Marijn Heule, Christiaan Hartman, Kees Kwekkeboom and Alain Noels systematically searched the entire space of 10-by-10 patterns with fourfold rotational symmetry, finding a Garden of Eden with 92 specified cells (56 live, 36 dead). Moreover, they proved the non-existence of Gardens of Eden within a 6-by-6 box.
If you asked a fellow Life enthusiast for the most important GoL discoveries in the 1990s, the Herschel track must surely feature. With a few elementary conduits, it is possible to design tracks capable of moving a signal to anywhere in spacetime (as long as there is enough 'manouevring room' and sufficient time), and placing it in any orientation. Herschel tracks underpin all but two of the known stable reflectors, and support the construction of glider guns for every period greater than or equal to 62.
Firstly, what is so special about the Herschel? Is it really so much more useful than any other transient objects? It appears that the answer is both yes and no: other objects can be used, but they must eventually decay into Herschels. This is illustrated rather eloquently by a simple matrix. The row represents the input; the column represents the output. A red blob indicates if a primary (one-stage) conduit exists to transform the input into the output. Clicking on the matrix will enable you to download a complete collection of primary conduits. (A collection of all conduits, primary and composite, is provided later in this article.)
Some of these conduits are new discoveries. The Pi-to-R converter was discovered by Guam on the conwaylife.com forums, published in the form of a quaternary Herschel conduit: H-Pi-R-B-H. The completed conduit takes 309 generations to turn a Herschel anticlockwise, so is designated L309. In terms of the number of intermediary objects, L309 is the most complex Herschel conduit to date. Indeed, its 309-tick delay is rather rapid for a quaternary conduit.
Not content with a single new conduit, Guam proceeded to discover a pi-to-Herschel converter capable of attaching to a handful of 'pre-Herschel' conduits. Moreover, the symmetry of the pi heptomino means that Guam discovered not only one conduit, but two 'isomers'! The conduits are designated F266 and Fx266 for the translation and glide-reflection variants, respectively. However, the restrictions upon which conduits may follow F*266 severely limit its use in practical Herschel circuitry.
From an earlier posting of mine, you may remember the contributions of a certain 'MikeP', again from the conwaylife.com forums. Matthias Merzenich has utilised a particular catalyst of his in a few conduits, including the periodic R135 conduit and a stable Pi-to-century converter. To process the resulting century, Matthias proceeded to find two unique century-to-Herschel converters, one of which is sufficiently compatible to yield a new tertiary Herschel conduit, the Lx496.
Matthias even found a use for this new inundation of Herschel conduits; he has incorporated them into glider guns with smaller dimensions than the current record-holders. Specifically, they are a p421 gun derived from the L309 and p685 gun based on the Lx496.
In addition to his spectacular Herschel conduits, Guam also found two reactions in which gliders collide with constellations of still lifes to form extra junk. We already have a glider-to-beehive and glider-to-block converter, virtue of Paul Callahan, but we can now add loaves and bi-blocks to the collection. The latter is especially interesting, as a glider in the same direction as the original can liberate the bi-block in the form of a Herschel. The gliders are separated by 4 half-diagonals, unlike in previous receivers, where the separation must be 2, 5 or 6; hence, this new receiver could function where others fail. Also, it transpires that the bi-block functions as a LWSS eater, which can be toggled by incoming gliders.
Finally, he noted that two gliders separated by 4 or 5 half-diagonals can be reflected into a single glider. Here it is demonstrated as part of a stable reflector and related pulse divider. Alas, this reflector does not break any records, unlike the next subject of discussion -- the rectifier. This fast reflector, subject of a previous article on LifeNews, can be used in various conduits for transforming Herschels into gliders, by either modifying the output or assisting in the cleanup of surplus blocks.
To summarise this article, here is a collection of the 30 distinct Herschel conduits (including four adjustable ones), and a comprehensive collection of every (sufficiently simple) conduit, transceiver and converter known, as of the time of writing.
Matthias Merzenich has discovered a c/7 diagonal spaceship -- the first of its speed. This raises the total to thirteen reasonably-low-period spaceship velocities, specifically eight orthogonal (c/2, c/3, c/4, c/5, 2c/5, c/6, 2c/7, 17c/45) and five diagonal (c/4, c/5, c/6, c/7, c/12). Of course, an infinite number of spaceship velocities are known, as the Gemini can be adapted accordingly.
Moreover, Matthias has actually discovered an infinite family of such spaceships, as one of the frontal components can support itself to yield an extensible spaceship.
As we've been busy recently, there have been no LifeNews postings for the last couple of months. Nevertheless, there have still been miraculous discoveries in Life that need reporting; this is one such example.
'Triller' from Nathaniel's forum has engineered an impressive construction: a fully functional display using pulsars to represent individual pixels. The display is continually refreshed at regular intervals, updating it with the data contained within several memory loops.
As he/she has included a comprehensive description of the mechanism, it would violate Occam's razor for me to describe it in great detail. However, there are some interesting features worth mentioning:
Nathaniel Johnston's website, conwaylife.com, is now back online, so the forums are accessible once again. Moreover, the homepage is updated with relevant blog entries from his blog, b3s23life, and even LifeNews itself!
Since time immemorial, there has been a desire to find a c/5 orthogonal anteater. The motivation arises from the existence of a c/4 diagonal antstretcher, which produces a compatible oblique line of ants. By fusing the two components seamlessly, a growing spaceship could be created.
Matthias Merzenich has finally found this long-awaited anteater, enabling completion of the antstretcher.
Eight copies of this can be combined to create another example of a 'space-nonfiller' (a term coined by Jason Summers), the earliest such example being discovered by him in 1999, which expanded at the vacuum speed limit.
At right is a diagram shows what the full pattern looks like, with a sample section of the generating line of cells expanded to explain the mechanism used to construct the breeder. Line sections are arranged to produce exactly-timed two-glider salvos, which collide to produce LWSSes, which in turn collide to build the breeder. A multi-step reaction at the X axis produces the second glider in each pair with an exactly-timed delay relative to the first one.
Paul Rendell has now completed his Universal Turing Machine, by adjoining two stack constructors to the bounded version of the pattern. The patterns are available here, including the c/2 orthogonal and c/5 diagonal variants of the pattern.
The new Turing machine emulator is period-23040, which makes it more HashLife-amenable than the previous UTM and TM emulators (p18960 and p11040, respectively). Additionally, the diagonal stacks run faster in Golly than the oblique analogues. The c/2 version outperforms the c/5 version, apparently, despite having a larger growth coefficient.
This is the first universal computer in Life to emulate a Turing machine in linear time (Chapman's URM takes exponential time; my UCC takes quadratic time), and therefore the most efficient to date.
To celebrate Paul Tooke's 50th birthday, this article is dedicated to one of his recent discoveries. Happy birthday, Paul!
Paul has recently been assembling patterns to defy common intuition about breeders, and thus help to determine a valid definition for what constitutes a breeder.
He has used the principles behind Gemini -- glider loops and universal construction -- to build unusual breeders with obscure properties. For example, he has engineered a SSS breeder, which amounts to a slide puffer (slide gun with stationary output) constructing more slide puffers. Moreover, he has designed it to have O(t^1.5) growth, rather than the O(n^2) typical of most breeders.
Paul's breeders, including a related SMS breeder, are available on the relevant forum thread. Rather than using the original Gemini construction arm, he has used an alternative construction arm known as the 'Pianola'. In the SMS version, this lays down block-laying switch engines; in the SSS version, this produces slide puffers instead.
Something amazing has happened. Two completely unrelated recent discoveries from 2010 have been elegantly unified into a single pattern.
First: a summary. Last month, LifeNews published an article detailing Paul Rendell's stack constructor, a pattern capable of extending the tape of his Turing machine without limit, thereby endowing his UTM with computation universality. The original version of his stack constructor comprises two orthogonal c/2 rakes, which stretch out an expanding wave of gliders, coalescing to form the tape.
Despite working as planned, Paul was somewhat unsatisfied with his stack constructor. Its population soon explodes due to the waves of gliders, an artefact of using two orthogonal rakes to build a diagonal structure. The natural solution would be to use a diagonal puffer to extend the stack; this is precisely what he has accomplished.
The second version of his stack constructor uses c/12 rakes to provide the influx of gliders necessary for stack construction. However, there is a catch. The tape has a spatial period of 90, so the temporal period of the Cordership fleet must be twelve times that -- or 1080. Corderships, on the other hand, have a temporal period of 96n, so the smallest multiple of the desired period is 4320. This means that four rakes are required to emulate the rakes. This is accomplished in a somewhat similar way to how gliders are interleaved in pseudo-period guns, but with rakes instead.
Additionally, because his rakes use a kickback reaction, the size of the rakes are increased by another factor of two. Here is the RLE file for the completed rake. The resulting stack constructor is truly immense, similar in population count to the Caterpillar, and the corresponding RLE file is 21.8 megabytes in size!
You may be wondering where the second piece of 2010 technology comes into the equation. Recall that four rakes are interleaved for each glider in the stack constructor, which is rather sub-optimal. Clearly, then, there are two options: redesign the tape to have an offset of (say) 96 cells, instead of 90; or forget c/12 technology completely.
Adam P. Goucher suggested the latter, advocating the use of the emerging field of c/5 diagonal technology. He modified Matthias' c/5 rakes to work at p450 instead, resulting in an immense glide-reflective rake, not much smaller than the c/12 pseudo-rakes! However, after several successive optimisations, and a sudden insight from Matthias, he managed to produce a substantially smaller rake:
Paul modified the rake yet again, to form a variant capable of inserting gliders using the kickback reaction. Instead of the naïve approach of using two rakes, he used the same engine to produce both incident gliders for the kickback rake. This cuts down on the population count by a factor of two.
The stack constructor based on the c/5 rake is almost five times smaller by population count than its c/12 counterpart (RLE file). What follows is a diagram of the constructor, similar to the previous one, giving an idea of the dimensions of this beast:
David Bell has reduced the minimum repeating population of any known sawtooth from 262 to 260 cells. The sawtooth, similar to his 262-cell variant (from 2005), uses a 4-engine Cordership and stationary p256 gun. The space between the dynamic and static ends of the sawtooth is alternately filled with gliders and emptied.
The population plot confirms the sawtooth behaviour of the pattern, showing that it is an exponential sawtooth. This is not the only possible envelope for a sawtooth, as Dean Hickerson built a parabolic sawtooth, so named because the convex hull of the population graph is a parabola.
David's sawtooth, despite having a very low population, still may not be optimal. There are two figure-8 oscillators between the p256 gun and four-engine Cordership, which could potentially be removed. Apparently, he searched for a sawtooth in which the interior of the gun performs this task, but to no avail.
We leave it as a challenge to the reader to create a sawtooth with a smaller minimum repeating population.
UPDATE: Thanks to Matthias Merzenich for noticing that I had confused David Bell and Paul Tooke; the article has now been amended.