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In a discussion in the newsgroup de.sci.mathematik, Klaus Nagel (see links) described a bijection P: G -> H between the grid points of two Cartesian grids G{Z X Z} and H{Z X Z} rotated against each other by Pi/4 around the only common point (0,0). This is a variation of the marriage problem asking for a matching in the infinite bipartite graph of the vertices of G U H with small distance d=|P(g)-g| for all points g in G.

Points within the grids are addressed by (i,j) in grid G and by (k,m) in grid H.

The plane is divided into horizontal strips of width cos(Pi/8) = sqrt(sqrt(2)+2)/2, with the x-axis as centerline of strip 0. Grid G is rotated by Pi/8, grid H by -Pi/8.

Assuming proper boundary conditions, there is exactly one grid point of G per grid line i=const and one grid point of grid H per grid line k=const inside each strip.

The intersections of the grid lines i=const from the rotated grid G and of lines k=const from the rotated grid H with the centerline of the strip are determined. The grid points inside the strip are paired such that the distance of the intersection points of lines i=const of grid G and of lines k=const of grid H with the strip centerline is minimized.

This bijection achieves a maximum of all mutual Euclidean distances of all pairs of cos(Pi/8)=0.9238795... (the strip width).

It is conjectured that the least possible maximum distance within pairs can be reduced to sqrt(5)*sin(Pi/8)=0.855706... (A386241), but not further, and that this can be achieved by "local repairs" of the result of the strip bijection, i.e. by reassigning the connections in groups of 4 pairs, one of which being the pair with d>0.8557... and 3 pairs in the vicinity of the violating pair, but potentially addressing points in neighbor strips. The conjecture is supported by extensive numerical results, but an announced proof by Klaus Nagel remained unpublished.

For the current sequence no repair is applied. The first repairs are required beyond i^2+j^2=40. The affected sequence terms for n>=124 are visible in the b-file of A307731.

The results of the matching are shown by enumerating the grid points of grid G according to the sequence pair A305575(n) for i and A305576(n) for j.

After finding the indices of the bijection partners (k,m) in grid H using Klaus Nagel's method, the position L where A305575(L)=k and A305576(L)=m is determined by table lookups, and the unique result is a(n)=L.

The sequence is a permutation of the natural numbers.

The strip bijection of A307110 assigns each grid point in one grid to a unique grid point in the rotated grid. The mapping therefore corresponds to a permutation of the nonnegative integers. Approximately 2/3 of the grid points are mapped in such a way that 4 points that form a unit square in the original grid also form a unit square after being mapped onto the rotated grid. We call this a stable (grid) cell under the bijection map. The method differs from that used in A307731 in that for each stable cell it is tried whether the maximum of the 4 pair distances resulting from the application of strip bijection can be reduced by a cyclic rotation of the connections. The one of the two assignments by cyclic connection change is selected that provides a smaller maximum of the 4 distances in the pairs assigned to each other. In contrast, a cyclic rotation of the connections is only carried out in the method of A307731 if the maximum of the 4 distances exceeds the upper limit of the bijection distance of sqrt(5)*sin(Pi/8)=0.855706... .

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.