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In a private communication in 2009, Klaus Nagel described a conjectured bijection between the grid points of two quadratic lattices G and H such that the second lattice H has an offset of 1/2 in both coordinate directions and is rotated by an angle of Pi/6 against the first lattice G. If all straight-line connections between points of G and H not exceeding the Euclidean distance d=sqrt(1/2) are drawn (see Nagel link), several types of connected components in the infinite bipartite graph of the vertices of G U H containing equal numbers of grid points from G and H arise.

One of them is an "octopus" with 4 contiguous arms identical up to symmetry. To achieve a bijection between the points of G and H, the four arms must not be finite. Otherwise the central grid point of H, which is equidistant to its 4 closest neighbors in G, but can only be used once, would lead to a violation of the condition of equal number of points from G and H in any connected component.

The coordinates (i,j) of the grid points of lattice G in one of the arms are provided in this sequence (i) and in A307447 (j). Instead of applying the offset of (1/2,1/2) between lattices G and H, all distances are scaled by a factor of 2. Only grid points with odd coordinates are used from G and grid points with even coordinates from H.

The construction of the selected one arm proceeds as follows: Start with the edge (1,1)->(3,1) in G. Determine the point with even coordinates in H (rotated) that is closest to the midpoint (2,1) of the considered edge of G. After rotation by Pi/6 the coordinates of point (2,0) in H become (1.732051,1.000). A grid point from H after rotation is only accepted if its distances from both end points of the edge in G are less than sqrt(2). The distance from (1,1) is 0.732051 and from (3,1) 1.267949, both < sqrt(2).

To determine the continuation of the path, excluding U-turns, the rotated grid points of H closest to the midpoints of the 3 candidate edges starting at the end point of the edge selected in the previous step are subjected to the tests of not exceeding distance sqrt(2) from both ends of the respective candidate edge of G. If more than one of the 3 candidate edges passes the checks, the one with least distance of its midpoint to the closest grid point in H (rotated) is selected.

A visualization of the selection process is provided in the plot "Construction of A307446 and A307447", see link. The selected path on the grid points of G is marked in red. Grid points of H (rotated) are marked in blue. The checked candidate edges are in cyan. A distance of the closest point in H from the start point of the candidate edge exceeding sqrt(2) is marked in orange, and a distance from the end point exceeding sqrt(2) is marked in brown. Vectors in purple are used for the distance between midpoints of edges in G to the closest point in H (rotated).

The infinite length of the 4 arms has yet to be proved.

For a description of the construction and visualizations see A307446.

The repeated application of the bijection function Q described in A367147, which maps a pair of triangular coordinates [i,j] to an image point [m,n], returns to the starting point after a number of steps dependent on the starting point. One mapping step leads to a location that approximately corresponds to a rotation of Pi/6, so that often, but not always, the lengths of the orbits created are multiples of 12. The situation is very similar to that described in the comment to A363760 for the analogous process applied to the square grid. As the lengths of the cycles increase, remarkable self-similar structures emerge; see the visualization of a cycle with a length L > 6*10^8.

Apparently, a(n) == 0 (mod 4) for n > 1. For cycles, whose lengths are multiples of 8, the visited points form 8 separated islands.

Larger terms are 4293336, 4462104, 5787768, 11050488, 28333080, 38414184, 72397248.

Apparently, a(n) == 0 (mod 4) for n > 1. For cycles, whose lengths are multiples of 8, the visited points form 8 separated islands.

Larger terms are 1660752, 4293336, 4462104, 5787768, 6647916, 11050488, 28333080, 38414184, 45366204, 184427544.

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.

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.

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.

Upper bound of the wobbling distance S of two rotated square lattices. See A307110 and A307731 for the special case of rotation angle Pi/4. According to Jan Fricke (1999), the angle Pi/4 is the most unfavorable case, i.e., smaller bounds can be found for all other angles.