The first few '''triangular numbers can be build from rows of 1, 2, 3, 4, 5, 6 items.

* | * | * | * | * | * |The| * * | * * | * * | * * | * * | | | * * * | * * * | * * * | * * * | | | | * * * * | * * * * | * * * * | | | | | * * * * * | * * * * * | | | | | |* * * * * *|

- P
_{2}(n) = ½n(n + 1) for triangular numbers; - P
_{3}(n) = 1/6 n(n + 1)(n + 2) for tetrahedral numbers; - P
_{4}(n) = 1/24 n(n + 1)(n + 2)(n + 3) for pentatopic numbers.

Another label for these numbers is **Pythgorean geometry**, since Pythagoras is credited with initiating them, and the notion that these numbers are generated from a Gnomon or basic unit.

For example, the gnomon of the square number is the Odd number, of the general form 2n + 1, n = 1, 2, 3, ... , easily demonstrated as follows.

To transform from n-square to (n + 1)-square (say, 7-square to 8-square):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 the n-square (say 7-square), adjoin (n + 1)-rows (8-rows) of elements;
- also adjoin (n + 1)-columns (8-columns);
- so the adjunctions total (n + 1) + (n + 1) = 2n + 2 = 2(n+1) elements (say, 2x8=16);
- the square pattern is completed by adjoining a corner element (numbers along diagonal, from upper left to lower right),say 8);
- thus, to the n-square, has been adjoined 2(n+1) + 1 elements (say, 16+1=17), which is an odd number;
- so the gnomon of any square is the odd number: 1, 3, 5, ... .

Conversely, one can calculate the square root of any number by **subtracting odd numbers**. Thus, 64 - 1 = 63; 63 - 3 = 60; 60 - 5 = 55; 55 - 7 = 48; 48 - 9 = 39; 39 - 11 = 28; 28 - 13 = 15; 15 - 15 = 0. The subtraction of the first 8 odd numbers from 64 yields 0; hence, the square-root of 64 is 8.

The tedium of increasing number of subtractions as number grows is bypassed by a method similar to the standard way of square-rooting taught in school. For example: 1225 = 35 x 35, Note the sum of the digits of this square root: 3 + 5 = 8. This square-root shortcut reduces 35 subtractions to only 8 ubtractions. Etc. The shortcut involves two "tricks": a markoff trick, and resumptive trick.

The markoff trick is already known from the familiar squareroot algorithm. One marks off the target number in pairs of digits, from the right, as in marking 1225 as 12'25; then, calculation begins with the first digit-pair to the left. The reason is that squaring a one-digit number results in a 1- or 2-digit square. Thus, 1, 2, 3 have, respectively, the 1-digit squares of 1, 4, 9. But 4 has the 2-digit square of 16; and numbers 5, 6, 7, 8, 9 have 2-digit squares. To allow for this, one begins with two digits to provide one digit at each process-stage.

The resumptive trick (unique to this present algorithm) shifts from one pair of target number digits to its next (rightward) two digits, explained in calculating the square root of 1225.

- Mark off 1225 as 12'25; begin calculation with left pair of digits, namely, 12.
- Begin subtracting odd numbers: 12 - 1 = 11; 11 - 3 = 8; 8 - 5 = 3; but the next odd number, 7, cannot be subtracted from Difference 3, so the resumptive trick is needed.
- The left-most digit of the square root, 3, actually representing 30, because the second digit from the right in Decimal numeration is the "tens digit".
- To difference 3 (= 8 - 5), adjoin next two marked off digits (25), obtaining 325, and resume odd number subtraction.
- The last "successful" subtrahend was 5; but the next odd number, 7, cannot be subtracted, so
**interpolate**between 5 and 7 for number 6. (This is "first part" of the resumptive trick.) - Since (noted above) the successful 3 subtractions actually represent the 2-digit 30, treat the interppolated 6 as 60; resume odd number subtraction with the first odd number in the sixities, namely, 61. (This is "seond and final part" of the resumptive trick or subalgorithm.)
- Result: 325 - 61 = 264; 264 - 63 = 201; 201 - 65 = 136; 136 - 67 = 69; 60 - 69 = 0.
- Having passed from 325 to 0 by five subtractions, the second digit is 5: and 30 + 5 - 35, that is, the square root of 1225 is 35, obtained in exactly 3 + 5 = 8 subtractions by applying the markoff and resumptive tricks or ubalgorithms.

- Markoff 144 as 1'44.
- Starting with left-most pair, start subtraction 1 - 1 = 0; so leftmost digit of square-root is 1, representing 10.
- Bring down second pair of digits: 0 + 44 = 44 and beging subtracting odd numbers.
- Interpolate between "successful" 1, "failed" 3, namely, 2, to represent the twenties, whose first odd number is 21, resuming the subtraction with 21.
- 44 - 21 = 23; 23 - 23 = 0, resulting in two subtractions, so second digit is 2: 10 + 2 = 12 as square-root of 144, obtained by 1 + 2 = 3 sbubtracting of odd numbers.

Cubes of natural numbers or positive integers can be generated from S = 1, 3, 5, 6, 7, ..., 2n + 1, ...., n = 1,2, 3, ..., by "moving sums", similar to the "moving averages" of Statistics:

- First member of S: 1 = 1
^{3}. - Mext two members of S: 3 + 5 = 8 = 2
^{3}. - Next three members of S: 7 + 9 + 11 = 27 = 3
^{3}. - Next four members of S: 13 + 15 + 17 + 19 = 64 = 4
^{3}. - Next five of S: 21 + 23 + 25 + 27 + 29 = 125 = 5
^{3}. - Next six of S: 31 + 33 + 35 + 37 + 39 + 41 = 216 = 6
^{3}. - Next seven of S: 43 + 45 + 47 + 49 + 51 + 53 + 55 = 343 = 7
^{3}. - Etc.
- * | * * * --> * * * | * * --> * * * * *
- * --> * * * --> * * * * * *
- * * * *
- *

This procedure (taking many words to explain, but quickly executed) is not restricted to calculating suqare roots of natural numbers or positive inegers. It can even be applied toward calculating the irrational square root of 2, to any number of decimal places. Etc.

School children construct figurate numbers from pebbles, bottle caps, etc. As a bonus, children can use figurate numbers to discover the Commutative law and associative law for addition and multiplication -- laws usually dictated to them -- by buiding rows and tables of dots.
Using asterisks in place of dots or bottle caps or pebbles the **additive commutativity** of 2 + 3 = 3 + 2 = 5 becomes:

And the **multiplicative commutativity** of 2 X 3 = 3 X 2 = 6 becomes:

Besides the subtractive method, the additive method can also approximate square roots of positive integers and solve quadratic equations (see "prairiestate" reference).

The concepts of figurate numbers and gnomon implicitly anticipate the modern concept of Recursion.

A related type of geometry is found under Pick's theorem.

**Gnomon, From Pharaohs to Fractals**. Midhat J. Gazalé, Princeton University Press, Princeton, 1999.

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