![]() Which is the larger area? There's not much in it, but the area (length x width) of the box on the right is just that little bit smaller than that on the left. They both contain the same number of circles. In the example above, the box packing the circles on the left is taller than that on the right, but the one on the right is a little bit wider. However, as the circles get smaller (or box gets bigger!), even with these extra voids, the closer packing of the rest of the circles results in a better fit. When the size of the circles to be packed is large (compared to the box), these additional voids are significantly large enough to over-rule the higher packing density. In hexagonal packing, every other row has to miss out of part of a circle from either end. Square packing does not suffer from edge-effects. Imagine we needed to pack tubes of cylindrical candy into a shipping carton (or pack cans of soda into a tray). Hexagonal packing is a much more efficient user of space if we had an infinite plane to pack, but reality adds constraints. This arrangement is called hexagonal packing with an efficiency of approx 90.69% By staggering every other row, and nesting them tightly together we create an arrangement that packs more efficiently. It isn't, however, the only way to tightly tessellate circles in a plane. Square packing achieves a packing ration of π/4 (approx 78.54%), which is pretty good. And so on …Īn other interesting consequence of this is that the two shaded shapes here (the quarter circle and the circle with half the radius) have the same area. Whatever the ratio of the square to the circle, the (2x2) is made entirely of building blocks that are the same ratio. Each of the more complex versions of the shape are self-similar to the fundamental shape. When there are n circles on an edge, there are a total of n 2 circles, each of area π(1/2n) 2. I think you can see it does not matter how many square-packed tessellated circles we put into the unit square, the answer will always be π/4. The Area of each square is, therefore, one unit squared. Let's define the square to have side length of one unit. The answer is, of course, that all three are the same. It's not really a trick question, it's an interesting result. Some people see it right away, others work it out on paper. In which of these three pictures do the circles cover the highest percentage of the square?
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