Introduction | Running patterns | Still-lifes | Pseudo-still-lifes | P2 oscillators | P2 pseudo-oscillator | P3-6 oscillators | P8n oscillators | P140 and P280 oscillators | Spaceships | Constellations | Methuselah | Guns | Puffers | Breeder | Rule variant 0

This is actually several closely-related rules. While the birth conditions are totalistic (and, in fact, the same as Life's), the survival conditions are not totalistic - i.e. they depend not only on the number of neighbors, but also their relative positions. Birth occurs on any 3 neighbors, while survival occurs on 2-4 neighbors that are orthogonally connected to the original cell (i.e. diagonal neighbors are allowed only if they are themselves adjacent to one or two orthogonal neighbors). Most common patterns are shared by all variants. The variants arise by tweaking three of the survival conditions:

- 1: Three on one side, plus a fourth on a diagonal. If allowed,
this enables a (5, 6)
*c*/19 clean oblique puffer; otherwise, it enables a related (2,3)*c*/15 clean oblique puffer. - 2: Two opposite neighbors. If allowed, this enables the Blinker;
otherwise, it enables a 2
*c*/9 diagonal glider (the Tadpole). - 4: Three on one side, plus both other diagonals

Furthermore, variant 0 (i.e. none of the above) has a period 28 *c*/14
non-monotonic orthogonal glider. Variant 1 (i.e. 1 but not 2 or 4) has a
period 280 (mod 70) shuttle with useful sparks. Variant 5 (i.e. 1 and 4 but
not 2) has a 7*c*/27 orthogonal glider. Variants 1 and 5 (i.e. 1 but not 2;
4 is irrelevant) have a natural replicator. All of the variants also possess
two other natural spaceships of different velocities (*c*/3 diagonal,
*c*/3 orthogonal).

Due to the survival rule favoring ominos (and almost nothing else), it follows that almost all still-lifes are composed entirely of ominos or groups of disconnected ominos. It is also interesting that most of the oscillators and spaceships, and even the replicator, have phases that consist entirely of ominos.

It is also curious that most of the common objects are very closely related by structure:

- The smallest phase of the replicator differs from the block in only one bit.
- The smallest phase of the replicator differs from the orthogonal glider in only one bit.
- One phase of the tadpole differs from the orthogonal glider in only one bit.
- One phase of the puffer engine differs from the orthogonal glider in only two bits.
- One phase of the puffer engine differs from the diagonal glider in only two bits.
- The period 280 shuttle differs from the orthogonal glider in only one bit.
- The period 8 eater is composed of two colliding replicators.

The patterns on this page can be run by Golly, but require the
following rule files to first be downloaded and installed.
They can either be installed for all users (in the Rules directory directly
under where the Golly application is installed), or in the current user's
personal Rule directory. (To find out where this is, run Golly, select
*File/Preferences...*, look at the *Control* tab.
It is displayed right next to the *Your Rules...* button.

Golly versions 2.5 and later can use the single rule file (Niemiec1.rule).

Earlier versions of Golly require two files: the rule table (Niemiec1.table) and the corresponding color table (Niemiec1.colors).

These are all the still-lifes up to 10 bits, plus a few others for which syntheses are known. Several show techniques that can be used to make many still-lifes infinitely extensible.

Block on block; Bi- block [2] | Block on carrier [6] | Block below carrier [6] | Block pair on block pair [6] | Block pair on pair of stools [4] | Two pairs of stools [2] |

Beacon [2] | Beacon tie block [4] | Semi- phoenix [4] | Bridged beacons [x] | Cyclone [x] | Phoenix [x] | Beacon tie beacon [4] | 17-bit P2 oscillator [x] | 2x1 Phoenix [x] | 21-bit P2 oscillator [x] |

Block on beacon [5] |

P3 corner [x] | P3 Gumball [x] | P3 Jacob's ladder [x] | 10-bit P4 [x] | 24-bit P5 [x] | 19-bit P6 [x] |

P8 eater [x] | P16 Ping pong [x] | P40 Ping pong [x] | P64 Ping pong [x] |

The period 8 oscillator is one of the two commonly-occurring oscillators with periods above 2. It is called the Eater, because it can eat all of the spaceships, which is very convenient, since the simplest glider guns are based on the period 280 shuttle, and 280 is multiple of 8.

The oscillator can also reflect diagonal gliders, allowing for simple
glider-loop oscillators of all periods 8+24*n*, *n*≥0. The first three of
these are shown.

P140 (mod 70) Two interleaved shuttles [4] | P280 (mod 70) Shuttle [2] |

There is a natural 7-bit shuttle that generates huge clean plumes of debris and rotates 90 degrees every 70 generations, yielding a period 280 oscillator. Two of these can occupy the same area, effectively making a period 140 oscillator. Because the central core is so small, it is shown with the smallest population of the core (7) rather than the smallest total population (27) that occurs 19 generations later. The large debris plumes are useful, in that they can eat gliders, and even rub against others to create glider guns.

c/3 Diagonal glider [2] |
2c/9 Diagonal Tadpole [2] |
(6, 5)c/19 Clean oblique puffer [2] |
c/3 Ortho- gonal glider [2] |
Inward glider on glider [4] | Outward glider on glider [6] |

This rule includes 4 basic common spaceships, all of which have different periods and velocities. The orthogonal one is the most common, and forms the basis for all syntheses. It can also form pseudo-spaceships, the smallest two of which sometimes occur naturally, and are shown.

These are all the constellations that can be formed from two gliders (excluding those that include one or more gliders), and several constellations consisting of blocks that can be formed from three gliders.

P; P190 P replicator [2] |

Run P... |

This rule features a very common 5-bit replicator, the P pentomino. Every 95
generations, it turns into two copies of itself rotated 90 degrees, 7 spaces
forward, and 17 to either side, so the corners expand diagonally at a speed of
12*c*/95, with diagonal glide symmetry, similar to Life's switch engine. It
starts out as a clean binary sawtooth, but unfortunately, on the fourth and
subsequent cycles, the interior consists of irregular debris. It emits large
numbers of gliders of all types in all directions, and its corners tend to
expand in a regular manner. It does occasionally expand at a faster rate than
12*c*/95, as most gliders escape at *c*/3 or close to it, and sometimes these
collide to create new replicators far away from the original one.

It is theoretically possible that this might eventually stabilize, but after 22000 generations with a population of almost 2.5 million, it shows no indication of doing so, having spawned new copies of itself over 500 spaces beyond its corners, and multiple others much closer to itself, so the ultimate fate of this object is unknown. Unfortunately, since its behavior is irregular, it is not conducive to rapid computation using hashing algorithms. (Contrast with the breeder below, which is extremely regular. At generation 1e497, Golly computes 1e495 generations every 8 seconds, on an 2.2 GHz CPU)

This rule is frequently explosive, as many random patterns develop replicators near an edge, causing that edge to rapidly explode, sometimes apparently forever.

Many other mechanisms in this rule closely resemble the replicator, including the shuttle, puffer, and eater.

As this replicator's behavior is erratic, it is considered a methuselah with an unknown (presumably infinite) census. Its smallest predecessor has 4 bits. Most other methuselahs in this rule appear to be replicators surrounded by other things; they are far too numerous to list individually.

There is one simple multi-purpose glider gun, consisting of two period 280 shuttles in tandem, with 180-degree rotational symmetry, rubbing their sparks together. Each cycle, this releases two orthogonal gliders, and two diagonal ones, all in different directions. Normally, one suppresses three of these, yielding a gun that makes either an orthogonal glider, or a diagonal one. (There are also several other similar known guns that make only pairs of one kind of glider or the other, but it's easier to just use one kind of gun for everything).

Two orthogonal guns can shoot gliders at each other, in such a way that the gliders collide to either make any of the four kinds of spaceships. "tee-guns" are useful at inserting a glider stream close to another.

This is an SMM breeder that uses three of the four basic glider types. It consists of five basic sections:

The west section consists of five orthogonal glider guns. The westmost one makes one eastbound glider A. The next two form a tee-gun producing another eastbound glider B southwest of A. The next two form another tee-gun making a third eastbound glider C southwest of B.

The east section is similar arrangement of five orthogonal glider guns. The eastmost one makes one westbound glider D. The next two form a tee-gun making another eastbound glider E southwest of D. The next two form a tee-gun making another eastbound glider F southwest of E. These are much closer together than the western guns.

The north section is similar arrangement of five orthogonal glider guns. The northernmost one makes one southbound glider G. The next two form a tee-gun making another southbound glider H slightly east and substantially north of G. The next two form a tee-gun making another southbound glider I west and slightly south of G.

The northeast section contains three guns. The southmost of these makes a a diagonal glider J heading southwest. The other two form a tee-gun that makes an oblique puffer engine K heading south-southwest. (Due to the placement of the two guns, the topmost eaters suppressing north-bound diagonal gliders can be removed, as the two spurious diagonal gliders harmlessly annihilate each other.) Because the two streams have directions that are very similar but not identical, these guns must be placed unusually far away for the rest of the breeder to allow the two streams to converge on the same area.

The middle section is where all the components come together. The three pairs of gliders A+D, B+E, C+F from the east and west make three synchronous oblique puffer engines heading southwest. One engine by itself is a clean puffer. Three together form a rake that shoots one orthogonal gliders westward every three cycles (i.e. every 57 generations), and also leaves two spurious blocks behind. Puffer engine K comes up behind the new three-engine rake, and uses its smoke to clean up these blocks at its own leisure.

Unfortunately, two engines together form a dirty puffer that produces much toxic debris. For the short duration when there are only two engines in place, gliders J and G respectively hit the toxic debris clouds from the first two cycles, suppressing them.

The first time the three-engine combination puffs, it leaves behind two additional spurious blocks. Subsequent cycles do not do this, because the new smoke interacts with the previous cycle's smoke to suppress these. Glider H removes the northernmost of these two blocks, and puffer engine K just happens to cleanly eliminate the other one as it passes by.

The breeder creates a wave of four-engine rakes heading south-southwest, with waves of gliders heading west. Unfortunately, the first five glider waves would normally destroy the west-side guns. The last four of these are easily suppressed by eaters. The first cannot be eaten, because the eater would be in the path of the west-side gliders, so glider I is used to remove it.

As *n* approaches infinity, at generation *n*, the population is
asymptotic to 0.00022452 *n*² (43 *n*²/191520).
There are around 3.13283e-5 *n*² (*n*²/31920)
generated gliders in the field, and around 0.0142857 *n* (*n*/70)
generated puffer engines.

The static parts of the breeder consists of 36 shuttles and 56 eaters, all of which are glider-constructible, so in theory, it should be possible to synthesize the breeder from gliders, although the synthesis would be rather large (184 gliders, plus possibly a few more needed to suppress output from components constructed before other components are ready).

This image (reduced to half size) shows the breeder at generation 1120, after four 280-generation cycles. Four rakes can be clearly seen, with the front three emitting waves of gliders.

Rule variant 0 is generally less interesting than the above, so it is just being mentioned briefly. The oblique puffer, replicator, and rotor no longer work. However, there are two new notable spaceships.

The patterns on this page can be run by Golly, but require the
following rule files to first be downloaded and installed.
They can either be installed for all users (in the Rules directory directly
under where the Golly application is installed), or in the current user's
personal Rule directory. (To find out where this is, run Golly, select
*File/Preferences...*, look at the *Control* tab.
It is displayed right next to the *Your Rules...* button.

Golly versions 2.5 and later can use the single rule file (Niemiec0.rule).

Earlier versions of Golly require two files: the rule table (Niemiec0.table) and the corresponding color table (Niemiec0.colors).

(3, 2)c/15 Clean oblique puffer [2] |
c/14 Ortho- gonal moth [2] |

**Other rules:**
B3/S23 (Conway's Life),
Multi-colored Life,
B2/S2 (2/2 Life),
B34/S34 (3/4 Life),
Niemiec's Rules.

**See also:**
definitions,
structure,
search methodologies,
other rules,
news,
credits,
links,
site map,
search,
expanded search,
search help,
downloads.

Copyright © 1997, 1998, 1999, 2013, 2014 by Mark. D. Niemiec.
All rights reserved.

This page was last updated on
*2015-02-19*.