|
%I #10 Aug 07 2021 01:44:36
%S 1,4,4,9,9,8,4,3,1,3,4,7,6,4,9,5,8,4,8,9,2,1,1,6,2,5,6,0,0,6,2,3,7,9,
%T 1,5,6,6,3,5,7,3,4,7,8,4,1,2,3,3,8,1,8,9,2,5,3,1,5,4,1,2,5,8,6,3,1,1,
%U 4,6,7,4,6,9,6,3,2,5,8,6,0,6,4,1,7,9,1,8,9,9,8,4,3,0,9,1,3,9,9,3,8,5,8,8,0
%N Decimal expansion of the Hausdorff dimension of 4 X 2 carpets with rows of 3 and 1 sub-parts.
%C Bedford (page 100 figure 34) gives this type of carpet as an example where the Hausdorff dimension differs from the capacity dimension (which is 3/2).
%C +---+---+---+---+ Fractal carpet with each S
%C | | S | S | S | a shrunken copy of the whole.
%C +---+---+---+---+ Any 3 parts in one row and
%C | S | | | | 1 part in the other row.
%C +---+---+---+---+
%H Timothy Bedford, <a href="http://wrap.warwick.ac.uk/50539/">Crinkly Curves, Markov Partitions and Dimension</a>, Ph.D. thesis, University of Warwick, 1984, chapter 4.
%H Curtis T. McMullen, <a href="https://doi.org/10.1017/S0027763000021085">Hausdorff Dimension of General Sierpinski Carpets</a>, Nagoya Mathematical Journal, volume 96, number 19, 1984, pages 1-9, see page 1 dim(R) for the case n=4, m=2, t_0 = 1, t_1 = 3.
%F Equals log_2(1+sqrt(3)).
%e 1.4499843134764958489211625600623791...
%t RealDigits[Log2[1 + Sqrt[3]], 10, 100][[1]] (* _Amiram Eldar_, Aug 04 2021 *)
%Y Cf. A346639 (3 X 2 carpets), A090388 (1+sqrt(3)).
%K cons,nonn
%O 1,2
%A _Kevin Ryde_, Aug 04 2021
|
