A343620 Decimal expansion of the Hausdorff dimension of 4 X 2 carpets with rows of 3 and 1 sub-parts.

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, 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, 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

Offset: 1

Author

Kevin Ryde, Aug 04 2021

Keywords

• cons
• nonn

Comments

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).

+---+---+---+---+ Fractal carpet with each S

| | S | S | S | a shrunken copy of the whole.

+---+---+---+---+ Any 3 parts in one row and

| S | | | | 1 part in the other row.

+---+---+---+---+

Examples

			1.4499843134764958489211625600623791...
		

Links

• Timothy Bedford, Crinkly Curves, Markov Partitions and Dimension, Ph.D. thesis, University of Warwick, 1984, chapter 4.
• Curtis T. McMullen, Hausdorff Dimension of General Sierpinski Carpets, 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.

Crossrefs

Cf. A346639 (3 X 2 carpets), A090388 (1+sqrt(3)).

Programs

Mathematica

RealDigits[Log2[1 + Sqrt[3]], 10, 100][[1]] (* Amiram Eldar, Aug 04 2021 *)

Formula

Equals log_2(1+sqrt(3)).