### 2005-05-05

## New Transcendental Patterns

Dean Hickerson has presented a pair of new transcendental patterns he's created. These consist of puffers and guns, which grow in what appear to be unpredictable ways.

The first is a "Ruler" generator:

This produces groups of LWSSs headed west, with gaps of fixed size between them. The lengths of the groups form the 'ruler' sequence, 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 5 ... (Sloane's A001511). The first group of length n is emitted about generation 96*2^n. The pattern uses a Corderman eater puffer found by Paul Tooke (Jan 2004), a p48 glider gun by Noam Elkies (Jun 1997), a p8 glider reflector by Noam Elkies (Sep 1998), and a p24 LWSS puffer (source unknown).

Shown are the starting pattern at Gen 0 and later at Gen 850, after 1-2-1-3-1 as been emitted. (Note: They've been rotated 90° to better fit the page.)

The second object is "Jagged Lines":

Jagged lines of gliders, formed by a drifting collision of two Lightweight Spaceships (LWSS) streams, crash to form an approximately vertical jagged line of pairs of blocks. I don't know if the line stays within a bounded distance of the center line, or extends infinitely far to the left, or to the right, or both.

Shown are the starting pattern at Gen 0 and later at Gen 850, shortly after the second block pair has been created. (Note: They've been rotated 90° to better fit the page.) Hickerson also simulated the placement of the block pairs and presented a plot showing the first 11,426,769 Twin Blocks produced by Gen 4,113,636,213. (The horizontal:vertical scale is 488:1 to emphasize the shifting locations.) The dimensions of the jagged line of Twin Blocks are cells 140,480 wide with a tail 685,605,960 cells long.

Hickerson says that he doesn't think it's a random walk:

There are some large portions of it that are almost symmetric across horizontal lines. Also, the transitions between successive minimal and maximal x-coordinates are rather brief. I.e. it spends a long time far to the left of the center line, then moves quickly to a point far to the right, spends a long time there, etc. I think there's an approximate scale-invariance; if you expand the picture by appropriate factors horizontally and vertically it'll look almost the same. But I don't understand it well enough to say what those factors are.

Gabriel Nivasch points out that if the Pre-Block (shown in red), which is responsible for the asymmetry of the pattern, is removed, then the pattern generated is one generated by a growing sequence which starts out with zero and adds four new items at the end while sequentially reading the digits already laid out. The additions are

0 -> 1 0 1 0

1 -> 1 0 1 1

which gives the initial sequence of "0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 1".

(Disclaimer— I have no association with anyone or any organization, and speak only for myself. Links and quotes are provided for information only.)