An oscillator of infinitely extensible size and period in which patterns like anchors travel along a track built of two evenly spaced lines of dots. The term can also refer to a component of a spaceship or puffer with the same mechanism. The first few iterations have periods, respectively, of 12, 28, 28, 124, 252 and 60. All iterations have a period of the form 4 × 2n - 4, as described below. Larger anchor tracks have increasingly low temperature because of the empty areas of the track, making them the coldest known oscillators other than the dot. Because their periods can in some cases increase proportionally to 2n, where n is the number of slots, anchor tracks can have an arbitrarily high ratio of period to minimum population: for example, a 39-slot anchor track has a minimum population of 80 and a period of 2,199,023,255,548. Anchor tracks are the only known oscillators with arbitrarily high period. With the exception of p12, no other oscillators are known for any period that an anchor track can have.

The following is referred to as a two-slot anchor track and is extended by placing another pair of dots four cells to the right of the first. With a minimum population of six cells, the two-slot anchor track is the smallest oscillator that is not a phoenix, in terms of bounding box, and ties with the ant and monster in terms of minimum population. It is also the smallest oscillator that doesn't consist of a single polyplet in any phase.

anchor track.png

x = 6, y = 5, rule = s0/b2
bo3bo$o2$o$bo3bo!

Uses

Anchor tracks are extremely useful in flotillas. They allow the creation of spaceships and puffers of arbitrarily high periods of the form 8 × 2n, using anchor eaters, anchor biters, and anchor nibblers. The infinite anchor track allows a puffer to leave a trail which is aperiodic, by constantly increasing the size of the track. In these cases, an anchor track is created parallel to the flotilla's direction of travel.

The term anchor track flotilla refers to a flotilla in which a combination of puffers and spaceships create anchor tracks (or just anchors) perpendicular to the flotilla's direction of travel. Anchor track flotilla technology allows the creation of puffers and rakes that leave behind objects other puffers can't produce, including monsters, radar, and free ships. Large freedom fighter flotillas, which may have any period of the form 6 × 2n through successive doubling of their period, also use anchor track flotilla technology.

Period

Effectively, an infinitely long anchor track implements a one-dimensional elementary cellular automaton (Rule 90); four generations in Live Free or Die correspond to one generation in the anchor track's automaton, and the presence or absence of anchors in slots corresponds to cells being on and off. The edges of an anchor track exhibit different properties: if we consider the space beyond the track to be a dead cell, the last slot's behavior is described by Rule 6, or a wide variety of other rules (because four states cannot occur.) The combination of the two rules generates complex behavior.

Multiple unique anchor tracks exist with the same number of slots if each one's oscillation doesn't include every possible pattern of anchors. For a track with a number of slots s that begins with a single anchor at one end, the period is 4 × 2n - 4 where n is the multiplicative suborder of 2 (mod 2s+1), or sord(2, 2s+1). In other words, it is the smallest value of 4 × 2n - 4 such that 2n == +- 1 (mod 2s + 1). (1)