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 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 packing. There are only 9 ratios of two coins allowing this:
In each of the 9 cases above, packing has been shown to maximize density. There are also 164 triplets of sizes allowing compact packing (full list, exact radii and applet to play with). The one below, for example, has been proven to maximize density for these coin sizes.
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|