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

Author

Kevin Ryde, Jul 26 2021

Keywords

• cons
• nonn

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.

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

Examples

			1.3496838201955775731155390813143199...
		

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

• 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.)

Crossrefs

Cf. A346640 (metric dimension).

Programs

Mathematica

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

PARI

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

Formula

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