Introduction |
Object Counts |
Still-lifes |
Small P2 oscillators |
Large P2 oscillators |
P3 oscillators |
P4 oscillators |
P6 oscillators |
P8 oscillators |
P10 oscillators |
P12 oscillators |
P20 oscillators |
Spaceships |
*c*/3 Orthogonal spaceships |
Constellations |
Methuselahs |
Links

One of the earliest variant Life rules to be studied and discussed was B34/S34, which was simply known as 3/4-Life. This is similar to Life, except both birth and survival occur on exactly 3 or 4 living neighbors. This rule has the following interesting properties:

- Due to the birth-on-4 rule, pseudo-objects are not possible.
- Still-lifes are possible, but except for the 4-bit block they are all extremely large and rare. They must all resemble large fortresses, whose outer walls must have corners resembling Life's ships or long-ships, and connected by straight crenelations. While the interiors of such still-lifes are more flexible, the exterior must necessarily be constructed in this way. The smallest three have 4, 36, and 44 bits respectively.
- Since the birth and survival rules are identical, a cell does not
participate in its subsequent state. I.e., no information is transferred
directly from a cell to its child. Information can only be transferred
from a cell to its subsequent generation by moving to an alternate cell
and coming back later. This principle attaches parity to information,
and it is no surprise that most known oscillators have a
period of 2 and fill the niche left
vacant by the rare still-lifes. Most higher-period oscillators also have
even periods (except for a few that are
period 3).
Furthermore, all three natural spaceships
(as well as most known larger ones) have a period of 3 and a velocity
of
*c*/3, demonstrating the same parity principle. - Unlike Life, where "hot" interiors of broths have a tendency to quickly over-populate everything and mostly die (causing broths to quickly fragment), the birth-on-4 rule allows large random broths to thrive. As a result, there are many patterns that exhibit chaotic infinite growth. In terms of object synthesis, the practical result of this is that it is harder to collide gliders together expecting them to die, leaving behind only small usable sparks. Similarly, small perturbations of existing objects frequently cause them to explode into uncontrolled growth.

Computer searches have counted still-lifes up to 45 bits, period 2 oscillators up to 17 bits, and period 3 oscillators up to 7 bits.

Numbers in **bold face** have been confirmed by computer search. Other
given numbers are believed to be complete, but have not yet been verified.
Lists with large numbers of objects that have not yet been counted are shown
with "many".

Status of object lists on sub-pages is shown by background color:

- White (or light blue) indicates small complete object list, with syntheses for all (or at least most).
- Red indicates complete object count; may have syntheses for some.
- Grey indicates objects not counted; may have syntheses for some.

Bits |
4 | 5 |
6 | 7 |
8 | 9 |
10 | 11 |
12 | 13 |
14 | 15 |
16 | 17 |
18-35 | 36 |
37-43 | 44 |
45 | 46-49 |
50 | 51 |

Still-lifes |
1 |
0 | 0 | 0 |
0 | 0 |
0 | 0 | 0 |
0 | 0 |
0 | 0 |
0 | 0 |
1 | 0 | 1 |
0 | 0 | 2 | 1 |

P2 oscillators |
3 |
1 | 0 | 5 |
11 | 6 |
25 | 75 |
173 | 301 |
1452 |
4859 | 12789 |
44621 | many | |||||||

P3 oscillators |
0 |
1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 18 | 0 | 157 | 10 | 0 | 2675 | 0 | 0 |

P4 oscillators |
0 |
0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 3 | 0 | many | |||||||

P6 oscillators |
0 |
0 | 1 | 0 | 0 | 1 | 0 | 0 | 3 | 0 | 2 | 0 | 3 | 0 | many | |||||||

P8 oscillators |
0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | none known | |||||||

P10 oscillators |
0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | none known | |||||||

P12 oscillators |
0 |
0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 26 | 0 | 0 | 0 | 0 | 0 | 305 |

P20 oscillators |
0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |

Spaceships |
0 |
0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | many |

These are all the still-lifes up to 51 bits. Except for the block, it is unlikely that any will every be synthesized from gliders.

Block [3] | 36-bit fortress [x] | 44-bit fortress [x] | 50-bit fortress #1 [x] | 50-bit fortress #2 [x] | 51-bit fortress [x] |

Although not generally considered a still-life, the empty field (i.e. death) technically fulfills all the criteria, and it occurs sufficiently frequently to warrant being mentioned. For example, it is the most common result of two gliders colliding.

These are all the period 2 oscillators 10 bits and smaller. All have syntheses, except eight 10-bit ones (roughly 1/3 of them).

These are some period 2 oscillators 11 bits and up that have syntheses. Note that the last one has a still-life fortress-structure on top, with oscillating bits on the bottom.

Star [3] | Light- weight beetle [12] | Middle- weight beetle [14] | Heavy- weight beetle [16] | Super diamond* [x] | Spur* [x] | Frog diamond* [x] |

Cis stars hassling block [13] | Trans stars hassling block [19] |

The stars hassling blocks and the super diamond are infinitely extensible, by using one star to hassle multiple oscillators. Blocks can also be hassled by super diamonds, spurs, and frog diamonds, and these are also infinitely extensible. Note that the frog diamond has corners resembling fortress-like still-lifes.

There appear to be many oscillators of period 4 with structures similar to ones of period 2 (including 1/3 of those shown). Note that the half-cross has a corner resembling fortress-like still-lifes.

The bi-hexafrob is infinitely extensible. There appear to be many oscillators of period 6 with structures similar to ones of period 2 (including half of those shown). Note that frog chain ripple has corners resembling fortress-like still-lifes, and has a pseudo-period of 6 (a composite of a period 2 exterior and a period 3 interior).

Rot8or [7] | 12-bit P8 #1 [4] |

Cyclone [7] | Webster's P10 [6] |

Loaf [3] | Prop- eller [3] | 18-bit P12 #1* [x] | 3 loaves hassling 2 M1s [x] | 4 loaves hassling 3 M1s [x] |

2 loaves hassling M1 [x] |

The M1-hassler is infinitely extensible. Two examples of this are shown.

Figure 8 [6] | Dual figure 8 [23] | Cis triple figure 8 [40] | Trans triple figure 8 [40] |

Figure-8s are infinitely extensible by sharing sparks, and can be arranged in any arrangement resembling large diagonally-aligned ominos. Three examples of this are shown.

There are only three common spaceships, all with a period of 3 and a
velocity of *c*/3. One moves diagonally, and the other two orthogonally.
The smallest one is the most common, and the basis for all syntheses. The other
two are extremely rare, but have been seen to escape from broths. The diagonal
one is the most rare, and is only commonly seen escaping along the line of
symmetry of some diagonal broths. It does, however, on occasion escape
asymmetric broths, but this happens much more rarely.

With the advent of computer search programs, several large spaceships
have been found; most also *c*/3 orthogonal, and 3 of other velocities. Due
to the large number of *c*/3 orthogonal spaceships, these are shown in the
next section. The alpha fast ship is infinitely extensible, by adding 8
additional sections on top for every 5 on the bottom; the first instance is
shown next to it.

Alpha fast ship; Glider 18266* [x] | Expanded alpha fast ship [x] |
200-bit c/5 orthogonal spaceship; Glider 21027 [x] |

c/3 Diagonal glider; Glider 563 [7] |
c/4 Diagonal glider; Glider 19358 [x] |

Note ¹: These spaceships can drag plows*, or act as drifters behind a trailing line of 4* or a trailing line of 2.

Note ²: These spaceships have front ends that resemble the basic glider, allowing them to be used as drifters in one of 6 different ways. (If this is mentioned twice, the spaceship has 2 front ends, also allowing 10 additional configurations where the spaceship is dragged by two smaller engines.

Since Delta ships can both drag drifters, and be used as drifters, they can form infinitely-extensible chains of Delta ships dragging one another.

Note ³: These spaceships can drag tag-alongs*, optionally with infinitely-extensible wick fragments between them. These can also be turned into wick-stretchers by anchoring the back end of the wicks.

Due to the large number of such variants, all the variants are grouped with their respective spaceships, and are not shown individually above (Gliders as drifters are shown with their draggers, and act as prototypes. Other drifters are shown with the spaceships being dragged.) The number of variants is shown in parentheses after the cost.

There are infinitely many constellations. These are some of the most common ones.

Two Ys [6] | Two yoyos [3] | Two clocks [2] |

Since many patterns in this rule grow forever, the term Methuselah is used to refer to such patterns, rather than merely long-lived ones, as it does in Life. These are some simple common ones.

M1; Methu- selah #1 [3] | M2; Methu- selah #2 [14] | M3; Methu- selah #3 [2] |

Run M1... | Run M2... | Run M3... |

Jack Eisenmann discovered most of the non-trivial spaceships, and many of the larger oscillators of several periods (marked with * in the above tables). He has much useful information about this rule on his web site.

**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*.