Exploring configurations of colour-based logic.
So far we have
Summing of two primary colours that yield a single secondary. We can also find the roots of a given secondary to decompose and yield the factors of it's composition.
0 = Black
1 = White
2 = Red
3 = Orange
4 = Yellow
5 = Green
6 = Blue
7 = Purple
Hmm, a colorification logic that yields a base eight numbering system.
HOLY MOSES!!!
..(Primary + Primary) ÷ 2 = Secondary...
Mathematical mapping operations:
Red + Blue = Purple
(Blue + Yellow) ÷ 2 = Green
(Yellow + Red) ÷ 2 = Orange
Solved.
For a base six numbering system:
1 = Red
2 = Orange
3 = Yellow
4 = Green
5 = Blue
6 = Purple
HEY! ..our mapping still applies
Red + Blue = Purple
(Blue + Yellow) ÷ 2 = Green
(Yellow + Red) ÷ 2 = Orange
Axiom of colour based mathematics?
The simplest six-colour form, for a base six colour-wheel based numbering system...
There are two FIRST ORDER primary colours (Red, Yellow) and one secondary (Orange).
There are two SECOND ORDER secondary colours (Green, Purple) and one primary (Blue).
WARNING ..experimental mathematics ahead
What if... there is need for such a combinatory system that contains more than six elements?
..what mathematical pattern will demand consideration and make itself apparent?
....
Here is a truth, in any odd numbered series of evenly scaled values
..(num1 + num3) ÷ 2 = num2
Hmm...
..(num1 + num5) ÷ 2 = num3
..(num1 + num7) ÷ 2 = num4
..(num1 + num9) ÷ 2 = num5
The nitty gritty of combinatorial routines and factoring them back to their constituent components.
Cool Stuff :)
Thank you Colour-wheel.
post script...
..when adding a primary of the first order to a primary of the second order ..the inclusion of a second order value negates the need for division by the number of values summed.
Maybe they are not ONLY first and second order colours, these are also organized in a natural symmetry...
First Order (PSP) red, orange, yellow
Second Order (SPS) green, blue, purple
Six colour layout with mathematical mixing of primarys that fill the scale of secondaries.
P-S-P-S-P-S
I know nature made this puzzle, attempting to scale this to more than six combinatorial elements is not immediately evident.
CONCLUSION:
The conclusion, a ring of colours from which combinations of every second element yield that which exists between them. The ring/loop configuration of colours (colourwheel) is handled mathematically as a series who's end of list function ties it with the beginning ..a sequential numerical representation of a circle, hexagon or wheel.
For a base six numbering system:
1 = Red
2 = Orange
3 = Yellow
4 = Green
5 = Blue
6 = Purple
(Blue + Yellow) ÷ 2 = Green
(Yellow + Red) ÷ 2 = Orange
Red + Blue = Purple
The 3 choices from which can be made 3 combinations, enumerated 1 through 6.
For a 10 increment wheel... 5 choices and 5 combinations, enumerated 1 through 10.
***** Wheelwright closes the ring, adding the first choice to the second last in the set. This allows the finite sequence to be interpreted as a continuous circle configuration.*****