Gnomon (figure)

For history and other uses, see gnomon.

A gnomon

In geometry, a gnomon is a plane figure formed by removing a similar parallelogram from a corner of a larger parallelogram. More generically, the term gnomon denotes the form that is to be added to a figure to produce a larger figure of the same shape.

Building figurative numbers

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 2n + 1, n = 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.

Note that this gnomonic technique also provides a 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.

See also

• Golden gnomon
• Pythagorean
• Pythagorean triple
• Figurate number