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

%I A327620 #63 Feb 16 2025 08:33:58
%S A327620 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 A327620 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 A327620 9,9,6,7,7,2,4,6,1,5,3,6,7,3,4,6,6,6,5
%N A327620 Decimal expansion of the Hausdorff dimension of the boundary of the tame twin-dragon curve.
%C A327620 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 A327620     . The Levy dragon: A191689
%C A327620     . The Heighway dragon: A272031
%C A327620     . The twin-dragon: A272031
%C A327620     . The tame twin-dragon: this sequence.
%C A327620 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 A327620 Jean-Paul Delahaye, Mathématiques pour le Plaisir, Belin Pour la Science, Paver des pavés, 2010, pp. 58-65.
%H A327620 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 A327620 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rep-Tile.html">Rep-Tile</a>.
%H A327620 Wikimedia, <a href="https://commons.wikimedia.org/wiki/File:TameTwindragontile.png">Tame twindragon tile</a>.
%H A327620 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hausdorff_dimension">Hausdorff dimension</a>.
%H A327620 Wikipedia, <a href="https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension">List of fractals by Hausdorff dimension</a>.
%F A327620 Equals 2 * log_2((1+sqrt(78)/9)^(1/3) + (1-sqrt(78)/9)^(1/3)).
%e A327620 1.2107605332885233950258675064294643888668202387553...
%p A327620 evalf(2*log((1+sqrt(78)/9))^(1/3)+(1-sqrt(78)/9))^(1/3))/log(2),50);
%t A327620 RealDigits[2 * Log2[(1 + Sqrt[78]/9)^(1/3) + (1 - Sqrt[78]/9)^(1/3)], 10, 100][[1]] (* _Amiram Eldar_, Sep 19 2019 *)
%o A327620 (PARI) 2 * log((1+sqrt(78)/9)^(1/3)+(1-sqrt(78)/9)^(1/3))/log(2) \\ _Michel Marcus_, Sep 21 2019
%Y A327620 Cf. A191689 (Levy dragon), A272031 (Heighway dragon and twindragon).
%K A327620 nonn,cons
%O A327620 1,2
%A A327620 _Bernard Schott_, Sep 19 2019