In the Pythagorean understand of number, all numbers have a corresponding geometric value. Each number corresponds to a set of points, with the points themselves connected through lines. These lines, in themselves must also be numbers. The problem occurs as to how one converts a number in a geometric structure, when the number itself as points, must contain a separate number as lines that must connect to each corresponding point in order to exist as a geometric form. In simpler terms each point of x number must connect to each point that composes x number, which each point synonymous to value of 1.
The following observes this problem:
2 points requires 1 line
3 points requires 3 lines
4 points requires 6 lines
5 points requires 10 lines
6 points requires 15 lines
7 points requires 21 lines
8 points requires 28 lines
9 points requires 36 lines
We can observe further that each set of lines differs by an numerically ascending value of 1
1 and 3 differ by 2
3 and 6 differ by 3
6 and 10 differ by 4
10 and 15 differ by 5
15 and 21 differ by 6
21 and 28 differ by 7
28 and 36 differ by 8
So we see a dual numerical sequence that corresponds with the original one composed of points.
The question occurs as to calculate each numbers inherent connections, or the connects which in themselves approximate the points. This approximate, as the observation of a connection delves somewhat in chaos theory in these regards.
In these respects the equation (α-1) * α(1/2) = ε → {α,ε} appears to provide the solution where "α" equal the number as point and "ε" equals the corresponding lines which approximate the points. In these respects, from a geometric perspective all numbers are conducive to sets in themselves.
Thoughts?