| Glimmering Garden
LifeViewer-generated soup (refreshes with page)
Glimmering Garden (Chinese name: 熠熠种花) is an isotropic non-totalistic Life-like cellular automaton devised by GUYTU6J on August 28, 2019.
Same as the classic Game of Life, Glimmering Garden is a 2-state rule on a square grid with range-1 Moore neighbourhood (i.e. Life-like). A cell will be ON in the next generation if its neighbourhood matches any of the (non-totalistic) configurations above up to reflection and/or rotation (isotropic). In Hensel notation this is expressed as B3-jknr4ity5ijk6i8/S23-a4city6c7c, which is used by Golly and LifeViewer.
Despite differing by many non-totalistic transitions, Glimmering Garden have shown similar dynamics and engineering capabilities to Life.
Without transition S3a, a block dies instantly, while the existence of B4i6i prohibits longer induction coils and B5k renders loaf unstable. Other than those the rule exhibits larger varieties of still lives compared with Life; the smallest asymmetric still life, for instance, is the 6-cell bobo$ob2o$bo! (xs6_2da, domino at tub) which does not persist in Life.
The most common still lives are:
|Name / rle||Apgcode||Major predecessors||Approximate relative frequency
out of all objects (out of 1.00)
|Cis-boat at boat||xs10_25bkc||3o$bo$3bo!||0.05178|
|Long ship||xs8_35ac||bo$3o$2bo$2bo$3bo! giving two copies of 2b2o$bobo$o2bo$3o!||0.04397|
|Boat||xs5_253||- (Modified from tub, long ship, etc)||0.04382|
|bo$ob2o$bobo$b2o!||xs8_2da6||3o$bo$3bo! (Interrupted halfway)||0.00856|
|domino at tub||xs6_2da||-||0.00706|
The listed tub and cis-boat at boat predecessor arises highly frequently; in 134 generations it develops a D4_x4-symmetric constellation consisting of two copies of each said still life aligned diagonally.
The period-240 oscillator cloverleaf is the most common oscillator as well as object[note 1] arising from soups. Several natural evolutionary sequences such as 3o$bo$3bo! (before annihilating) and bo$3o$o2bo$bobo! mentioned above give rise to copies of its predecessor, namely, R-pentomino or its 2-generation descendant B-heptomino (possibly with some spark cells tied to both). It is a reflectorless rotating oscillator of loopability 1 and 3 (but not 2) with ⌘-shaped motion. It was this oscillator, found in other rules, that led to the discovery of Glimmering Garden.
The period-28 oscillator Julia is the third most common oscillator. The envelope resembles a Julia set, hence the name. An octomino predecessor for it, 2bo$b3o$2obo$bo!, can be seen above at generation 41 of cloverleaf.
The period-44 oscillator butterfly is the fourth most common oscillator. It arises from a heptomino bo$3o$2b2o$2bo! or the namesake butterfly sequence o$2o$obo$b3o! in Life. There is a common non-trivial dimer of it with a 7-cell predecessor b3o$obo$2o! as well. Without S4c in the rule, this pattern becomes a replicator wave.
|Stamp collection for 16 still lives and 747 oscillators (period＜1000) as of June 10, 2022 (click above to open LifeViewer)|
RLE: here Plaintext: here
The periods for which statorless oscillator exists are 4, 12 (also strictly volatile), 14, 24+12n (p12 T-ship reflector), 28 (see above, also strictly volatile), 30, 44 (see above), 56+28n (p28 reflector), 60+30n (p30 reflector), 80 (see above, also strictly volatile), 88+44n (p44 gun-based reflector, not shown), 240 (see above, also strictly volatile). [note 2]
The glider from Life does not work in this rule. Spaceships from official apgsearch symmetries are listed below:
|c/2 orthogonal||(1,0)c/2||xq2_1b1||T-ship, the second most common object from C1. Some trivial flotillae are not separated by apgsearch. See section below for details|
|(2,0)c/4||xq4_0gn20uz133||Two T-ships with period-4 block tagalong, first found from Rust Life Search|
|(12,0)c/24||xq12_6e6x7oghewehgo7x6e6 and xq12_y2gom3as88sa3mogz1b1yg1b1||Two T-ships escorting two B-heptominoes, from even bilateral symmetries|
|(24,0)c/48||xq48_37... and xq48_y3...|
|(48,0)c/96||xq96_yz...||Two B-heptomino pair puffers|
|(24,0)c/48||xq48_277zw27d6y5be4zw277||Two T-ships escorting a 4-tick descendant of 3o$bo$3bo!, making a would-be p12 cloverleaf puffer from C1|
|8c/17 orthogonal||(16,0)c/34||xq34_y08... and xq34_y0g...||Two engines that alone would be a 32c/68 backrake, from bilateral symmetries; the latter one with even symmetry is the smallest known|
|(32,0)c/68||xq68_yk... and xq68_yu...||Engines are the same as above but with larger spatial distance, from C2 symmetries|
|(64,0)c/136||xq136_y1...||Engines are the same as above, from even bilateral symmetries|
|c/4 orthogonal||(1,0)c/4||xq4_nmkz31||The second smallest and fastest spaceship with distinctive speed, being non-monotonic as well, which has a 5T synthesis that is intrinsically a T+blinker+two dots reaction|
|3c/20 orthogonal||(3,0)c/20||xq20_dadz373||Quite similar to the fireship in Life: symmetric, non-monotonic, with wide and sparky rear end|
x = 68, y = 61, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c 20bobo$19bob3o$22bobo$24b2o$21bob2o$21b2o$21bo$b3o$o2bo$2o2bo$23b3o$ 24bo$24bobo2$23bobo2$17b3o5bo$17bo$17bo2bo$18b3o3$39bobo$38bob3o$41bob o$43b2o$40bob2o$40b2o$40bo4$42b3o$43bo$43bobo2$13b3o26bobo$12bo2bo$12b 2o22b3o5bo$36bo$36bo2bo$37b3o21bobo$60bob3o$63bobo$65b2o$62bob2o$62b2o $62bo3$44bo$41bobob3o16b3o$41b2obo2bo17bo$44bo2bo17bobo$44b3o$35b3o26b obo$34bo2bo$34b2o22b3o5bo$58bo$58bo2bo$59b3o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART GPS 10 TRACK 7/46 -7/46 ZOOM 4 WIDTH 480 HEIGHT 480 LOOP 92 ]]
x = 56, y = 53, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c bo11b3o$4o9bo$o2bo10bob3o$bobo7b3obo3bo$b4o5b2ob2o4bo$b4ob3o2bo2bo2b2o bo$7bo2bo3bo4b2o$5b2o4bob3o2b3o$8b3o8b2o$9bo7bobobo$11b2o3bo3b2o$13b2o $12bo10bo$12b3o4b4obo$14b2o3bo4bo$15bob2o5bo$17b2o4bobo$18b2o2bobobo$ 22bo3bo$17b2ob4o2bo$17bobo2bo$17b2o7bo$17b2o5b2obo$17b2o5bo$20bo13bo$ 19bobo10b3o$20bobo4bobob2o$21bobo2b2ob2o2b2o2b4o$22bobo2b2o4bo2bob2o2b o$23b2o4bo7b2obobo$30bob2o5bobobo$31bobo6bobo$32bobo8bo$33bobo8b2o$34b obo3b2o$35bob4obob2o$36bobobobobobo$37bo2bo2bo2bo$41bob2ob2o$42bob2o2b o$45bo2b2o$42bo5bobo$42b3o2bobo$44bo2bo2bo$45bob2obobo$46bo3b3o$51bo$ 52bo$51b3o$48b3o$48bo$54b2o$50b4o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART GPS 2 TRACK -0.25 -0.25 ZOOM 4 LOOP 4 ]]
Some non-exhaustive searches for c/2 and 2c/4 components have been carried out with Rust Life Search. Moreover, searching with ikpx2 has been done for c/4 diagonal and found 27 spaceships including tagalongs and extensible components as of February 2021; the smallest one is shown to the left.
FWKnightship constructed the first engineered spaceships from rake and puffers of simplified speed 4c/11, 8c/17 orthogonal and 7c/46 diagonal. Later, 16c/34 spaceships have also emerged from symmetric soups. Of these, the 4c/11 (unsimplified 24c/66) spaceships are the largest; even the minimal 2-engine form has 167×56 or at least 457 cells.
x = 174, y = 152, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c o4bo3bobo3bobobo2bo3bo3bo3bo4bo4bobo$80bobobobo3bo2bo6bo3bo49bobobo$ob o2bo2bo3bo4bo4bo3bo3bo3bobo2bo2bo$80bo2bo2bo3bo3bobo3bo55bo$o2bobo2bo 3bo4bo4bobobo3bo3bo2bobo2bo2bobo$80bo5bo3bo6bo2bo3bo51bo$o4bo2bo3bo4bo 4bo3bo3bo3bo4bo2bo4bo$80bo5bo3bo3bobo4bobo52bo$o4bo3bobo5bo4bo3bo3bo3b o4bo4bobo$156bo10$9bo4bo144bo5b2o$6bob2o4b2obo138bob2o4b3o$9bo4bo26bo 50bo66bo5b2o2$41bo50bo$40b3o48b3o4$14bo$9bo4b2obo21bo49bo$6bob2o4bo21b ob2o46bob2o$9bo29bo49bo$163b3o$163b3o$164bo3$14bo$14b2obo$9bo4bo144bo$ 6bob2o146bob2o$9bo34bo114bo2$44bo$43b3o2$14bo$14b2obo$14bo$9bo29bo101b obobobobobobobobobobobobobobobobo$6bob2o26bob2o$9bo29bo49bo$149bobobo 3bobobo2bo$89bo$88b3o58bo4bo2bo6bo$14bo144bobo$14b2obo131bo4bo2bo6bo$ 14bo$149bobobo3bobobo2bobobo$9bo79bo$6bob2o76bob2o$9bo35bo43bo68bo2$ 45bo112bo$44b3o110b3o4$15bo$9bo5b2obo20bo119bo$6bob2o5bo20bob2o116bob 2o$9bo29bo119bo6$15bo$15b2obo$9bo5bo$6bob2o$9bo150bo$38b3o$38b3o119bo$ 39bo119b3o2$15bo$15b2obo$15bo$9bo29bo119bo$6bob2o26bob2o116bob2o$9bo 29bo119bo7$15b2o$9bo4b3o$6bob2o5b2o$9bo153bo$41b3o$41b3o119bo$42bo119b 3o3$15b2o$14b3o$9bo5b2o22bo119bo$6bob2o26bob2o116bob2o$9bo29bo119bo$ 90b3o$90b3o$91bo2$15b2o$14b3o$15b2o$9bo79bo$6bob2o76bob2o$9bo79bo$42b 3o114b3o$42b3o114b3o$43bo116bo$15b2o$14b3o$15b2o2$9bo29bo119bo$6bob2o 26bob2o116bob2o$9bo29bo119bo11$44b3o$44b3o$45bo5$39bo49bo5bo$36bob2o 46bob2o5b2obo$39bo49bo5bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ ZOOM 1.8 ]] #C [[GPS 8]]
The bullet heptomino is the smallest and fastest spaceship. Its other phase is a t-tetromino plus one dot spark at the back, hence the name T-ship. Its significance for Glimmering Garden is to the same extent as the glider for Game of Life.
There are 29 distinct two T-ship collisions, 19 of which creates nothing, 6 creates a single T-ship (include 4 fly-by deletions, one 90-degree kickback and one 180-degree kickback; both kickback involve 3o$obo$bo!), one creates two T-ships via a teardrop-like sequence and another one creates two parallel blinkers. The remaining two is messy.
FWKnightship has demonstrated that a blinker may serve as a construction elbow by finding a set of PUSH, PULL and FIRE (sideways) elbow operations with 4~6 synchronized T-ships each. The minimum separation is 51 ticks, suitable for simple signal circuitry built with tubs and boats (see below).
Infinite growth patterns
x = 11, y = 8, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c 5bo$4bobo$4b3o3$bo7bo$obo5bobo$bo7bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART GPS 10 ZOOM 20 Y -2 THUMBSIZE 4 ]]
x = 27, y = 26, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c 10bo5bo$9b2o5b2o$8b2o7b2o3$13bo$12b3o$12bobo$12bobo$4bo2bo4bobo4bo2bo$ 5b2o6bo6b2o$2o23b2o$o25bo$3obo17bob3o$4b2o15b2o$3b2o17b2o7$9bo2bobo2bo $8b3o5b3o$9bo3bo3bo$10bob3obo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART GPS 10 TRACK 0 -8/17 ZOOM 7 Y 7 LOOP 68 ]]
The most common linear growth pattern is a 32c/68 orthogonal double T-ship backrake, as shown on the right. The shorter gap between adjacent T-ships is 20 ticks, while the longer gap is 112 ticks. It is the basis of many other linear growths collected in Catagolue, where the outputs are filtered by oscillators or crashed into other objects. The smallest linear growth polyplet predecessor is 3o$bo$bo$bo$2bo!, producing two opposite copies of this backrake with only 7 cells. Other ways to form the backrake include a collision between two copies of cloverleaf or w-pentomino (essentially the 3-tick descendant of 3o$bo$3bo!). The backrake is repeatedly generated and extinguished in some high-period oscillators such as the two p408 and the p828, and it can be reflected 180-degrees to make a p670 shuttle.
The third most common one is a 14c/92 diagonal puffer arising from a double interaction between a cloverleaf and a Julia, making the outputs solely Julias. Inrerestingly, these can be burnt at exactly the same speed, leading to a natural linear growth with the highest occured period.
x = 61, y = 51, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c 45b3o$46bo2$46bo2$7bo13bo$5bo3bo9bo3bo2$7bo13bo$5bo3bo9bo3bo$6bobo11bo bo$7bo13bo$5bo3bo9bo3bo$7bo13bo$7bo13bo$2bo4bo4bo3bo4bo4bo$5bo3bo4bo4b o3bo$obo9bobobo9bobo$6b3o5bo5b3o3$4bo5bo7bo5bo$36b3o15b3o$36bobo15bobo $22b3o12bo17bo$23bo13bo17bo$23bo13bo17bo$22b3o10bo3bo13bo3bo$5bo7b3o 20bobo15bobo$4b3o6b3o5bobobo$4b3o7bo6bobobo10bobo15bobo$23bo11b5o13b5o $4b3o15b3o11b3o15b3o$4bobo16bo13bo17bo$5bo$4bobo48bo$55bo$54b3o$16bo 38bo$13bob2o29bo7b3o$16bo28b3o7bo$26bo18b3o$23bob2o28bo$26bo26b2ob2o$ 36bo$33bob2o$36bo15b3ob3o$46bo3b3ob3ob3o$43bob2o3bo2bo3bo2bo$46bo3bo3b obo3bo$51b3o3b3o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART GPS 10 ZOOM 8 THUMBSIZE 2 WIDTH 600 HEIGHT 480 Y 4 TRACK 0 -3/20 LOOP 120 ]]
Combining 3c/20 spaceships gives T-ship backrake, siderake and forerake of multiple periods. On the right is a forerake composed of a 2-engine p20 backrake by FWKnightship, a 2-engine 90-degree (back-to-side) reflector, and a p120 core sustained by a single dot spark.
Other growth rates
A sqrtgun and caber tossers have been built. A T-ship to 24c/48o rake converter may be employed to build super-linear growth patterns. Sawtooth with a minimum repeating population of 188 has been found. A naturalistic primer was constructed by FWKnightship.
For demonstrations of this part, refer to the oscillator stamp collection above.
Giving off similar century-like sparks, the oscillators cloverleaf, Julia and butterfly could be excellent for periodic technologies. For Julia, T-ship 90-degree reflector, 180-degree reflector, Heisenburp (cf. p56 #9) and AND gate were found early on. However, later research focused on stable technologies and so p44 and p240 are lesser seeked.
Periodic 90-degree reflectors are also known with different mechanisms. Firstly, a head-on domino spark converts a bullet heptomino to a B-heptomino immediately, which can be turned to a new T-ship (type 4n+1) with a dot spark; this can be done at p8 (cf. p24 #5) with relatively tight bounding box, and the minimum working period for the domino sparker is p6 (cf. p18 #8). Secondly, the domino spark can be replaced with a xs12_2523ad2 (see below), providing new timing (but still type 4n+1) and thus allowing for p3 (cf. p27 #1) and p5 (cf. p30 #8) reflectors. Thirdly, bullet heptomino hitting a dying spark of 2×3 rectangle on the symmetrical axis gives another spark resembling Life's traffic light sequence, which evolves into a T-ship (type 4n+3) if suppressed with a tub (cf. p25 #5 and p34 #5).
There are a number of T-ship shuttles with periodic 180-degree reflectors (cf. p16 #2, p22, p24 #3, p30 #4, p33 #2, p36 #2, p40 #6, p48 #2).
A somewhat general gun construction scheme exists for oscillators emitting the 2×3 rectangle spark. Two copies of them face-to-face can make a blinker, and a third one dots its edge to create a T-ship. (cf. p30 #3) Another method involves dotting a "power button" blinker directly and using the resulting T-ship to regenerate original blinker as well as emitting another T-ship. (cf. p41 #3, p63 #4, p70, p124 #2; also see the p120 forerake above) This can also function as a parity-specific Heisenburp device for edgy dot-producing spaceships.
A decent application of periodic technologies is FWKnightship's p36960 OTCA metapixel, released surprisingly in January 2022.
Table 1: a brief summary of key repeat times
|Item||Repeat time in ticks|
|B-heptomino loop without extra signals||27|
|B-heptomino loop with extra T-ship||34|
|B-heptomino loop with extra B-heptomino||34|
|90-degree T reflector||41|
Table 2: catalysts
|Tub||Development of stable technologies uses tub as rock extensively.||Many|
|Boat||* As a replacement of tub when it loses one cell and becomes a t-tetromino;
* The first known bait for a T-ship giving two B-heptominoes at repeat time 51.
|* p3 #3, p40 #5, p66 #2, p126 #1, etc |
* p51 #2, p59 #5, p66 #3, p142 #1
|Ship||Its two-cell surface bites another two-cell surface via B4i, most commonly applied to a century spark to get a blinker; can be substituted with a tub at tub.||p41 #3, p109 #1, p124 #2, p133 #4|
|Barge||A flintstone that, upon hitting certain patterns, two B-heptominoes emerge like boat.||p23 #1, p32 #1, p56 #1, p64 #2, etc|
|Beehive at tub||A eater that works like CGoL fishhook, using the side near the tub; can be substituted with a xs8_2da3.||p3 #13, p46 #5, p56 #7, p68 #2|
|Snake||* As a replacement of tub, but cells connected to it survive longer; can be substituted with a fishhook for better clearance;
* A rock that works as in CGoL; can be substituted with a beehive.
|* p16 #3, p43 #1, p84 #2, p92 #1, etc |
* p23 #3, p38 #1, p56 #2
|Domino at tub||* As a replacement of tub when it gains one extra cell to be a boat;
* Providing a two-cell obo! surface, like CGoL hook with tail. More or less equivalent to a snake in some cases.
|* p28 #8, p30 #7, p34 #1, p92 #3, etc |
* p5 #24, p6 #6, p15 #2, p40 #3, etc
|Boat at tub||As a replacement of tub when it loses one cell and becomes a triplet spark.||p43 #2, p48 #1, p55, p56 #8, etc|
|xs12_2523ad2||The core of second-phase development, with very robust reaction site.||Many|
|xs16_at26z9m9 or xs17_2da32da23||A modified version of xs12_2523ad2 serving as the initial catalyst of T-to-B at repeat time 41.||p41 #4, p82 #4, p97|
|xs10_5rkc||The initial catalyst of T-ship reflector at repeat time 43.||p43 #1|
|xs20_2da32523ad2||A dimerized version of xs12_2523ad2 but less robust.||p10 #1, p38 #2, p40 #2, p78 #1, etc|
Stable technologies are of particular interest. Concentrating on tubs and boats, the first-phase elaboration has resulted in exact-timing (synchronized) Boolean logic AND, OR, XOR, ANDNOT gates, and eventually, a stable Rule 110 unit cell proving the rule Turing-complete. In the second phase starting from June 2021, much more catalysts were manually designed and tested against reactions, boosting development of the rule generally. As an example, the most compact 90-degree T-ship reflector uses a single 23-cell still life, which fits in a 12×15 bounding box and has a repeat time of 44 ticks, as shown in p44 #8.
x = 69, y = 37, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c 65bo2$65bo$64b3o7$27bo39bo2$27bo39bo$26b3o37b3o16$3bo14bo24bo14bo$ob2o 11bob2o21bob2o11bob2o$3bo14bo24bo14bo3$23b2obo37bobo$23bob2o35b2ob2o$ 62bobo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART GPS 10 ZOOM 8 LOOP 64 PAUSE 2 T 42 PAUSE 2 T 52 PAUSE 2 T 63 PAUSE 2 ]]
Memory cells that require destructive read can be built from a snake, while permanent switch where information can be read non-destructively is realized with the still life xs8_5rk. Meanwhile, still-life factorys making tub, boat, beehive, etc are used in demultiplexer, regulator and pulse divider.
In January 2022, a fangtian (Chinese: 方田) has been completed that is the counterpart of p1 megacell in this rule, with a unit cell size of 8192×8192 and a recommended period of 262144 (218). Next month, a 327680×327680, period-16777216 (224) fangtian-2 extended the simulatable range to anisotropic range-1 3-state rules on a square grid. Both have a couple of sample patterns registered in Golly's collection under Patterns/HashLife/Metacell folder.
- Taking the hundreds of thousands of combinations of cloverleaves that apgsearch does not separate into account, it is overwhelmingly more than the T-ship.
- This two-cloverleaf p480 is almost strictly volatile if it were not including two cells that turn on once per 240 ticks; check Oscillizer for that.
- GUYTU6J (June 12, 2019). Re: Reflectorless Rotating Oscillators (RRO) (discussion thread) at the ConwayLife.com forums
- 熠熠种花 - Glimmering Garden (B3-jknr4ity5ijk6i8/S23-a4city6c7c) (discussion thread) at the ConwayLife.com forums
- Glimmering Garden at Adam P. Goucher's Catagolue