A346639 Decimal expansion of the Hausdorff dimension of Hironaka's curve and equivalent carpets.

1, 3, 4, 9, 6, 8, 3, 8, 2, 0, 1, 9, 5, 5, 7, 7, 5, 7, 3, 1, 1, 5, 5, 3, 9, 0, 8, 1, 3, 1, 4, 3, 1, 9, 9, 0, 0, 4, 9, 7, 9, 3, 6, 1, 4, 2, 9, 1, 8, 8, 7, 6, 7, 7, 4, 9, 4, 8, 4, 4, 1, 5, 3, 7, 5, 4, 2, 2, 2, 6, 1, 3, 5, 1, 8, 3, 0, 4, 9, 9, 0, 3, 9, 9, 8, 9, 9, 6, 1, 6, 3, 1, 2, 0, 2, 4, 2, 3, 6, 5, 2, 2, 4, 3, 5

Offset

1,2

Comments

McMullen calculates the Hausdorff dimension of various carpets, with the present constant being 3 parts in a 3 X 2 grid.

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

| | S | | Fractal carpet with each S

+---+---+---+ a shrunken copy of the whole.

| S | | S | Any 3 parts not all in one row.

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

References

Gerald Edgar, Measure, Topology and Fractal Geometry, second edition, section Hironaka's Curve, pages 232-234, where exercise 7.2.17 is to find McMullen's result.

Links

Table of n, a(n) for n=1..105.

Robert Dickau, Hironaka's Curve, describing the curve construction.

Curtis T. McMullen, Hausdorff Dimension of General Sierpiński Carpets, Nagoya Mathematical Journal, volume 96, number 19, 1984, pages 1-9, see start of page 2.  (Also author's image gallery showing Hironaka's M curve.)

Formula

Equals log_2(1 + 2^log_3(2)).

Example

1.3496838201955775731155390813143199...

Mathematica

RealDigits[Log2[1 + 2^Log[3, 2]], 10, 105][[1]] (* Amiram Eldar, Jul 27 2021 *)

Prog

(PARI) log(1 + 2^(log(2)/log(3)))/log(2) \\ Michel Marcus, Jul 27 2021

Crossrefs

Cf. A346640 (metric dimension).

Sequence in context: A021290 A016656 A319021 * A249187 A084425 A365480

Adjacent sequences: A346636 A346637 A346638 * A346640 A346641 A346642

Keyword

cons,nonn

Author

Kevin Ryde, Jul 26 2021

Status

approved