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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.

Similar to A283307, but with secondary sorting by polar angle.

Similar to A283308, but with secondary sorting by polar angle.

From Peter Munn, Jul 22 2021: (Start)

The points of the spiral are equally the points of a hexagonal lattice, the points of an isometric (triangular) grid and the center points of the cells of a honeycomb (regular hexagonal tiling or grid). The coordinate system can be described using 3 axes that pass through spiral point 0 and one of points 1, 2 or 3. Along each axis, one of the coordinates is 0.

a(n) is the signed distance from spiral point n to the axis that passes through point 2. The distance is measured along either of the lines through point n that are parallel to one of the other 2 axes and the sign is such that point 1 has positive distance.

This coordinate can be paired with either of the other coordinates to form oblique coordinates as described in A307012. Alternatively, all 3 coordinates can be used together, symmetrically, as described in A345978.

There is a negated variant of the 3rd coordinate, which is the conventional sense of this coordinate for specifying (with the 2nd coordinate) the Eisenstein integers that can be the points of the spiral when it is embedded in the complex plane. See A307013.

(End)

Cartesian coordinates (x,y) of the grid points are converted to barycentric coordinates (i,j,k) by i = x - y/sqrt(3), j = 2*y/sqrt(3), k = x + y/sqrt(3). The sequence gives i. j is given in A307016, k is given in A307017.

The sorting by polar angle affects the grid points in the shells of size A035019, starting at indices given by A038590.

Cartesian coordinates (x,y) of the grid points are converted to barycentric coordinates (i,j,k) by i = x - y/sqrt(3), j = 2*y/sqrt(3), k = x + y/sqrt(3). The sequence gives k. i is given in A307014, j is given in A307016.