OFFSET
1,1
COMMENTS
From Jianing Song, Oct 12 2019: (Start)
log_3(16) is the Hausdorff dimension of the 4D Cantor dust. In general, the n-dimensional Cantor dust has Hausdorff dimension n*log_3(2).
Also, 1 + log_3(16) = log_3(48) is the Hausdorff dimension of the 4D analog of the Menger sponge. In general, let S_n = {(Sum_{j>=1} d_(1j)/3^j, Sum_{j>=1} d_(2j)/3^j, ..., Sum_{j>=1} d_(nj)/3^j) where d_(ij) is either -1, 0 or 1, Sum_{i=1..n} |d_(ij)| >= n-1 for all j}, then the image of S_n is the n-dimensional Menger sponge, whose Hausdorff dimension is log_3(2^n+n*2^(n-1)) = (n-1)*log_3(2) + log_3(n+2). n = 2 gives the SierpiĆski carpet, and n = 3 gives the original Menger sponge. See pages 10-12 of the arXiv link below, which gives an alternative construction of the n-dimensional Menger sponge and an illustration of the 4-dimensional Menger sponge. (End)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Yun Yang, Yanhua Yu, The generalization of Sierpinski carpet and Sierpinski triangle in n-dimensional space, arXiv:1702.04901 [math.DG], 2017.
EXAMPLE
2.5237190142858297483981084573710434171983425605275217114826...
MATHEMATICA
RealDigits[Log[3, 16], 10, 120][[1]] (* Vincenzo Librandi, Aug 29 2013 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
N. J. A. Sloane, Oct 30 2009
STATUS
approved