# Symmetry

16 17 stAckinG And pAckinG fruit, froth, foams and other spacefillers Finding the easiest and most efficient way of stacking a pile of oranges in a given area is one of those deceptively simplesounding tasks that have farreaching mathematical ramifications. The problem is simple enough to begin with. The most obvious ways of packing spherical objects together are the triangular and square arrangements, 13 these configurations obviously relate to the regular division of the plane see Appendix. Having laid out the fruit in either of these patterns it is difficult to stack a second layer other than in the interstices formed by the first. They tend to fall, literally, into a pattern of minimum energy. There are three distinct cubic arrangements 4,5,6 , but the facecentred assembly has been shown to be the most efficientalthough a final proof came only 400 years after Kepler first proposed it. In many other circumstances, however, threeway junctions of 120 provide the most economical systems. Beecells, of course, are the classic example. They use the minimum amount of wax to create storage containers for their honey 7. It is also the case that small groups of soap bubbles with free boundaries pull themselves into this efficient angular formation, known as the Plateau border 8 . When it comes to larger clusters of soapbubbles, however, an entirely different magic angle is involved, namely 109 28 16. In any froth or elastic foam 9 the interior surfaces tend to meet at this anglewhich is exactly that formed by a line from the centre to the corner of a tetrahedron 10. Interestingly, as a solid figure, the tetrahedron by itself will not completely fill spacealthough it will in combination with the octahedron. 1. The triangular arrangement 2. Sucessive layers of this arrangement lie on different centers of the triangular grid 3. The square arrangement 4. Simple cubic packing. These various forms of spherical closepacking relate to the 3dimensional Bravais lattice structures in crystal formation see next page 3. 1. 2. 4. 5. 6. 7. 10. 9. 8.