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propensity to lay surfaces of revolution,
which is a consequence of the tubular geometry of its egg-laying apparatus, comes to the aid of both archaeologist and mathematician. There is a relatively simple formula for the area of an ellipsoid of revolution:
where
    A = the area
a = half the long axis
c = half the short axis
e = the eccentricity, which equals

    How to rotate the ellipse.
    Putting this all together, using measurements from modern ostrich eggs and intact ancient ones, led to an average figure of 570 square centimetres for one egg. This seemed quite large, but experiments with a modern egg confirmed it. The sums then indicated that at least six eggs had been deposited in Structure 07, the largest concentration of ostrich eggs in any single Predynastic deposit.
    You never know when mathematics will be useful.
    For the archaeological details, see:
    www.archaeology.org/interactive/hierakonpolis/field07/6.html

Order into Chaos
    Many puzzles, indeed most of them, lead to more serious mathematical ideas as soon as you start to ask more general
questions. There is a class of word puzzles in which you have to start with one word and turn it into a different one in such a way that only one letter is changed at each step and that every step is a valid word. 7 Both words must have the same number of letters, of course. To avoid confusion, you are not allowed to rearrange the letters. So CATS can legitimately become BATS, but you can’t go from CATS to CAST in one step. You can using more steps, though: CATS-CARS-CART-CAST.
    Here are two for you to try:
    • Turn SHIP into DOCK.
    • Turn ORDER into CHAOS.
    Even though these puzzles involve words, with all the accidents and irregularities of linguistic history, they lead to some important and intriguing mathematics. But I’ll postpone that until the Answers section, so that I can discuss these two examples there without giving anything away here.
     
    Answers on page 283

Big Numbers
    Big numbers have a definite fascination. The Ancient Egyptian hieroglyph for ‘million’ is a man with arms outstretched - often likened to a fisherman indicating the size of ‘the one that got away’, although it is often found as part of a symbolic representation of eternity, with the two hands holding staffs that represent time. In ancient times, a million was pretty big. The Hindu arithmeticians recognised much bigger numbers, and so did Archimedes in The Sand Reckoner, in which he estimates how many grains of sand there are on the Earth and demonstrates that the number is finite.

    The million that got away . . .
    In mathematics and science the usual way to represent big numbers is to use powers of 10:
    10 2 = 100 (hundred)
10 3 = 1,000 (thousand)
10 6 = 1,000,000 (million)
10 9 = 1,000,000,000 (billion)
10 12 = 1,000,000,000,000 (trillion)
    There was a time when an English billion was 10 12 , but the American usage now prevails almost universally - if only because a billion is now common in financial transactions and we need a snappy name for it. The obsolete ‘milliard’ doesn’t have the right ring. In this age of collapsing banks, trillions of pounds or dollars are starting to be headline material. Billions are passé.
    In mathematics, far bigger numbers arise. Not just for the sake of it, but because they are needed to express significant discoveries. Two relatively well-known examples are:
    10 100 = 10,000, . . . ,000 (googol)
with one hundred zeros, and
    10 googol = 1,000, . . . ,000 (googolplex)
which is 1 followed by a googol of zeros. Don’t try to write it down that way: the universe won’t last long enough and you won’t be able to get a big enough piece of paper. These two names were invented in 1938 by Milton Sirotta, the American mathematician Edward Kasner’s nine-year-old nephew, during an informal discussion of big numbers (Cabinet, page 213). The official name for googol is ten duotrigintillion in the American system and ten thousand sexdecillion in the