piles of oranges

Everyone is familiar with the stacking of oranges in a pyramid on Market stalls and in green grocery shops.
The simplicity and familiarity of this hides an interesting question about stacking theory.
In two dimensions, if one wants to place the maximum number of circles e.g. coins of the same denomination around a central circle or coin, the maximum number is always six, the kissing number.. That is a hexagonal lattice.
Oranges, when stacked in a pyramid have a kissing number of 12, which is the three dimensional extension of this system. Each orange enclosed inside this stack is in contact with 12 neighbours. The ranking of kissing numbers for spaces and dimensions is 1, 2, 3, 8 and 24.
All to do with sphere packing and binary codes.
And what has this to do with nutrition. Well oranges are said to be healthy foods to eat. So what better object to study sphere packing and binary codes. Which all began with Kepler’s book in 1611 on why a snow drop has a 6 membered symmetry.
Stewart, (2003) Nature 424, 895-6

Martin Eastwood
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