Larger than Life
Larger than Life (abbreviated as LTL or LtL) is an algorithm that supports a family of rules with an extendable neighbourhood, as defined by Kellie Michele Evans in her 1996 thesis.
Contents
Notation
Larger than Life rules are supported by Golly 3.0b1 and onwards, using the following notation, created by Mirek Wójtowicz for MCell:
- Rr,Cc,Mm,Ss_{min}..s_{max},Bb_{min}..b_{max},Nn
Here:
- Rr specifies the range (r is from 1 to 500 in Golly^{[note 1]}; 1 to 10 in MCell).
- Cc specifies the number of states (c is from 0 to 255 in both Golly and MCell^{[note 2]})
- Mm specifies if the middle cell is included in the neighborhood count (m is 0 or 1).
- Ss_{min}..s_{max} specifies the count limits for a state 1 cell to survive.
- Bb_{min}..b_{max} specifies the count limits for a dead cell to become a birth.
- Nn specifies the extended neighborhood type (n is M for Moore or N for von Neumann in Golly; NM or NN respectively in MCell).
This diagram shows the extended Moore and von Neumann neighborhoods for range 3:
If the number of states (specified after C) is greater than 2, then states 1 and above don't die immediately but gradually decay. Note that state values above 1 are not included in the neighborhood counts and thus play no part in deciding the survival of a state 1 cell, nor the birth of an empty cell. C0 and C1 are equivalent to C2.
Examples
The Patterns/Larger-than-Life folder included with Golly 3.0b1 contains a number of example patterns (mostly from the MCell collection). The following table shows a number of example rules along with their commonly used names:
Rule | B/S equivalent | Name | Remarks |
---|---|---|---|
R1,C0,M0,S2..3,B3..3,NM | B3/S23 | Life | the default rule for this algorithm in Golly. |
R5,C0,M1,S34..58,B34..45,NM | — | Bugs | a chaotic rule by Kellie Evans. |
R10,C0,M1,S123..212,B123..170,NM | — | Bugsmovie | a chaotic rule by David Griffeath. |
R8,C0,M0,S163..223,B74..252,NM | — | Globe | an expanding rule by Mirek Wójtowicz. |
R1,C0,M1,S1..1,B1..1,NN | B1/S0V | Gnarl | an exploding rule by Kellie Evans. |
R4,C0,M1,S41..81,B41..81,NM | — | Majority | a stable rule by David Griffeath. |
R7,C0,M1,S113..225,B113..225,NM | — | Majorly | an expanding rule by David Griffeath. |
R10,C255,M1,S2..3,B3..3,NM | — | ModernArt | a chaotic rule by Charles A. Rockafellor. |
R7,C0,M1,S100..200,B75..170,NM | — | Waffle | an expanding rule by Kellie Evans. |
Alternative rule notations
Golly also allows rules to be entered using the notation defined by Kellie Evans in her thesis. The range, birth limits and survival limits are specified by five integers separated by commas:
- r,bmin,bmax,smin,smax
Catagolue uses a related notation in which the letter t ("to") is used to indicate birth/survival condition ranges:
- rRbB_{min}tB_{max}sS_{min}tS_{max}
These notations assume an extended Moore neighbourhood in which a live middle cell is included in the neighbourhood count. For example, Life can be entered as 1,3,3,3,4 in Evans' notation.
Generalizing LtL rules to different ranges
Larger than Life rules can be generalized ("converted") to different ranges. To convert the range-r_{0} rule Rr_{0},Cc,Mm,Ssmin_{0}..smax_{0},Bbmin_{0}..bmax_{0},Nn to a range-r_{1} rule:
- Compute N = (2 · r_{1} + 1)^{2} / (2 · r_{0} + 1)^{2}.
- Multiply the original rule's minimum and maximum birth/survival conditions by N and round off, to wit:
- Compute smin_{1} = round(smin_{0} · N).
- Compute smax_{1} = round(smax_{0} · N).
- Compute bmin_{1} = round(bmin_{0} · N).
- Compute bmax_{1} = round(bmax_{0} · N).
The converted rule is Rr_{1},Cc,Mm,Ssmin_{1}..smax_{1},Bbmin_{1}..bmax_{1},Nn.
Example
For example, converting the range-2 rule R2,C0,M1,S5..9,B7..9,NM to range 7, we obtain:
- N = (2 · 7 + 1)^{2} / (2 · 2 + 1)^{2} = 225 / 25 = 9.
- smin_{1} = 5 · 9 = 45.
- smax_{1} = 9 · 9 = 81.
- bmin_{1} = 7 · 9 = 63.
- bmax_{1} = 9 · 9 = 81.
The converted rule is, therefore, R7,C0,M1,S45..81,B63..81,NM.
Also see
Notes
- ↑ 1 to 50 in Golly 3.0b1.
- ↑ The documentation for MCell specifies a maximum of 25 states; this is a typo.
References
- Cellular Automata rules lexicon: Family: Larger than Life at Mirek Wójtowicz's Cellebration page
- Kellie Michele Evans: Larger than Life: it's so nonlinear (1996 Ph.D. thesis)
Further reading
- Kellie Michele Evans: Larger than Life's Extremes: Rigorous Results for Simplified Rules and Speculation on the Phase Boundaries, in: Andrew Adamatzky (ed.), Game of Life Cellular Automata, Springer (London) (2010), ISBN 978-1-84996-216-2, OCLC 619946115
- Kellie Michele Evans: Larger than Life, in: Andrew Adamatzky, Genaro J. Martínez (eds.): Designing Beauty: The Art of Cellular Automata, Springer (2016), ISBN 978-3-319-27269-6, OCLC 934720008
External links
- Larger than Life (discussion thread) at the ConwayLife.com forums
- Golly 3.0b1 (discussion thread) at the ConwayLife.com forums