Life Variant: 3/4 Life

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



Introduction

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:


Object Counts

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:

Bits 45 67 89 1011 1213 1415 1617 18-3536 37-4344 4546-49 5051
Still-lifes 1 000 00 000 00 00 00 101 00 21
P2 oscillators 3 105 116 2575 173301 1452 485912789 44621many
P3 oscillators 0 100 00 00000 02018 015710 02675 00
P4 oscillators 0 010 01 001 00 030 many
P6 oscillators 0 01001 00302 030 many
P8 oscillators 0 00000 10100 000 none known
P10 oscillators 0 00000 00200 000 none known
P12 oscillators 0 010 10 000 00 00 02 2600 00 0305
P20 oscillators 0 00000 00001 0001 02000 00
Spaceships 0 01010 01000 000 many



Still-lifes

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.

B34/S34 still-lifes

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.


Small Period-2 Oscillators (4-10 bits)

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

B34/S34 period 2 oscillators 4-10 bits

Z (4.1) [3] Y (4.2) [3] Yoyo (4.3) [3] Blinker (5.1) [2] Needle (7.1) [3] Sema- phore (7.2) [5] N (7.3) [5] YY (7.4) [3] Clock (7.5) [2]
Guard (8.1) [5] Crown (8.2) [5] Candle (8.3) [5] Yoyo (8.4) [3] Cis yoyo on yoyo (8.5) [4] Trans yoyo on yoyo (8.6) [6] Yoyo by yoyo (8.7) [4] Turban (8.8) [13] Skewed Y on Y (8.9) [6]
Y tie cis Y (8.10) [6] Y tie trans y (8.11) [6] Acrobat (9.1) [3] 9.2 [33] 9.3 [14] Yoyo blinker (9.4) [3] Fork (9.5) [3] 9.6 [18]  
10.1 [20] 10.2 [25] 10.3 [22] 10.4 [19] 10.5 [17] 10.6 [9] 10.7 [21] 10.8 [11] 10.9 [8]
10.10 [8] 10.11 [9] 10.12 [4] Hat (10.13) [2] Sea- horses (10.14) [4] 10.15 [x] 10.16 [6] 10.17 [x] 10.18 [x]
10.19 [x] 10.20 [x] 10.21 [10] 10.22 [3] 10.23 [x] 10.24 [x] 10.25 [x]  


Large Period-2 Oscillators

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.

B34/S34 period 2 oscillators 4-10 bits

Small dollar (11.1) [19] 11.2 [8] 11.3 [22] 11.4 [10] 11.5 [50] Bell (11.20) [3] 11.21 [11] 11.23 [10] 11.24 [12] Clock skewed tie Y (11.25) [13]
11.30 [8] Clock tie cis Y (11.46) [5] Clock tie trans Y (11.47) [13] N tie cis Y (11.51) [5] YY tie cis Y (11.52) [6] Sema- phore tie cis Y (11.54) [8]   13.6 [6] 13.10 [29] 13.112 [11]
12.6 [28] 12.7 [17] 12.8 [4] 12.64 [4] 12.71 [4] 12.72 [4] 12.117 [8] Phoenix (12.144) [6] 12.158 [4]  
14.10 [20] 14.123 [3] 14.127 [22] 14.389 [22] 14.390 [22] 14.401 [12] Bi- clock (14.402) [5] 14.656 [20] 14.657 [12] 14.1420 [12]
15.1699 [9] 16.1573 [9] 16.6520 [8]   17.1158 [21] 17. 39101 [10]   18-bit flip- flop #1 [7] 18-bit flip- flop #1 [22] 19-bit flip- flop #1 [9]
20-bit flip- flop #1 [15] 20-bit flip- flop #2 [8]   26-bit flip- flop #1 [6]   35-bit fortress flip- flop [x]  


Period-3 Oscillators

B34/S34 period 3 oscillators

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.


Period-4 Oscillators

B34/S34 period 4 oscillators

Pinwheel [3] Flag [5] 12-bit P4 #1 [x] 16-bit P4 #1 [x] Octagon [2] Frog [41] 18-bit P4 #3 [x] 18-bit P4 #1 [x] 18-bit P4 #2 [x]
19-bit P4 #1 [29] 20-bit P4 #2 [x] 20-bit P4 #1 [31] Half Cross [x] 28-bit P4 #1 [x] Cross [x]  

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.


Period-6 Oscillators

B34/S34 period 6 oscillators

Hexafrob [2] W [3] Bi- hexa- frob [5] 12-bit P6 #1 [x] Yoyo line-4 yoyo [7] 14-bit P6 #1 [x] Y line-2 line-4 yoyo [x]   Frog chain ripple* [x]
Donut [x] 16-bit P6 #2 [x] Y line-2 line-4 line-2 Y [16] 18-bit P6 #1 [x] 19-bit P6 #1 [x] 20-bit P6 #1 [x] Chorine [x] Bakery [4]

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).


Period-8 Oscillators

B34/S34 P8

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


Period-10 Oscillators

B34/S34 P10

Cyclone [7] Webster's P10 [6]


Period-12 Oscillators

B34/S34 period 12 oscillators

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.


Period-20 Oscillators

B34/S34 period-20 oscillators

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.


Spaceships

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.

B34/S34 spaceships

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]


c/3 Orthogonal Spaceships

B34/S34 c/3 orthogonal spaceships

Glider; Glider 212² [3] Super glider; Asym- metric glider; Glider 1125 [4] Super glider w/ tag-along³* [x] (8(∞)) Glider and plow²* [x] (7) Broken titan glider¹* [x] (4) Titan glider¹* [x] (4) Gamma ship* [x] Theta ship* [x] Eta ship³* [x] (7(∞)) Zeta ship* [x]
Beta ship* [x] Epsilon ship²²* [x] (17) Nu ship* [x] Lambda ship* [x] Alpha ship* [x] Xi ship* [x] Sigma ship* [x]
Mu ship²* [x] (7) Kappa ship* [x] Delta ship¹²²* [x] (20(∞)) Iota ship²* [x] (7) Omicron ship* [x] Pi ship* [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.


Constellations

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

B34/S34 spaceships

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


Methuselahs

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.

B34/S34 spaceships

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



Links

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.

Home page | Life page

Copyright © 1997, 1998, 1999, 2013, 2014 by Mark. D. Niemiec. All rights reserved.
This page was last updated on 2015-02-19.