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The period 94 gun is 143 by 607; it’s based on the “AK47” reaction discovered independently by Dave Buckingham and Rich Schroeppel. A honey farm starts to form but is modiﬁed by an eater and a block. It emits a glider, forms a trafﬁc light, and then starts forming another honey farm in a different location. If you delete the trafﬁc light, the cycle repeats every ¢¾ generations. A close pair of AK47s can delete each other’s trafﬁc lights, so we can build a long row of them that is unstable at both ends. In the period 94 gun, two such rows, with a total of 36 AK47s, emit gliders that can crash to form MWSSs, which hit eaters, forming gliders that stabilize the ends of
a row. Here’s how to turn 2 MWSSs into a glider:
In Winning Ways, you (JHC) described how to thin a glider stream by kicking a glider back and forth between two streams. Using normal kickbacks, the resulting period must be a multiple of 8. I’ve found some eaterassisted kickbacks that give any even period. (Sadly, they don’t work with period 30 streams. Fortunately, there are other ways to get any multiple of 30.) It’s also possible to build a gun that produces a glider stream of any period ½. The gun itself has a larger period; it uses various mechanisms to interleave larger period streams. It’s fairly easy to get any period down 208 B. GOSPER to 18 this way (period 23 is especially simple), and Buckingham has found clever ways to get 15, 16, and 17. (I built a pseudo-period 15 gun based on his reactions; its bounding box is 373 by 372.) New Year’s Eve Newsﬂash: Buckingham has just announced “... I have a working construction to build a P14 glider stream by inserting a glider between two gliders 28 gens apart!... This will make it possible to produce glider streams of any period.” (Less than 14 is physically impossible.)
Returning to Hickerson’s account:
Many of the large constructions mentioned later require a period 120 gun. The smallest known uses a period 8 oscillator (“blocker”) found by Wainwright to delete half the gliders from a period 60 stream. Here it’s shown deleting a glider; if the glider is delayed by 60 (or, more generally,
Ò · ) generations, it escapes:
There are also combinations of two period 30 guns that give fairly small guns with periods 90, 120, 150, 10, 210, 300, and 360.
We can also synthesize light-, middle-, and heavyweight spaceships from
gliders, so we have guns that produce these as well. For example, this reaction leads to a period 30 LWSS gun:
Large Constructions. Universality implies the existence of Life patterns with various unusual properties. (At least they seem unusual, based on the STRANGE NEW LIFE FORMS: UPDATE 209 small things we normally look at. Even inﬁnite growth is rare for small patterns, although almost all large patterns grow to inﬁnity.) But some of these properties can also be achieved without universality, so some of us have spent many hours putting together guns and puffers to produce various results. (Figuring out how to achieve a particular behavior makes a pleasant puzzle; actually building the thing is mostly tedious.) The ﬁrst such pattern (other than glider guns) was Bill’s breeder, whose population grows like Ø ¾. Smaller breeders are now known.
Probably the second such pattern is the “exponential aperiodic,” versions of which were built independently by Bill and myself, and probably others. If you look at a ﬁnite region in a typical small pattern, it eventually becomes periodic. This is even true for guns, puffers, and breeders.
But it’s fairly easy to build a pattern for which this isn’t true: Take two glider puffers, headed east and west, ﬁring gliders southwest and northeast, respectively. Add a glider that travels northwest and southeast, using a kickback reaction each time it hits one of the glider waves. Cells along its ﬂight path are occupied with decreasing frequency: The gap between the occupations increases by a factor of 3 or 9 each time, depending on which cells you’re looking at.
Bill also built an “arithmetic aperiodic” in which the gap between occupations increases in an arithmetic progression.
Both of these patterns have populations tending to inﬁnity. I’ve also built some aperiodic patterns with bounded populations. These use a glider salvo to push a block (or blinker); the reaction also sends back a glider (or two) which triggers the release of the next salvo.
In Winning Ways you (JHC) describe how to pull a block three units using two gliders and push it three units using 30. I found a way to pull it one unit with two gliders and push it one unit with three gliders. Using this I built a sliding block memory register, similar to the one you described.
(It has one small difference: the “test for zero” is not a separate operation.
Instead, a signal is produced whenever a decrement operation reduces the value to 0.) Most of my large constructions are designed to achieve unusual population growth rates, such as Ø ½ ¾ Ø¿ ¾ ÐÓ ´Øµ ÐÓ ´Øµ¾ Ø ÐÓ ´Øµ and Ø ÐÓ ´Øµ¾, and linear growth with an irrational growth rate. In addition, there are several 210 B. GOSPER “sawtooth” patterns, whose populations are unbounded but do not tend to inﬁnity.
¿¼¼¼) that computes prime I also built a pattern (initial population numbers: An LWSS is emitted in generation ½¾¼Ò if and only if Ò is prime.
(This could be used to get population growth Ø ¾ ÐÓ ´Øµ, but I haven’t built that.) Paul Callahan and I independently proved that arbitrarily large puffer periods are possible. His construction is more efﬁcient than mine, but harder to describe, so I’ll just describe mine. Give a glider puffer of period Æ, we produce one of period ¾Æ as follows: Arrange 3 period Æ puffers so their gliders crash to form a MWSS moving in the same direction as the puffers.
The next time the gliders try to crash, there’s already a MWSS in the way, so they can’t produce another one. Instead, two of them destroy it and the third escapes; this happens every ¾Æ generations. (Basically this mimics the way that a ternary reaction can double the period of a glider gun; the MWSS takes the place of the stable intermediary of the reaction.) Dave Buckingham and Mark Niemiec built a binary serial adder, which adds two period 60 input streams and produces a period 60 output stream.
(Of course, building such a thing from standard glider logic is straightforward, but they used some very clever ideas to do it more efﬁciently.) —Dean Hickerson To clarify, Buckingham’s “awe-inspiring glider syntheses” are the constructions of prescribed, often large and delicate, sometimes even oscillating objects, entirely by crashing gliders together. The difﬁculty is comparable to stacking water balls in 1G. Warm ones. Following Dean’s update, we were all astounded when Achim Flammenkamp of the University of Bielefeld revealed his prior discovery of Dean’s smallest period 3 and 4 oscillators during an automated, months-long series of literally millions of random soup experiments. Thankfully, he recorded the conditions that led to these (and many other) discoveries, providing us with natural syntheses (and probably estimates) of rare objects.
STRANGE NEW LIFE FORMS: UPDATE 211
ººººººººººÓººººººººººººººººººººººººººConway called such oscillators billiard tables, and along with the rest of us, never imagined they could be made by colliding gliders.
Finally, after a trial run of the foregoing around the Life net, Professor Harold McIntosh (of the Instituto de Ciencias at Puebla) responded: “The humor in that proposed introduction conjures up images of untold taxpayer dollars (or at least hours of computer time) disappearing into a bottomless black hole. That raises the question, ‘Have other hours of computer time been spent more proﬁtably?’ (Nowadays, you can waste computer time all night long and nobody says anything.) “Martin Gardner is a skilled presenter of ideas, and Life was an excellent idea for him to have the opportunity to present. Not to mention that the time was ideal; if all those computer hours hadn’t been around to waste, on just that level of computer, perhaps the ideas wouldn’t have prospered so well.
“In spite of Professor Conway’s conjecture that almost any sufﬁciently complicated automaton is universal (maybe we can get back to that after vacations) if only its devotees pay it sufﬁcient attention, nobody has yet 212 B. GOSPER come up with another automaton with anything like the logical intricacy of Life. So there is really something there which is worth studying.
“It might be worth mentioning the intellectual quality of Gardner’s presentation of Life; of all the games, puzzles and tricks that made their appearance in his columns over the years, did anything excite nearly as much curiosity? (Well, there were ﬂexagons.) “Nor should it be overlooked that there is a more serious mathematical theory of automata, which certainly owes something to the work which has been performed on Life. Nor that there are things about Life, and other automata, that can be foreseen by the use of the theories that were stimulated by all the playing around that was done (some of it, at least).” The Life you save may not ﬁt on your disk.
In October 1983 I discovered hollow mazes. A. K. Dewdney presented the concept in Scientiﬁc American, September 1988. In this article I shall give a more detailed description of the subject. The ﬁrst part is about multiple silhouettes in general, the second part is about their application to hollow mazes.