Figurate Number - Gnomon

Gnomon

The gnomon is the piece added to a figurate number to transform it to the next larger one.

For example, the gnomon of the square number is the odd number, of the general form 2n + 1, n = 0, 1, 2, 3, ... . The square of size 8 composed of gnomons looks like this:


8 8 8 8 8 8 8 8
8 7 7 7 7 7 7 7
8 7 6 6 6 6 6 6
8 7 6 5 5 5 5 5
8 7 6 5 4 4 4 4
8 7 6 5 4 3 3 3
8 7 6 5 4 3 2 2
8 7 6 5 4 3 2 1

To transform from the n-square (the square of size n) to the (n + 1)-square, one adjoins 2n + 1 elements: one to the end of each row (n elements), one to the end of each column (n elements), and a single one to the corner. For example, when transforming the 7-square to the 8-square, we add 15 elements; these adjunctions are the 8s in the above figure.

This gnomonic technique also provides a mathematical proof that the sum of the first n odd numbers is n2; the figure illustrates 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 82.

Read more about this topic:  Figurate Number

Famous quotes containing the word gnomon:

    Like to the Artick needle, that doth guide
    The wand’ring shade by his magnetick pow’r,
    And leaves his silken Gnomon to decide
    The question of the controverted houre;
    Francis Quarles (1592–1644)