Recursive Subdivisions of the Plane Yielding Nearly Hexagonal Regions

Hexagonal regions are optimal for subdividing the plane in the sense that the regions are as close as possible to the disk shape while still providing a tessellation of the plane. It is however not possible to do recursive subdivisions, i.e. to divide a hexagon into smaller hexagons. In this paper, hexagon-like fractal regions will be presented. It is possible to decompose a hexagon-like fractal region into a number of smaller but equally shaped hexagon-like regions. Possible applications of hexagon-like fractal regions are cell organization in cell-phone systems and image coding algorithms, including algorithms with position variant resolution. 1 Hexagon-like Fractal Regions As a starter, let us introduce hexagon-like fractal regions which are composed of 7 identical but smaller regions, also called Gosper regions[1]. A bottom-up approach for generating Gosper regions is to recursively merge seven adjacent regions, starting from the grouping of hexagons shown in Figure 1. After each merging operation the regions still lie on a hexagonal grid, which is tilted by the same angle in each iteration. It is however possible to choose if the tilt is going to be in the positive or negative direction, so by choosing alternate positive and negative tilts, the coordinates of the regions will be much easier to handle. After a few iterations, the regions obtain the shape shown in Figure 2. In the general case of hexagon-like fractal regions, which has to our knowledge not been presented earlier in literature, the base tiling consists of S hexagons, see Figure 3. We can relate S to k, where k is the number of edge hexagons along one of the six sides, by S = 3(k2+ k)+1: (1) The ratio of areas of two nearly hexagonal regions at adjacent regions scale with S, while the lengths scale with R =pS. Hexagon-like fractal regions tile the plane, which can be proved e.g. by induction over k. Because of the recurFigure 1: A base tiling for subsampling a factor of seven, given an image of hexagonal representation. sive nature of the region generation, it is enough to prove that the base tiling (like the one shown in Figure 1) tile the plane. A Hexagon-like region having k = 2 (and thus S = 19) is shown in figure 4. From generating an Iterated Functions System[2], we can compute the fractal dimension of the border of the regions. The dimension D is given by D = 2 log(2k+1) log(3(k2+ k)+1) : (2) It can be noted that D approaches one as k increases, which for hexagon-like fractal regions means that the regions approach hexagons with increasing k. In Figure 6 an image that has been divided into nearhexagonal regions is shown. In the figure it is also shown how a region can be subdivided into smaller regions.