A stable reflector is a reflector composed entirely of still lifes. That is, it is a collection of still lifes that can reflect some type of spaceship (usually a glider) without suffering permanent damage. Stable reflectors are special in that, if they satisfy certain conditions, they can be used to construct oscillators of all sufficiently large periods. It was known for some time that stable reflectors were possible (see universal constructor), but no one was able to construct an explicit example until Paul Callahan did so in October 1996.
Types of stable reflectors
There are several distinct categories of stable reflector:
- 'Create-then-remove' reflector - A reflector that temporarily creates an unwanted still-life and later destroys it.
- 'Destroy-then-rebuild' reflector - A reflector that temporarily destroys a necessary still-life and later reconstructs it.
- Direct reflector - A reflector that does not contain a Herschel track.
The repeat time, also called the recovery time or compression of a stable reflector is the number of generations after the acceptance of one spaceship required for the reflector to be able to reflect the next spaceship.
The staged-recovery system was invented by Dave Greene, when he incorporated it into his highway robber and stable Heisenburp patterns. In 2009, Adam P. Goucher built two 'Create-then-remove' reflectors using a staged-recovery system.
Before 2013, all known stable reflectors were quite slow. Callahan's original reflector has a repeat time of 4840, soon improved to 1686 and then 894 and then 850. In November 1996, Dean Hickerson managed to reduce the repeat time to 747 using a specialised Herschel-to-glider conduit. David Buckingham reduced it to 672 in May 1997 using a somewhat different method; he used a boat as the 'bait' catalyst, which is converted, on impact, directly into a Herschel, which rebuilds the initial boat. In October 1997, Stephen Silver reduced the time to 623 by a method closer to the original; instead of using a boat and conduit 1 to convert the R-pentomino into a Herschel, Silver used a loaf. In November 1998, Callahan reduced this to 575 with a new initial reaction based on a beehive, rather than a block. A small modification by Silver a few days later brought this down to 497. In 2009, Adam P. Goucher used a staged-recovery system to delete the beehive much quicker, lowering the repeat time to 466.
In April 2001, Dave Greene found a 180-degree stable glider reflector with a repeat time of only 202 known as the boojum reflector. This reflector won a $100 prize that Dieter Leithner had offered in April 1997 for the first stable reflector to fit in a 50×50 bounding box, and an additional $100 that Alan Hensel had offered in January 1999 for the same feat.
Dave Greene offered two $50 prizes for stable reflectors:
- Find a stable glider reflector that fits inside a 50×50 bounding box.
- Find a stable glider reflector that fits inside a 35×35 bounding box.
Matthias Merzenich offered two similar $50 prizes for stable reflectors:
- Find a stable glider reflector with a repeat time of 100 generations or less.
- Find a stable glider reflector with a repeat time of 61 generations or less.
Mike Playle won all four prizes with his discovery of his small stable reflector in April, 2013.