The notion of RECURSION was implicit in the (DIGITAL) geometric theory of numbers created 2500 years ago by Pythagoras in giving us the concept expressed by the Greek word "gnomon". The Pythagoreans treated "gnomon" as a unit or building-block of a pattern; thus, the odd number is a gnomon of a triangular number pattern. The term also referred to a vertical metal triangle or pin on a sundial, whose projected shadow is an indicator of the time of day. The term also involves knowledge, as in the suffix, -gnomy: for example, physiognomy.

Using gnomon as a "building-block" of numbers, Pythagoras-L developed the geometric theory of numbers, giving us such labels as "squares" for 4, 9, 16, 25, etc., and "cubes" for 8, 27, 64, 125, etc., because he created these numbers as 2-D squares of dots or 3-D squares of dots.

Thus, Pythagoras-L showed that the odd number is the gnomon of the square, since recursive addition of odd numbers creates the sequence of square numbers. Thus, the successive squares of 1 = 1 x 1, 4 = 2 x 2, 9 = 3 x 3, 16 = 4 x 4, 25 = 5 x 5, etc., can be constructed by adding the successive odd numbers, 1, 3, 5, 7, 9, etc. Behold: 1; 1 + 3 = 4 = 2 x 2; 1 + 3 + 5 = 9 = 3 x 3; 1+ 3 + 5 + 7 = 16 = 4 x 4; 1 + 3 + 5 + 7 + 9 = 25 = 5 x 5; etc. (Children can easily be shown that, given a square of equal rows and columns of dots or bottle-caps or marbles or blocks, the adjunction of one more row and one more column, together with their corner closure, results in the next larger square.)

Once children have been taught this method of additive construction of number-squares, the "Kierkegard kickback" of recursive inversion-by-subtraction of odd numbers can be applied to extract square roots! Thus, consider 36 = 6 x 6. Now, 36 - 1 = 35; 35 - 3 = 32; 32 - 5 = 27; 27 - 7 = 20; 20 - 9 = 11; 11 - 11 = 0. Question: How many odd numbers were used to reduce 36 to 0? Answer: We used odd numbers, 1, 3, 5, 7, 9, 11 -- 6 odd numbers. Hence, the square root of 36 is 6!

("Hey!", you say. "What about bigger numbers? Won't that be a lot of work? And what about numbers that aren't 'perfect squares'?")

Shush! There's an easy trick shown in x that simplifies the work and applies to any "whole number". Take 1225 = 35 x 35. Instead of peforming 35 subtractions to discover that 35 is the square of 1225, this trick requires only 3 + 5 = 8 subtractions to elicit the answer, 35. And you can use it to determine the square root of 2 to 5 places, 1.xxxxx, by yyyyyyyy subtractions.)

Besides "squares" and "cubes", Pythagoras-L also talked about , etc. I have a very loconek Mathtivity entitled Bottle-Cap GeometrY, which teaches children how to construct numbers by bottle-caps.

As to the loconek of the sun-dial, Pythagoras built geometric models of numbers whose gnomon is 1 or 2 or 3 or 4, etc. That is, the numbers 1, 2, 3, 4, etc., are built from initiator 1 by the gnomon 1: 1; 1 + 1 = 2; 2 + 1 = 3; 3 + 1 = 4; etc. The numbers 1, 3, 5, 7, etc., are built from initiator 1 by the gnomon 2: 1; 1 + 2 = 3; 3 + 2 = 5; 5 + 2 = 7; etc. The numbers 1, 4, 7, 10, 13, etc., are built from initator 1 by the gnomon 3: 1; 1 + 3 = 4; 4 + 3 = 7; 7 + 3 = 10; 10 + 3 = 13; etc. The numbers 1, 5, 9, 14, 19, 24, etc., are built from initiator 1 by the gnomon 4: 1; 1 + 4 = 5; 5 + 4 = 9; 9 + 4 = 13; etc. And so on for other such constructions.

So? So, Pythagoras modeled such numbers by dots fanning out in a triangular segment. The model for 1, 2, 3, 4, etc. -- with gnomon of 1 -- occupies a single triangular segment. The model for 1, 3, 5, 7, etc. -- with gnomon of 2 -- occupies a triangular segment bisected into 2 triangular subsegments. The model for 1, 4, 7, 10, etc. -- with gnomon of 3 -- occupies a triangular segment trisected into 3 triangular subsegments. Etc. Put together, these begin to sweep out a circular sector resembling the face of a sun-dial.

Please NOTE!!! Sequences of numbers built by successively adding or subtracting the same number are known as arithmetic progressions, and gave rise to that form of average we know as the arithmetic mean -- the one most people call "THE AVERAGE", although there are many such. They appear in Euclid's Elements of Geometry as a display of line segments, which may be called the "analogic" form of what Pythagoras initiated in the "digital" form.

And the basic idea behind such constructions gave rise to the most powerful of all forms of proof, namely,mathematical induction. WARNING!!! This should be called proof by recursion -- to put alongside definition by recursion. This "misnaming" not only disconnects it from the recursive process, but gives rise to confusion with the use of "induction" in logic, where it is a sometimes questionable form of reasoning -- meaning generalizing from a few cases. for a well known example of this. The power of mathematical induction is shown by the fact that the simplest form of logic, the logic of statements or propositions -- also known as zero-order predicate logic -- needs only this method of proof!)

An important hiconek of gnomonics is found in the science of crystallography, which explains beautiful jewelry, the molecular structure of pharmaceuticals and other chemical achievements, the creation of synthetic insulin, and the transistor. (See the GNOMON-TABLE.)

WARNING!!! The foundational philosophy of Platonism WILL NOT ALLOW GNOMONS ("ATOMS") IN SET THEORY, SINCE THESE PRECLUDE THE TWO MOST POWERFUL NONCONSTRUCTIVE PROOF METHODS (PROOF-BY-CONTRADICTION AND AXIOM OF CHOICE). Hence, THESE HIGH PRIESTS OF MATHEMATICS TRY TO PRECLUDE ANY GENERATIVE COMPETITION TO THEIR AXIOMATIC DOGMA!

PYTHAGORAS DIGITALS RECURSION.