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In Tables of cellular automata, p.542, Wolfram defines the density $\delta$ of a rule to be the asymptotic density of nonzero sites when the initial configuration has density $1/2$. Wolfram quotes rational numbers for densities "whenever analytical arguments yield exact results. In a few cases, the rigour of these arguments may be subject to question."

For rule 110 he quotes $\delta_{110} = 4/7$ (so seems to have an analytical argument).

For rule 54 he quotes $\delta_{54} = 0.49 \pm 0.01$ (so seems to have no argument for $\delta_{54} = 1/2$).

Note that rules 110 and 54 both belong to Wolfram's class IV.

My question is two-fold:

  1. I could not find Wolfram's analytical argument for $\delta_{110} = 4/7$. Can anyone give me a reference, please?

  2. The best rational approximation of $0.49$ with denominator $d \leq 32$ is $15/31 = (2^4 - 1)/(2^5 - 1)$. On the other hand we have $\delta_{110} = 4/7 = 2^2/(2^3 - 1)$. Might there be an analytical argument that yields $\delta_{54} = 15/31$? Or is the finding spurious and/or numerology?

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  • $\begingroup$ I am pretty sure he doesn't have an argument for the 4/7. $\endgroup$
    – Ville Salo
    Commented Oct 27, 2022 at 11:15
  • $\begingroup$ @VilleSalo: But he seems to say so. $\endgroup$
    – Hans-Peter Stricker
    Commented Oct 27, 2022 at 11:20
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    $\begingroup$ Well it's probably the density of the periodic zones that always appear and that he describes. But surely he has not proved such zones almost always appear and cover most of the configuration, he just observes it. $\endgroup$
    – Ville Salo
    Commented Oct 27, 2022 at 11:30
  • $\begingroup$ @VilleSalo: I was just pointed to this blog post in the Wolfram community (filed under "recreation"): community.wolfram.com/groups/-/m/t/2843628. It confirms your assumption that there is no argument for 4/7. $\endgroup$
    – Hans-Peter Stricker
    Commented Mar 5, 2023 at 11:30

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