Chapter 10: Processes of Perception and Analysis

Section 5: Data Compression

Lengths of [number] representations

(a) n, (b) Floor[Log[2,n]+1], (c) Tr[FixedPointList[Max[0,Ceiling[Log[2,#]]]&,n+2]]-n-3, (d) 2 Ceiling[Log[3,2n+1]], (e) Floor[Log[GoldenRatio,Sqrt[5] (n+1/2)]]. Large n approximations: (a) n, (b) Log[2,n], (c) Log[2,n]+Log[2,Log[2,n]]+…, (d) 2 Log[3,n], (e) Log[GoldenRatio,n].

Shown on a logarithmic scale, representations (b) through (e) (given here for numbers 1 through 500) all grow roughly linearly:

Lengths of [number] representations image 1

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]