Saturday, July 02, 2005

What on earth is a ‘gnomon’?

A bit of magic, really.

The Oxford English Dictionary gives several definitions of the gnomon. Here are the two relevant ones:
  • An odd number
  • Something shaped like a carpenter's square; an L-shaped bar, etc.

  • Lets look at the odd numbers first:
    1, 3, 5, 7, 9, 11, 13...

    If we sum the first 1 gnomon, we get 1.
    If we sum the first 2 gnomons, we get 1 + 3 = 4.
    If we sum the first 3 gnomons, we get 1 + 3 + 5 = 9.
    If we sum the first 4 gnomons, we get 1 + 3 + 5 + 7= 16.

    To make life easier, we’ll arrange the sum of gnomons in a series, like so:
    1, 4, 9, 16, 25, 36, 49...

    Look! The sum of the first n gnomons correspond to the square of n. The sum of the first 3 gnomons is 9, which is 3 x 3. And the sum of the first 4 gnomons is 16, which is 4 x 4.

    Magical isn’t it?


    Magic, explained geometrically


    A geometric representation of the sum of gnomons.



    Start with the sum of the first 2 gnomons, 1 + 3 = 4. From the figure, 1 + 3 is represented as the red square and 3 grey squares. Collective, they form a larger 2-by-2 square.

    The sum of the first 3 gnomons is 1 + 3 + 5 = 9. It is represented by the red square, 3 grey squares and 5 purple squares. Collectively, they form a larger 3-by-3 square.



    This introduction to number theory brought to you by Tan Yee Wei.

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