Calcyman's compact highway robber, 27 Feb 2009
Highway-robber glider reflector, recovery time 1244 ticks
Calcyman has constructed a much smaller Herschel-based "highway robber" glider reflector. These patterns are named for their ability to "steal" a glider from the edge of a glider highway -- a diagonal region of the Life universe along which many gliders may be travelling on closely parallel lanes.

A good highway robber can absorb a glider and produce an output signal, without disturbing gliders on nearby lanes, even one cell farther away from the highway-robber device.

Calcyman's new construction rebuilds the loaf bait and is ready for another glider input in 1244 ticks.

A perfect highway robber could extract a glider without disturbing any other nearby gliders, even one following on the same lane at the minimum distance of 14 ticks. Gliders on the next lane would never be disturbed, even while the highway robber was in operation.

So far, no perfect highway robbers have been built. They are known to be technically possible, but with current technology they would be very large and very slow. The key discovery would be a "bait" still life that would collapse quickly away from the highway when struck by a glider, and that could be catalyzed directly from an input Herschel without the catalysts getting in the way of the highway.

2009 March 08

New c/6 Extensible Spaceship

Paul Tooke's c/6 front-sparking spaceshipWith a month-long search using a customized version of the 'gfind' search program in October of last year, Paul Tooke completed a new c/6 spaceship based on a flexible self-supporting pushalong component supplied by Hartmut Holzwart.

The pushalong component can follow behind another copy of itself and supply the necessary sparks to keep it going -- surprisingly, three different phases work equally well, as shown in the right-hand spaceship in the diagram.

A very unusual feature of this spaceship is that it has a strong central forward spark, and it travels slowly enough to allow this spark to interact with other objects without destroying the spaceship. Many types of spaceships have side sparks that can be used to modify or clean up other objects in passing. But this new c/6 can simply run right over the top of some still lifes, small oscillators, and constellations, and continue unharmed -- as the left-hand spaceship demonstrates. A sample constellation is shown at the top of the "junk column" for the left-hand spaceship, which produces a clean glider output that escapes in front of the spaceship. It is an open question what the smallest or easiest-to-construct still life or constellation is that has this property. Forward LWSS or other orthogonal spaceships, traveling ahead of the c/6 spaceship on the same track, could also be produced in this way. Sparks at the back of the spaceship can be used to delete following gliders; as of this writing, it is unknown if any glider- or spaceship-turning reactions are possible using these sparks.