OCA talk:Rule 120

how is this different from sierpinski apart from the fact its tilted a bit tommyaweosme 21:17, 14 May 2024 (UTC)

it isn't a tilted Sierpinski (the only rules that do form it are the set of equivalence classes {(26,82),(60,102),(154,120,166,180),(18),(22),(90),(126),(146,182)}), the Sierpinski triangle that they approach can be defined as the limit S_∞ of a sequence of shapes where S₀ = {(0,0)} and S_i+1 = S_i ∪ {(x,y+2ⁱ): x,y ∈ S_i} ∪ {(x-2ⁱ,y+2ⁱ): x,y ∈ S_i}; as such it triples in size for each doubling in length, so has fractal dimension log₂(3)

x = 35, y = 16 34bo$33b2o$32bobo$31b4o$30bo3bo$29b2o2b2o$28bobobobo$27b8o$17bo8bo7bo$16b2o7b2o6b2o$15bobo6bobo5bobo$14b4o5b4o4b4o$8bo4bo3bo4bo3bo3bo3bo$7b2o3b2o2b2o3b2o2b2o2b2o2b2o$3bo2bobo2bobobobo2bobobobobobobobo$ob2ob4ob8ob16o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]

(click above to open LifeViewer)

whereas the fractal this one approaches (if one ignores the lines that form an asymptotically infinitesimal proportion of its total "area") instead displaces each copy of S_i by (0,0),(0,4ⁱ),(0,2*4ⁱ),(0,3*4ⁱ),(-2ⁱ,3*4ⁱ), so has a fractal dimension of log₂(5) with respect to x and of log₄(5) with respect to y (so I suppose one could describe it as log_√8(5)-dimensional with respect to area)

x = 18, y = 32, rule = B3/S23 17bo$17bo$17bo$16b2o$17bo$17bo$17bo$16b2o$17bo$17bo$17bo$16b2o$15bobo$15bobo$15bobo$14b4o$8bo4bo3bo$8bo4bo3bo$8bo4bo3bo$7b2o3b2o2b2o$8bo4bo3bo$8bo4bo3bo$8bo4bo3bo$7b2o3b2o2b2o$8bo4bo3bo$8bo4bo3bo$8bo4bo3bo$7b2o3b2o2b2o$3bo2bobo2bobobobo$3bo2bobo2bobobobo$3bo2bobo2bobobobo$ob2ob4ob8o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]

(click above to open LifeViewer)

in general, the nth rule's fractal is log_b(2ⁿ+1), where b=2 is horizontal, b=2ⁿ is vertical and b=2ⁿ⁺¹² is areawiise

also note, I intend to either rewrite much of this page or maybe upload the LaTeX writeup I've been working on, in which I have proven considerably more than is explained here, but that led to further questions that I have not yet answered, but feel would be necessary before it may be published. DroneBetter (talk) 23:02, 14 May 2024 (UTC)

cyclic tapes: single cell's period is not maximal

A small note for future investigators:

from dronery import*
cycshift=λ l,s,i: (s<<i%l|s>>(l-i%l))&~(~0<<l)
lexmin=λ l,s: min(map(λ i: cycshift(l,s,i),range(l)))
oscs=Y(λ f: λ o,i: λ t: (f(o[:-1]+((n,),),n+1)(t) if (n:=next(filter(λ i: not any(map(rcontains(i),o)),range(i,len(t))),-1))>=0 else o[:-1]) if o[-1]==() else f(o[:-1]+(o[-1]+(n,),()) if (n:=t[o[-1][-1]]) in o[-1] else o[:-1]+(o[-1]+(n,),),i)(t))(((),),0)
m=0 #depending on whether cyclically-shifted states are equivalent
print(stratrix(tap(λ n: sorted(sap(λ o: len(o)+~o.index(o[-1]),oscs(tap(λ s: (λ s: lexmin(n,s) if m else s)(s^cycshift(n,s,1)&cycshift(n,s,2)),range(1<<n))))),range(2,12)),dims=2))

one obtains tables of possible periods

wid| ordinary              | cyclic
 1 | 1                     | 1
 2 | 1                     | 1
 3 | 1                     | 1
 4 | 1,  4                 | 1
 5 | 1, 15                 | 1, 3
 6 | 1                     | 1
 7 | 1, 49                 | 1, 7
 8 | 1,  4,120             | 1,15
 9 | 1,  9, 54             | 1,18
10 | 1, 15,410             | 1, 3,41
11 | 1,176                 | 1,16
12 | 1,  4, 56, 60         | 1, 5,28
13 | 1, 10, 26,143,403,416 | 1, 2,10,11,31,32

the maxima go

       n|1,2,3,4, 5,6, 7,  8, 9, 10, 11,12, 13
  cyclic|1,1,1,1, 3,1, 7, 15,18, 41, 16,28, 32
ordinary|1,1,1,4,15,1,49,120,54,410,176,60,416
 A334505|1,1,1,2,15,1,49, 15,54,205,176, 1,403

(with A334505 being the eventual period of a single cell in an n-cell universe which is cyclic but not invariant under cyclic shift)

so the easy-to-conjecture idea that a single cell always becomes a maximal-period oscillator for odd n breaks down at n=13