Maxima and minima

The problem, which we met in Section 1.2, of determining the rectangle of smallest perimeter among all those with area 1 is but one of the problems of maxima and minima in geometry. Problems of this kind were studied by Greek geometers before the birth of Christ. Of course, it is uncertain who were the first people to pose problems involving maxima and minima, but many arise quite naturally and might have, and might yet occur to people in a primitive culture. For example, what is there about the shape of a circular cylinder that causes many flower stems, tree trunks, and many other natural objects to take its shape, why are small water droplets and bubbles that float in air approximately spherical, and why does a herd of reindeer form a circle if attacked by wolves? Admittedly, these problems involve mathematics only indirectly, but they are capable of stimulating mathematical thought. There are problems which are more directly mathematical. For example, what shape should a plot of ground be so that a given length of fence will enclose the greatest area, and what are the dimensions of a cylindrical container so that it will contain the greatest volume for a given surface area? Can you think of other examples? The Greeks were mostly interested in natural phenomena such as the hexagonal arrangement of cells in honeycomb, but they also had practical problems.