polygonal number (thing)

The concept of a polygonal number is a generalisation of triangular numbers, square numbers, etc..

To conceptually count a polygonal number: First, start with a dot. This will be a vertex of the polygons. Construct an appropriate polygon using the dot at one corner (for a triangular number, the appropriate polygon is a triangle, etc.). The length of the sides of the triangle should be unity (one centimeter, one inch, whatever - just choose one of something). Draw a dot on each corner of the shape. Now draw one twice as big, using the same vertex, drawing two of the sides along the same sides as the smaller shape. Draw one dot per unit length along each side, i.e. one dot in each corner and one dot in the middle of the side (and the dots in the middle of the two sides coming out of the origin will overlap dots previously drawn). Now draw a polygon three times as big as the first, using the same vertex and two of the same sides again, with dots on the corners and 2 dots in the middle each side (and the 2 dots on the sides from the origin will overlap previous dots). After you have drawn as many polygons as you want, count all the dots.

For example, here are drawings of the 4th triangular, square, and pentagonal numbers:

*--*--*--*
| /  /  /
|/  /  /
*  *  *
| /  /
|/  /
*  *
| /
|/
*

*--*--*--*
|  |  |  |
|  |  |  |
*--*  *  *
|     |  |
|     |  |
*--*--*  *
|        |
|        |
*--*--*--*

         *
        / \
       /   \
      *     *
     / \   / \
    /   *-*   \
   *           *
  / \         / \
 /   *       *   \
*     \     /     *
 \     *-*-*     /
  *             *
   \           /
    *         *
     \       /
      *-*-*-*

The rth n-gonal number is given by the expression:
pⁿ_r = ((((n* (r - 1)) - (2 * (r - 2))) * r) / 2)