As I said, the pattern's surface obviously approaches a fractal, a quadratic variety of Koch curve.silversmith wrote: ↑May 20th, 2022, 10:09 amThe pattern grows indefinitely, so the details are “infinitely downscalable” relative to the entire pattern.Yoel wrote: ↑May 19th, 2022, 2:00 pm
1. It is true that fractals don't need to be self-similar. However, unless such structures are recursively proven to act like fractals, they may not be fractals. Technically speaking, true fractals are impossible in cellular automata, because by definition they must be infinitely downscalable. Even Sierpinski triangles in CA are not fractals as such, but approach true fractals at their limit.
2. The surface of this self-expanding structure approaches a true fractal, i.e. a Sierpinski triangle. The ever-growing central part, however, does not exhibit fractal properties. It's a non-repetitive spacefiller resembling Penrose's tiles.
Regarding the pattern itself (the inner part), you are technically right. At its limit, its fractal (Hausdorff) dimension is 2 (square-like), but still a fractal. Maybe more like a variety of Peano or Hilbert curve rather than Penrose's tiling, because hashlife easily captures and predicts its inner structure at any scale. BTW, Penrose tiling can also be viewed as a fractal.
However, usually people don't call such structures fractals, because at its limit the details become invisible. If you zoom it out, it looks like a plain filled space.
EDIT:
An obscure Sierpinski triangle:
Code: Select all
x = 38, y = 40, rule = R2,C17,M0,S2..3,B4..4,NM
10$11.3A5.J3.J$10.A3B3.J2.3K4.G$10.AB2.A3.2K2M2K3H$10.AB2CA3.2KM.3K2H
$9.D.2EDC.B.K2M2NHK2H$10.E2FEDC4.O2P2N.LG$9.2E2GF2CA2.2O2.P.N$8.2EG2H
F3A.2O3.2P2N.M$2.3A2.2EG.2IHGF.2O5.P3N$2.2AC.2EGHIJIHNF2O7.N.N$.A3C2E
2GIJ2K2F2O9.POM$.AC.DFGHIJK3LOP10.O$.3ADFGHIJKLEMOP6.O2.3O2LK$2.2A.D
2FHIJL.MNO2P2.P6.O5.I$3.2AD.2FHI2HKMNMJ2.P4.O.3L3J$6.4D2HEGE2MJ2P8.2N
L2J$5.B.2DGF2EGLG3J5.3ANJ.LKJB$8.2DE.2EB.DA7.2BJ2LKJDIC$7.C2.C.B.BD.B
A4.ABCBAJKJB2D$12.B.BD.BA3.ABC2DA2J2HGED$12.A2.2BA3.ABCD2EGHIHCG.D$
14.3A4.ABCDEFG2HEGE2C$22.AB2C2E3GE2C$23.3AC.3E.C$25.2A.CE.DC.B$26.2A
2CDCA$28.BOCB$27.P2.P!