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The most iconic soccer ball is probably the telstar. It is made of 12 pentagons and 20 hexagons, with no two adjacent pentagons (this can be taken as a definition). What if we allow adjacent pentagons? Then we can get different balls. These are specific fullerenes or buckyballs, and there are 1812 different ones. The picture below shows three of them that I assembled with real parts of soccer balls (you can even try to play with).
Continuous transformation between binary disc packings with a triangulated contact graph, designed to break as few contacts as possible. You can spot (can you?) 7 of the 9 binary disc packings depicted here. The way disc packings are deformed is explained in Section 2 of this paper.
The steps are laser-cut from MDF and glued together like a dome (there are supporting arches underneath). One filling is uniformly randomly drawn for each size (10, 20, 30, 40 and 70 small cubes on each side), the other is the complementary filling (which allows them to be combined to make a cube that is easy to transport).
Everyone knows the best way to arrange identical coins on a table, without overlapping, so as to minimize the area left free between the coins. This is the hexagonal compact packing: the coins are centred on a triangular grid. What if there are two different coins? More? Some coin size ratios are particular: they allow to have only "triangular" spaces between the coins: this is called compact or triangulated packing. There are only 9 ratios of two coins allowing this. There are also 164 triplets of sizes allowing compact packing (full list, exact radii and applet to play with). For each of the 9 ratios which allow a binary triangulated packing, the density has been shown to be maximized by such a packing. This also holds for 16 of the 164 cases of ternary triangulated packings. These are the packings depicted above.
Packings which maximize the density, for each ratio which allow a binary triangulated packing. | Packings which maximize the density, for 16 of the 164 ratios which allow a ternsary triangulated packing. |
These patterns also seem to appear in this simulation of the assembly of nanoparticles on a diethylene glycol " ice rink " (common work with chemists of the LPCNO).
Consider the tilings of a grid by two squares, respectively of side 1 and c>1 (for c=2 this is related to the king problem, and close to the fibonacci 2D subshift problem). How many are them? How to efficiently random sample one of them? For which value of c does a typical tiling contains the more small squares (i.e., which of the below pictures is the whitest one)? What about random tilings by squares of any size?
Sides 1 and 2 | Sides 1 and 3 | Sides 1 and 4 | Sides 1 and 5 | Sides 1 and 6 | Integer sides |
Codimension 1 | Codimension 2 | Codimension 3 |
Typical tiling | Maximal tiling | Concentric rings |
Math-Info 2010 | Allouche's 60th birthday | Tilings and Tesselations | TransTile |