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In addition, the "cook's cup" above is not the same as a "coffee cup", which can vary anywhere from 100 to 200 mL (3.5 to 7.0 imp fl oz; 3.4 to 6.8 US fl oz), or even smaller for espresso. In Australia, since 1970, metric utensil units have been standardized by law, and imperial measures no longer have legal status.
For most of these numbers (with the exceptions only of 5 and 10), the packing is the natural one with axis-aligned squares, and is ⌈ ⌉, where ⌈ ⌉ is the ceiling (round up) function. [ 2 ] [ 3 ] The figure shows the optimal packings for 5 and 10 squares, the two smallest numbers of squares for which the optimal packing involves tilted ...
A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. In simplest terms, this is the ratio of the volume of bodies in a space to the volume of the space itself.
Random close packing (RCP) of spheres is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when they are packed randomly. For example, when a solid container is filled with grain, shaking the container will reduce the volume taken up by the objects, thus allowing more grain to be added to the container.
In all of these arrangements each sphere touches 12 neighboring spheres, [2] and the average density is π 3 2 ≈ 0.74048. {\displaystyle {\frac {\pi }{3{\sqrt {2}}}}\approx 0.74048.} In 1611, Johannes Kepler conjectured that this is the maximum possible density amongst both regular and irregular arrangements—this became known as the Kepler ...
People are given n unit squares and have to pack them into the smallest possible container, where the container type varies: Packing squares in a square: Optimal solutions have been proven for n from 1-10, 14-16, 22-25, 33-36, 62-64, 79-81, 98-100, and any square integer. The wasted space is asymptotically O(a 3/5).
Of these, solutions for n = 2, 3, 4, 7, 19, and 37 achieve a packing density greater than any smaller number > 1. (Higher density records all have rattles.) [10]
For all these radius ratios a compact packing is known that achieves the maximum possible packing fraction (above that of uniformly-sized discs) for mixtures of discs with that radius ratio. [9] All nine have ratio-specific packings denser than the uniform hexagonal packing, as do some radius ratios without compact packings.