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Sierpinski's Gasket and Dihedral SymmetryThe Sierpinski gasket could be obtained in a variety of ways. The applet below implements a finite automaton - a step by step procedure - on a square grid which initially contains a 1 in the upper left corner and zeros everywhere else. On every step, the content
The change occurs simultaneously on the whole grid. (1) is similar to the creation of the Pascal triangle that is produced by replacing
In the applet below, (1) and (2) correspond to the checkboxes "Add" and "Replace", respectively. Observe that, taken modulo 2 on a grid of size 2n, (1) and (2) lead to them result. However, on grids of size which is not a power of 2, the pictures are quite different. The most curious property of (1), in my view, is that, at every step, the picture has the dihedral symmetry of a regular triangle. I.e., assuming that the trangle has been sheared into an equilateral one, the pattern has 6 different symmetries: identical, 2 rotations around its center, and the reflections in each of its three, say, medians. In the applet, both (1) and (2) are identified is being 2-pronged. A 3-pronged analogy proceeds according to
and, respectively,
where the symbol ":=" relates the values of the grid (the left hand side) to those on the previous step (the right hand side.) (1') results in a square pattern that also features the dihedral symmetry: identical, three rotations, 2 reflections in the midlines, and 2 reflections in the diagonals. Given an apparent asymmetry of the definitions (1) and (1'), the presence of the dihedral symmetry on every step comes as a surprise. Does this phenomenon have a simple explanation?
Let C(n, m) denotes the binomial coefficient - "n choose m." It's not hard to surmise the formulas for a generic term of the tables resulting form (1) and (1'). Let n and m be the row and column indices starting from 0 for the table obtained after N steps. Denote the table entries as
For other values of n and m, we have
It is quite obvious that (3') has the dihedral symmetry, and just a little less so for (3). 
Copyright © 1996-2008 Alexander Bogomolny |
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