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Packing
Tiling is the filling of a 2-dimensional space without gaps or overlaps. A similar phenomenon in 2 or 3 dimensions is called packing. Packing is filling as much as possible of a 2-dimensional closed planar figure or a 3-dimensional bounded space with a single 2- or 3-dimensional object. That is, how can as many copies of the object as possible be packed inside of the bounding object? For example, how many circles of radius 1 will fit inside a square of side length 2? This packing is trivial, since it is clear that only one circle will fit in such a square. But now, what is the smallest side length of a square that will
fit 3 circles of radius 1? With quite some calculation, the answer
can be found to be 2+sqrt(1/2)+sqrt(3/2).
Now you try! Print and cut out this template, and try to find out how to pack five and six circles into a square. Then, click here for the answers!
Thought that was hard? Try packing circles on a sphere! Even more amazing than the math involved, these intellectual questions have real applications. Suppose that you are a canner, you work for Campbell's soups, and have to make deliveries of millions of cans of soup. Wouldn't you like to know just how may cans you can fit in each box? No point in having wasted space. Or if you need to ship by the case, how big should your custom made boxes be? If you know the radius of the cans, and how many you need to fit in a box, the question dissolves into the one we were working with originally: how large must the square be to fit x circles with radius y? Although in this case, we can use a rectangle instead of a square. Additionally, perhaps you work for the producer of those giant rubber bouncy balls, and he needs to ship 2,000 to KayBee Toy Stores. How many spheres can you pack in a rectangular prism?
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