Editing Figurate number (section)

== Gnomon ==
{{See also|Gnomon (figure)#Building figurate numbers}}
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 {{math|2''n'' + 1}}, {{math|''n'' {{=}} 0, 1, 2, 3, ...}}.  The square of size 8 composed of gnomons looks like this:

:<math>\begin{matrix}1&2&3&4&5&6&7&8\\2&2&3&4&5&6&7&8\\3&3&3&4&5&6&7&8\\4&4&4&4&5&6&7&8\\5&5&5&5&5&6&7&8\\6&6&6&6&6&6&7&8\\7&7&7&7&7&7&7&8\\8&8&8&8&8&8&8&8\end{matrix}</math>

To transform from the ''{{mvar|n}}-square'' (the square of size {{mvar|n}}) to the {{math|(''n'' + 1)}}-square, one adjoins {{math|2''n'' + 1}} elements: one to the end of each row ({{mvar|n}} elements), one to the end of each column ({{mvar|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 {{mvar|n}} odd numbers is {{math|''n''<sup>2</sup>}}; the figure illustrates {{nowrap|1 + 3 + 5 + 7 + 9 + 11 + 13 + 15}} = 64 = 8<sup>2</sup>.

There is a similar '''gnomon''' with [[centered hexagonal number]]s adding up to make cubes of each integer number.