Gnomon (figure)

In geometry, a gnomon is a plane figure formed by removing a similar parallelogram from a corner of a larger parallelogram.

In the Pythagorean theorem

Figurate numbers were a concern of Pythagorean geometry, since Pythagoras is credited with initiating them, and the notion that these numbers are generated from a gnomon or basic unit. The gnomon is the piece which needs to be added to a figurate number to transform it to the next bigger one.

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

nowrap|8 8 8 8 8 8 8 8
nowrap|8 7 7 7 7 7 7 7
nowrap|8 7 6 6 6 6 6 6
nowrap|8 7 6 5 5 5 5 5
nowrap|8 7 6 5 4 4 4 4
nowrap|8 7 6 5 4 3 3 3
nowrap|8 7 6 5 4 3 2 2
nowrap|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 2"n" + 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.

Note that this gnomonic technique also provides a proof that the sum of the first "n" odd numbers is "n"²; the figure illustrates 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 8².

ee also

*Pythagorean
*Pythagorean triple
*Figurate number