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A367147 Index of matching grid point in the bijection between two infinite triangular grids with one grid rotated by Pi/6 around the common point (0,0), using an enumeration of the grid points by A307014 and A307016.

%I A367147 #17 Jan 06 2024 18:55:04
%S A367147 0,1,2,3,4,5,6,12,14,15,9,17,18,29,7,8,23,10,11,30,13,20,21,22,33,24,
%T A367147 16,26,27,28,36,42,19,38,39,25,41,31,32,57,34,35,60,54,43,44,45,46,47,
%U A367147 48,49,50,51,52,53,72,37,63,66,40,69,55,73,74,56,76,77,58,79,80,59
%N A367147 Index of matching grid point in the bijection between two infinite triangular grids with one grid rotated by Pi/6 around the common point (0,0), using an enumeration of the grid points by A307014 and A307016.
%C A367147 The methods used to achieve a distance-limited bijection of the points of two square grids (see A307110) are applied here to triangular grids. The two grids, which are rotated by 30 degrees = Pi/6 from each other, are assigned the colors red and blue to distinguish them, which are also used in the illustrations. The blue triangular grid is turned clockwise by 15 degrees = Pi/12, all points are lined up on parallel lines with inclination Pi/12 towards the vertical axis. These are called blue lines. The vertical distance between adjacent points is cos(Pi/12). The same is done for the red grid with a CCW rotation of Pi/12. The whole plane is divided into stripes with a width of cos(Pi/12) ~= 0.9659. Every blue line and every red line contains exactly one grid point of its color in each stripe. The blue and red lines alternately intersect the horizontal centerline of a stripe. The distance between two intersections of the same color is d = sqrt(3)/(2*cos(Pi/12)). The bijection maps the section of a blue line in a stripe to the section of the unique red line, that intersects the centerline less than d/2 away. The grid points on these two line sections are the partners of the tile bijection.
%C A367147 While the method described only finds a minimum of the maximum distance of approximately 0.9659 by assigning the bijection partners using tiles, applying the Hopcroft-Karp algorithm to the bipartite graph corresponding to a sufficiently large section of the two infinite grids achieves significantly lower maximum distances. We conjecture that an upper bound for the maximum distance is sqrt(2)/2~=0.7071. See the corresponding link.
%C A367147 A method that reduces the maximal occurring bijection distance to its conjectured minimum, and only requires local rearrangements, as described for the square grids in A307731, is currently not known in the present case of the triangular grids.
%H A367147 Klaus Nagel, <a href="/A367147/a367147.png">Stripes and tiles used for mapping</a>.
%H A367147 Hugo Pfoertner, <a href="/A367147/a367147_1.png">Numbering of points in red and blue grid</a>.
%H A367147 Hugo Pfoertner, <a href="https://www.randomwalk.de/sequences/MinimumDistance.pdf">Notes on the minimum achievable bijection distance</a>, Nov 2023.
%H A367147 Hugo Pfoertner, <a href="/A367147/a367147.gp.txt">PARI program</a>.
%H A367147 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hopcroft%E2%80%93Karp_algorithm">Hopcroft-Karp algorithm</a>
%e A367147    n  A307014(n)        Bijection partner
%e A367147    |  |  A307016(n)     in rotated grid
%e A367147    |  |  |                          rotated by Pi/6
%e A367147    |  |  |   x    y     i  j  a(n)   u      v   Distance([x,y],[u,v])
%e A367147    0  0  0  0.0  0.0    0  0   0    0.0    0.0  0.0
%e A367147    1  1  0  1.0  0.0    1  0   1    0.866  0.5  0.51764
%e A367147    2  0  1  0.5  0.866  0  1   2    0.0    1.0  0.51764
%e A367147    3 -1  1 -0.5  0.866 -1  1   3   -0.866  0.5  0.51764
%e A367147    4 -1  0 -1.0  0.0   -1  0   4   -0.866 -0.5  0.51764
%e A367147    5  0 -1 -0.5 -0.866  0 -1   5    0.0   -1.0  0.51764
%e A367147    6  1 -1  0.5 -0.866  1 -1   6    0.866 -0.5  0.51764
%e A367147    7  1  1  1.5  0.866  2 -1  12    1.732  0.0  0.89658
%e A367147    8 -1  2  0.0  1.732  0  2  14    0.0    2.0  0.26795
%e A367147    9 -2  1 -1.5  0.866 -2  2  15   -1.732  1.0  0.26795
%e A367147   10 -1 -1 -1.5 -0.866 -2  1   9   -1.732  0.0  0.89658
%e A367147   11  1 -2  0.0 -1.732  0 -2  17    0.0   -2.0  0.26795
%e A367147   12  2 -1  1.5 -0.866  2 -2  18    1.732 -1.0  0.26795
%e A367147   13  2  0  2.0  0.0    3 -2  29    2.598 -0.5  0.77955
%e A367147   14  0  2  1.0  1.732  1  1   7    0.866  1.5  0.26795
%e A367147   15 -2  2 -1.0  1.732 -1  2   8   -0.866  1.5  0.26795
%o A367147 (PARI) \\ See linked file; function call to output data:
%o A367147 a367147(70)
%Y A367147 Cf. A307014, A307016, A307110, A307731, A367148.
%K A367147 nonn
%O A367147 0,3
%A A367147 _Klaus Nagel_ and _Hugo Pfoertner_, Nov 06 2023