Colossal Gardner, ch. 10


Packing Spheres. If you have a crate and a bunch of balls all of the same size, how many balls will be needed to fill up the crate? The answer can depend on whether each layer is centered over the layer below it, or over the gaps between the balls of the below layer, or if you use random packing. This raises the questions of what’s the densest packing you can get, and the opposite, for what’s the least dense rigid packing possible?

As an experiment, you can take ping pong balls and coat them with rubber cement. Wait until the cement dries and you can stick them together to make both square and triangular pyramids. Flat 2D triangular pyramids are characterized by triangular numbers (each new row of the three-sided pyramid is one larger than the row above it: 1, 2, 3, 4, 5…). To add a new “row” to a square, you add the balls of two of the sides, plus one for the corner (the “row lengths” are 1, 3, 5, 7, 9…) For a triangular pyramid of 5 rows, the total number of ping pong balls is 1+2+3+4+5 = 15, and for a 5-row square the total is 1+3+5+7+9 = 25.

Additionally, a square can be said to be made up of 2 right triangles. If you have a 7×7 ping pong ball square, it can be divided into a 7-row and a 6-row triangle. The 7-row triangle has 1+2+3+4+5+6+7 = 28 balls, and the 6-row has 1+2+3+4+5+6 = 21 balls. 21+28 = 7×7 = 49. This links square and triangular numbers.

To return to the above questions, each ball in the crate is going to touch 12 other surrounding balls, giving a ratio of the volume of the spheres to the total space of pi/sqrt(18) approx = 75%. In 1958, H.S.M Coxeter suggested that the most dense packing may not yet have been found, which as of the time of Gardner’s book hadn’t yet been solved. The loosest packing was found in 1933 by Heesch and Laves to have a density of only 0.0555 (each sphere only touching 4 others).

Puzzles:
Someone wants to make a courthouse memorial using cannon balls. They lay the balls out in a square first, then pile the balls into a square pyramid with no balls left over. What’s the smallest number of cannon balls used?

A grocer stacks oranges into two tetrahedral pyramids (base has 3 sides). By combining the two into one big tetrahedral pyramid, what’s the smallest number of oranges he needs if the two smaller pyramids are the same sizes? If they are different sizes?

 

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