Polygonal Number

A polygonal number is a whole number that can be represented by dots arranged in the shape of a regular polygon, one of the two-dimensional figurate numbers. Ancient Greek mathematicians studied these numbers as far back as the sixth century BC, working out the oblong, triangular and square cases first. The number ten can be laid out as a triangle, nine as a three by three square, and thirty-six happens to be both a square number and a triangular number. By convention the number one counts as the first polygonal number for every polygon shape, and later numbers in a given sequence are built by extending two sides of the shape and filling in the new points created. Different polygon shapes give different sequences, so the triangular numbers run one, three, six, ten, fifteen, twenty-one and on, while the square numbers run one, four, nine, sixteen, twenty-five, thirty-six.