Elsewhere, we label as "squares" such numbers as 4, 9, 16, 25, etc., because Pythagoras of Samos (circa 580-496 B.C.) taught us to build these numbers as squares of dots or pebbles. We label as "cubes" such numbers 8, 27, 64, 125, etc., because Pythagoras taught us to imagine them thus in 3D. Pythagoras also spoke of "triangular numbers", "pentagonal numbers", "hexagonal numbers", etc.

Pythagoras taught us the concept of the gnomon as the building-block of number patterns. (The word also refers to the device on the sundial which casts "the shadows of the hours". The word "gnomon" is also related to words for "knowing". Thus "gnostic" means "secret knowledge".)

Children can emulate Pythagorean or Figurate Geometry by using bottle caps (or pebbles or pegs in a pegboard or tiles on the floor or wall). One BONUS is that children can DISCOVER THE COMMUTATIVE AND ASSOCIATIVE LAWS FOR ADDITION AND MULTIPLICATION -- usually just dictated to them -- by ROWS and TABLES of dots. I demonstrate with asterisks for dots or bottle caps or pebbles.

This demonstrates the ADDITIVE COMMUTATIVITY of 2 + 3 = 3 + 2 = 5:

*  * | *  *  *  -->  *  *  * | *  *  -->  *  *  *  *  * 

This demonstrates the MULTIPLICATIVE COMMUTATIVITY of 2 X 3 = 3 X 2 = 6:

*  *  -->  *  *  *  -->  *  *  *  *  *  *
*  *       *  *  *
*  *                                     

Pythagoras taught us that the odd number is the gnomon of the square. That is, squares are generated by summing odd numbers. Thus, 1 + 3 = 4 = 2 x 2, the "square of 2". 1 + 3 + 5 = 9 = 3 x 3, "square of 3". 1 + 3 + 5 + 7 = 16 = 4 x 4. 1 + 3 + 5 + 7 + 9 = 25 = 5 x 5. Etc.

*   |  *  *   |  *  *  *  | *  *  *  *  | *  *  *  *  *  |
    |  *  *   |  *  *  *  | *  *  *  *  | *  *  *  *  *  |
    |         |  *  *  *  | *  *  *  *  | *  *  *  *  *  |
    |         |           | *  *  *  *  | *  *  *  *  *  |
    |         |           |             | *  *  *  *  *  |

The triangular numbers are 1, 2, 3, 4, 5, etc., -- the "counting numbers" -- because building rows of one bottle cap (or dot), a row of two, a row of three etc., builds larger and larger triangles.

    *    |     *       |     *     |     *     |     *     |     *     |
         |    * *      |    * *    |    * *    |    * *    |    * *    |
         |             |   * * *   |   * * *   |   * * *   |   * * *   |
         |             |           |  * * * *  |  * * * *  |  * * * *  |
         |             |           |           | * * * * * | * * * * * |
         |             |           |           |           |* * * * * *|

(Do you recognize the middle triangle? Look at it upside down. Thanks how, as a bowler, you see the ten-pin set on a bowling alley.)

The pentagonal numbers are 7, 15, 26, 40, 57, etc., because dots or caps of these build up larger and larger pentagons.

And other constructions build those patterns which are also known as "figurate numbers". A Table of regular n-gons built by bottle caps, dots, etc. evokes fascinating implications

These figurate numbers I've discussed involve additve gnomons or building-blocks. For the study of figurate numbers with regard to their multiplicative structure, see sieving at this website.