SUDOKU Colorings of the Hexagonal Bipyramid Fractal

Abstract

The hexagonal bipyramid fractal is a fractal in three dimensional space, which has fractal dimension two and which has six square projections. We consider its 2nd level approximation model, which is composed of 81 hexagonal bipyramid pieces. When this object is looked at from each of the 12 directions with square appearances, the pieces form a 9×9 grid of squares which is just the grid of the SUDOKU puzzle. In this paper, we consider colorings of the 81 pieces with 9 colors so that it has a SUDOKU solution pattern in each of the 12 appearances, that is, each row, each column, and each of the nine 3×3 blocks contains all the 9 colors in each of the 12 appearances. We show that there are 140 solutions modulo change of colors, and, if we identify isomorphic ones, we have 30 solutions. We also show that SUDOKU coloring solutions exist for every level 2n approximation models (n ≥ 1).