A327620 Decimal expansion of the Hausdorff dimension of the boundary of the tame twin-dragon curve.

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

Offset

1,2

Comments

There are only six regular 2-reptiles in the plane, four of which have fractal boundaries. Listed below are the names of these four tiles, along with the numbers of the corresponding sequences that give the decimal expansion of the Hausdorff dimension of each dragon curve's boundary; the pictures of these four 2-reptile fractals are drawn in the Mathafou link.

. The Levy dragon: A191689

. The Heighway dragon: A272031

. The twin-dragon: A272031

. The tame twin-dragon: this sequence.

The Hausdorff dimension of the dragon curve's boundary is given by dim_H(Delta dragon) = 2 * log_2(lambda_max) where lambda_max is the largest eigenvalue of some characteristic polynomial associated to the dragon tile. The characteristic polynomial associated with this tame twin-dragon tile is x^3 - x - 2 (see [Ngai, Sirvent, Veerman, Wang] link, p. 15) whose only real root is (1+sqrt(78)/9)^(1/3) + (1-sqrt(78)/9)^(1/3) = 1.521379706804... Hence the formula.

References

Jean-Paul Delahaye, Mathématiques pour le Plaisir, Belin Pour la Science, Paver des pavés, 2010, pp. 58-65.

Links

Table of n, a(n) for n=1..87.

Sze-Man Ngai, Victor F. Sirvent, J. J. P. Veerman, and Yang Wang, On 2-Reptiles in the Plane, Portland State University, PDX Scholar, 1999.

Eric Weisstein's World of Mathematics, Rep-Tile.

Wikimedia, Tame twindragon tile.

Wikipedia, Hausdorff dimension.

Wikipedia, List of fractals by Hausdorff dimension.

Formula

Equals 2 * log_2((1+sqrt(78)/9)^(1/3) + (1-sqrt(78)/9)^(1/3)).

Example

1.2107605332885233950258675064294643888668202387553...

Maple

evalf(2*log((1+sqrt(78)/9))^(1/3)+(1-sqrt(78)/9))^(1/3))/log(2), 50);

Mathematica

RealDigits[2 * Log2[(1 + Sqrt[78]/9)^(1/3) + (1 - Sqrt[78]/9)^(1/3)], 10, 100][[1]] (* Amiram Eldar, Sep 19 2019 *)

Prog

(PARI) 2 * log((1+sqrt(78)/9)^(1/3)+(1-sqrt(78)/9)^(1/3))/log(2) \\ Michel Marcus, Sep 21 2019

Crossrefs

Cf. A191689 (Levy dragon), A272031 (Heighway dragon and twindragon).

Sequence in context: A357585 A284797 A316135 * A325872 A021896 A188835

Adjacent sequences: A327617 A327618 A327619 * A327621 A327622 A327623

Keyword

nonn,cons

Author

Bernard Schott, Sep 19 2019

Status

approved