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A346639
Decimal expansion of the Hausdorff dimension of Hironaka's curve and equivalent carpets.
2
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
Robert Dickau, Hironaka's Curve, describing the curve construction.
Curtis T. McMullen, Hausdorff Dimension of General SierpiƄski 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
KEYWORD
cons,nonn
AUTHOR
Kevin Ryde, Jul 26 2021
STATUS
approved