CCG

May 5 2000

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DEFINITION OF THE PYTHOGAM

Df.1. The Pythogram is a 2D-Image of an Abstract Number-Theoretical (NT) Object.

Df.2. NT-Object is a segment of the 1D-Series of Natural Numbers, n ³ 1, with a NT-Predicate P(n) defined on it.

Df. 3. The Sense of the Number Theory consists in a representation of the connection between the Additive and Multiplicative Properties of Natural Numbers.

The most basic CCG-Technique:

Color all natural numbers in according with the rule: if P(n) then White else Black.
Convert the 1D-series of natural numbers into the 2D-Image (Table).

Pythogram of a segment [1,54] of the Natural Number Series by Modudlus 11. Here, NT-Predicate P(n) = "n is a square of a natural number".

MAIN PYTHOGRAM PROPERTIES:

Additivity of n is modeled by its Colour.

Multiplicativity of n is modeled by its position in a j-coloumn of the Pythogram since "n ³1 (if n Î j-column, then n º j (mod L)).
So, any Pythogram as a whole visualizes the connection between two abstract properties of the Natural Numbers - Additivity and Multiplicativity Properties.
Changing Modulus we get a possibility TO SEE this Connection
(the sense of images sequences.
Any Pythogram has a Music-Invariant which consist of the visual image sequence after conversion to tones.