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%I #62 Jun 24 2023 13:08:38
%S 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,
%T 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,
%U 9,9,6,7,7,2,4,6,1,5,3,6,7,3,4,6,6,6,5
%N Decimal expansion of the Hausdorff dimension of the boundary of the tame twin-dragon curve.
%C 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.
%C . The Levy dragon: A191689
%C . The Heighway dragon: A272031
%C . The twin-dragon: A272031
%C . The tame twin-dragon: this sequence.
%C 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.
%D Jean-Paul Delahaye, Mathématiques pour le Plaisir, Belin Pour la Science, Paver des pavés, 2010, pp. 58-65.
%H Sze-Man Ngai, Victor F. Sirvent, J. J. P. Veerman, and Yang Wang, <a href="http://archives.pdx.edu/ds/psu/17802">On 2-Reptiles in the Plane</a>, Portland State University, PDX Scholar, 1999.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rep-Tile.html">Rep-Tile</a>.
%H Wikimedia, <a href="https://commons.wikimedia.org/wiki/File:TameTwindragontile.png">Tame twindragon tile</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hausdorff_dimension">Hausdorff dimension</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension">List of fractals by Hausdorff dimension</a>.
%F Equals 2 * log_2((1+sqrt(78)/9)^(1/3) + (1-sqrt(78)/9)^(1/3)).
%e 1.2107605332885233950258675064294643888668202387553...
%p evalf(2*log((1+sqrt(78)/9))^(1/3)+(1-sqrt(78)/9))^(1/3))/log(2),50);
%t RealDigits[2 * Log2[(1 + Sqrt[78]/9)^(1/3) + (1 - Sqrt[78]/9)^(1/3)], 10, 100][[1]] (* _Amiram Eldar_, Sep 19 2019 *)
%o (PARI) 2 * log((1+sqrt(78)/9)^(1/3)+(1-sqrt(78)/9)^(1/3))/log(2) \\ _Michel Marcus_, Sep 21 2019
%Y Cf. A191689 (Levy dragon), A272031 (Heighway dragon and twindragon).
%K nonn,cons
%O 1,2
%A _Bernard Schott_, Sep 19 2019
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