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.

Notation

Larger than Life rules are supported by Golly 3.0 and onwards and LifeViewer build 260 and onwards, using the following notation, created by Mirek Wójtowicz for MCell:

Rr,Cc,Mm,Ssmin..smax,Bbmin..bmax,Nn

Here:

  • Rr specifies the range (r is from 1 to 500 in Golly and LifeViewer; 1 to 10 in MCell).
  • Cc specifies the number of states (c is from 0 to 255 in Golly, LifeViewer and MCell[note 1])
  • Mm specifies if the middle cell is included in the neighborhood count (m is 0 or 1).
  • Ssmin..smax specifies the count limits for a state 1 cell to survive.
  • Bbmin..bmax 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. Golly and LifeViewer also support C for Circular neighborhood. LifeViewer additionally supports 2 for L2, + for Cross, X for Saltire and * for Star neighborhoods and many more).

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 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.

Notable patterns which are frequently found within Larger than Life rules include "bugs", which are patterns (usually spaceships, but sometimes oscillators) which are hollow, characterised by those from Bosco's Rule, "solid ships", which are typically extremely slow circular spaceships first noted in 2002 (such as this one), and "roomba bugs", travelling patterns that usually stabilise into low-period oscillators after intense amounts of generations but are infrequently found to be real spaceships.

Alternative rule notations

Golly and LifeViewer also allow 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, apgsearch and LifeViewer use a related notation in which the letter t ("to") is used to indicate birth/survival condition ranges; the initial gC is optional, with the number of states defaulting to two:

gCrRbBmintBmaxsSmintSmax

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 amounts of neighbors

Larger than Life rules can be generalized ("converted") to different ranges and even neighborhoods. To convert the range-r0 n0-neighborhood rule Rr0,Cc,Mm,Ssmin0..smax0,Bbmin0..bmax0,Nn0 to a range-r1 n1-neighborhood rule:

  1. Compute N to equal the ratio of the number of neighbors in a range-r1 n1-neighborhood to the number of neighbors in a range-r0 n0 neighborhood.
  2. Multiply the original rule's minimum and maximum birth/survival conditions by N and round off, to wit:
    1. Compute smin1 = round(smin0 · N).
    2. Compute smax1 = round(smax0 · N).
    3. Compute bmin1 = round(bmin0 · N).
    4. Compute bmax1 = round(bmax0 · N).

The converted rule is Rr1,Cc,Mm,Ssmin1..smax1,Bbmin1..bmax1,Nn1.

Example

For example, converting the range-2 rule R2,C0,M1,S5..9,B7..9,NM to range 7 with the same neighborhood, with the formula for the number of neighbors in a range-r Moore neighborhood being (2 · r + 1)2, we obtain:

  1. N = (2 · 7 + 1)2 / (2 · 2 + 1)2 = 225 / 25 = 9.
  2. smin1 = 5 · 9 = 45.
  3. smax1 = 9 · 9 = 81.
  4. bmin1 = 7 · 9 = 63.
  5. bmax1 = 9 · 9 = 81.

The converted rule is, therefore, R7,C0,M1,S45..81,B63..81,NM.

gollark: (frankly, that one should have been obvious!)
gollark: Oh, right, Tux1 is operated as a joint project with baidicoot.
gollark: Not ones you'd know, most of them lurk.
gollark: I have some other alt accounts, but I forgot exactly which.
gollark: We are in fact the same person, as is gibson, Esobot and Ducko.

See also

Notes
  1. The documentation for MCell specifies a maximum of 25 states; this is a typo.

References

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
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