This article describes stable reflectors and related constructions in Life. A stable reflector is a still life which can reflect a glider (through 90 or 180 degrees) without suffering any permanent damage. As Conway himself pointed out back in the 1970's, a suitable stable reflector can be used to construct oscillators of all sufficiently large periods. This is done by arranging the reflectors to form a track for gliders. The length of the track can be varied by moving the reflectors, and the period of the resulting oscillator can be varied still further by altering the number of gliders in the track. The first quarter of a century of the history of Life passed without any stable reflector being found. In 1996 Dave Buckingham achieved the goal of constructing oscillators of all large periods using Herschel, rather than glider, tracks. Buckingham's Herschel conduits proved to be the key to constructing stable reflectors, and so far no stable reflector has been found that does not rely on them.

The first stable reflectors

The first stable reflector was found by Paul Callahan in October 1996. It is based on a reaction he found in which a glider is converted into a glider, a beehive and an R-pentomino. The beehive is a problem, but the R-pentomino provides a means a getting rid of it: using Buckingham's conduits the R-pentomino is converted into a Herschel and moved to a position where it can fire a glider to delete the unwanted beehive. This process, almost incidentally, produces extra output gliders. It also makes the reflector very slow, 894 generations being required until the reflector is ready to accept another glider. Improved stable reflectors have centred on decreasing this time, which I call the recovery time. Callahan's reflector is shown below. As with all reflectors and converters in this article, it is shown together with an incoming glider.

The existence of this stable reflector meant not only that oscillators of all sufficiently large periods exist (which had already been proved by Dave Buckingham), but also that they need only have a few hundred live cells.

Another point to note about the above reflector is that the last two blocks are merely used to delete the Herschel after it has fired its glider. Removing these two blocks, and suppressing the unwanted output gliders, gives a stable glider-to-Herschel converter (shown below). Glider-to-Herschel converters can be used for further interesting constructions, as will be shown at the end of this article.

Dean Hickerson soon found a way to modify Callahan's original reflector to make it faster. The Herschel-to-glider mechanism that he used to achieve this turns out to be of use also in a later reflector. Hickerson's modified reflector is shown below. The non-standard still life used here is needed in order to allow the use of the 64 64 77 sequence of Herschel conduits. (In fact a smaller still life will do the job, as will be seen in a moment.)

In this reflector the Herschel is destroyed at the end, so we don't get a new glider-to-Herschel converter. Actually this is not strictly true. An arrangement of reflectors could be used to produce a Herschel by colliding two gliders, thus giving a glider-to-Herschel converter as fast as the reflectors. But this sort of construction is so large (at least with present stable reflectors) that it will be ignored for the remainder of this article.

Dave Buckingham found a faster stable reflector that does not use Paul Callahan's special reaction. Instead, the incoming glider hits a boat to make a B-heptomino, which is converted into a Herschel and moved round to restore the boat. A compact form of the 119-step Herschel conduit is needed here, as is a non-standard still life to cope with the 64 64 77 conduit sequence.

Again the Herschel is destroyed at the end, so no new glider-to-Herschel converter results.

A new R-to-Herschel conduit

In October 1997 I was using Paul Callahan's gencols program to look for interesting reactions. Setting it to search for reactions in which a loaf is hit by an R-pentomino and survives (not exactly a likely scenario) I found a new R-to-Herschel conduit. The loaf reacts with all the junk the R-pentomino produces as it naturally transforms into a Herschel, and miraculously reappears some time later leaving no debris at all. It is necessary to prevent the first Herschel glider from hitting the fading remnants of the reaction, and there is no room for an ordinary eater. But luckily a tub with tail and a block can be used instead. Here is the conduit:

It is a fact of Life that a given R-to-Herschel conduit almost never works with any given source of R-pentominoes. There is a good reason for this. While Herschels are usually produced by a standard evolutionary sequence which throws the Herschel away from its source, R-pentominoes (and most other common active objects) tend to be produced in relatively inaccessible places. I was, therefore, not very hopeful about the usefulness of the new conduit. But it turned out that there was one R-pentomino source with which it could be used: Paul Callahan's reflector reaction. There was therefore the possibility of an improved reflector, especially as the new conduit produced a Herschel that appears sooner and is closer to the beehive that must be deleted.

Putting the relevant data into my Herschel track program produced the following stable reflector with a recovery time of 623. Dean Hickerson's method of producing a forwards glider again proves useful.

At the time of writing this is the fastest and smallest known stable reflector. Nonetheless it is still very slow and very large. Major improvements can only come from methods that do not involve moving a Herschel round to fix some defect created by the glider impact, but all attempts to find such a method have so far failed.

Another track gave the following glider-to-pi converter making use of Paul Callahan's 95-step Herschel-to-pi conduit, which happens to fire a glider in just the right way.

This was nice, but there was no known way of converting the pi-heptomino into a Herschel, so it didn't result in an improved glider-to-Herschel converter. In fact the best I could manage was the following, only slightly faster than Paul's original, and also larger.

But later that same month Paul found a pi-to-Herschel conduit that can be added to his Herschel-to-pi conduit to make a 176-step Herschel conduit. This was just what I needed to produce a much faster glider-to-Herschel converter, also smaller than my last one:

It should perhaps be pointed out that all these reflectors and converters are not quite as bad as their large recovery times make them appear. It is possible, for certain periods, to pack multiple active objects into the reflector/converter with each one deleting the beehive (or, in the case of Dave Buckingham's reflector, replacing the boat) that would normally be dealt with by a later one. As an example of this, here is a p157 gun consisting of two copies of the latest stable reflector:

p157 glider gun using two stable reflectors.

(It can be shown that such a reflector loop can be packed to period k iff (r-1) mod k >= 114, where r is the recovery time. The time around the whole loop must, of course, be a multiple of k --PBC).

Glider-to-spaceship converters

Paul Callahan had used his stable glider-to-Herschel converter to produce stable glider-to-LWSS and glider-to-MWSS converters. Unaware of Paul's work I produced similar glider-to-spaceship converters, also based on his glider-to-Herschel converter. When we compared notes, Paul had the smaller glider-to-MWSS and I had the smaller glider-to-LWSS. But all of our constructions were very slow, being limited by the recovery time of 894 of the glider-to-Herschel stage.

The new faster glider-to-Herschel converter naturally led to faster glider-to-spaceship converters. New Herschel conduits and a (relatively) small method of duplicating a Herschel that I had not previously been aware of led to reductions in size as well. Here then are some of the better glider-to-spaceship converters that I have made, one for each of the three basic orthogonal spaceships. In each case the recovery time is 629 because of the glider-to-Herschel stage, the Herschel-to-spaceship stage being much faster (262 in all cases, the bottleneck being the Herschel duplicator).

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