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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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2007 April 03

Restricted Patterns
Life Digits update

Stable Digits Dean Hickerson has updated his digits webpage to include many new stable objects and pseudo objects. He also shows how it is possible to come up with a number which produces a Period 30 Glider Gun.

Shuttles Nicolay Beluchenko also found a few new, large stable objects, and also came up with some numbers which produced the Period 30 Queen Bee and the Period 46 Twin Bee Shuttle elements. Hickerson was able to come up with numbers that result in oscillators based on those shuttle elements.

2007 March 03

Restricted Patterns
Life Digits

Life Digits Dean Hickerson and Eric Angelini have investigated a "recreational Life" question, that of Life patterns from from the digits of numbers. The following is excerpted from some messages from Hickerson on this subject, and quoted with permission.

Although this is an unnatural restriction to put on patterns, and looking at such patterns is unlikely to lead to anything of great significance, I've had some fun playing around with them. Some of our discussion is on Angelini's website. [Note: much of the commentary is in French, while correspondence is in English.] More information can also be found at my web page.

1125344743766111111111947742 ...22222222222222... Eric first asked if infinitely many such patterns die. It's easy to see that the answer is "yes"; e.g. numbers of the form 1444...444 all die in 9 gens. Arbitrarily slow death is also possible; e.g. numbers of the form 1125344743766111...111947742, with an odd number of 1's, die by crashing a LWSS and a MWSS (Shown here at Gen 50). Numbers of the form 1125344743766189077900222...2220066748424 are even more amusing; here the number of 2's in the middle must be divisible by 3 and >= 9. The 2's in the middle form two lines of blinkers, which decay from opposite ends (Shown here at Gen 100).

Spaceship producers Eric also asked about numbers N which die in exactly N generations. A trivial case is N=10, and I'm not sure we'll ever see another example. However, I've pretty much convinced myself that there are larger ones: There are digit strings which can exist within a longer number and which produce spaceships traveling east or west. Here are the ones that I've found.

Using things like this, we can do "slow *WSS constructions", similar to the slow glider constructions which it is believed can build just about anything. (Or, if we find an appropriate collision, we can crash the spaceships into each other to form gliders, and then use those to do slow glider constructions.)

So presumably there's a number M that produces a computer that runs the particular program that I'll describe. Now form a larger number N consisting of the digits of M followed by a string of digits that produce a binary representation of M. (For example, a long string of 1's with an occasional 3 in the middle forms a pair of blocks at each 3; the presence or absence of such a pair represents one bit.) The computer is programmed to read this representation of M, use it to compute both M and the digit string that produced the pairs of blocks, concatenate them to form the value of N, and then self-destruct after N generations.

Objects from digits Spaceships from digits Turning to patterns that don't die, I've found numbers that produce many of the small named objects and patterns. (I'm especially fond of the Pentadecathlon.)

Switch engines The smallest number which shows infinite population growth is 154299. In Gen 539 it produces a block-laying Corderman Switch Engine. Also shown here is 4114073236, which produces the glider shooting Switch Engine. (Both shown here at Gen 1500.)

Hickerson's webpage on this subject gives more information on additional topics such as