What If...

[Capstone]

Check this out: incorporating 1-9 into an infinity loop with a single mathematical impetus can result in a nine-pointed star as follows:

1+4=5
5+4=9
9+4=4 (1+3)
4+4=8
8+4=3 (1+2=3)
3+4=7
7+4=2 (1+1)
2+4=6
6+4=1 (1+0)

The movement: 1, 5, 9, 4, 8, 3, 7, 2, 6, 1

Sorry about my sloppiness, but I think careful measurement would result in an inverted nonagon in the center.

Rodin's method is very cool.

 

[Capstone]

It turns out that careful measurement not only results in a central inverted nonagon, it also reveals a total of 3 nine-pointed stars, which I've illustrated in black and white in the following image.



Look deeper to see 3 overlapping triangles (shown in grey, black, and blue as follows):



And deeper still for a dead-on corner-view of a four-sided (!) pyramid (in grey):



I realize, at the moment, all this math and geometry stuff might seem irrelevant to the topic of universal oneness, but I intend to tie it all together in this thread ...just as soon as I've sorted out the connections in my own mind.

 

[Unkotare]

Classic example of a dimwitted buffoon taking himself too seriously.

 

[Capstone]

I've invested more than my fair share in ridicule over the years (especially as a kid), so to see it returning to me with interest as an adult ...feels as though some of the debt I've incurred in this life is being paid off a little bit at a time. That's a good feeling. Karma's only a bitch to those who fail to see her beauty.

That's not necessarily to say I'm not a dimwitted buffoon who takes himself too seriously, though.

Anyway, getting back to Rodin's Quantum Numerology, while the nonagram in my previous post seems to illustrate width (via the 3 nine-pointed stars that appear to emanate from the center outward), depth (with the stacked-up triangles), and height (with the inferred 4-sided pyramid), what it doesn't represent is the overlapping cross-directional travel of the previous formula.

The challenge then, is to create an opposing directional flow that remains true to the underlying geometry ...and, at least to some degree, to the movement by fours.

For this, I think some restraints on the numerological reduction thing (without completely suspending it) may be in order.

As always, starting the next axis where the previous one left off (at 1), but parenthetically restricting the numerological reduction here on out, the addition of 1 to all of the numbers 4 times around the perimeter would come to a close at 37 (1+1=2, 2+1=3, 3+1=4, ...and so on to 36+1=37). Notice that 37 would correspond to 1's position on the chart, which makes perfect sense (since 3+7=10, and 1+0=1). The same would be true for all of the non-reduced numbers that had preceded it (36 would correspond to 9, 35 to 8, 34 to 7, all the way back to 1).

From 37 (and position 1), still refraining from numerological reduction outside of the parentheses, we could start the next axis by moving in fours via subtraction and travel backwards along the very path used during the initial addition phase.

37-4=33 (3+3=position 6)
33-4=29 (2+9=11, 1+1=position 2)
29-4=25 (2+5=position7)
25-4=21 (2+1=position 3)
21-4=17 (1+7=position 8)
17-4=13 (1+3=position 4)
13-4=9 (position 9)
9-4=5 (position 5)
5-4=1 (position 1)

So, the subtraction axis would move: [1], 6, 2, 7, 3, 8, 4, 9, 5, 1

The synthesized movement would then flow:

1, 5, 9, 4, 8, 3, 7, 2, 6, [1], 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, [1], 6, 2, 7, 3, 8, 4, 9, 5, 1.

Addition of 4, addition of 1 by 4x around the perimeter, and subtraction of 4 ...would be the respective impetuses.

Notice the mirror symmetry between the first and third axises, despite the parenthetical restriction placed on the third.

This movement would result in 4 overlapping rotations in the same direction around the perimeter, as well as two perfectly overlapping nonagrams traveling in opposite directions.

 

[Capstone]

...oh, and there would no longer be any need to infer the other two sides of the pyramid.

 

[Capstone]

The 4 laps around the perimeter and the restriction of numerological reduction to chart positions only could be circumvented by simply switching from addition of 4 to addition of 5 at the outset of the second axis.

Addition of 5

1+5=6
6+5=2 (11, 1+1)
2+5=7
7+5=3 (12, 1+2)
3+5=8
8+5=4 (13, 1+3)
4+5=9
9+5=5 (14, 1+4)
5+5=1 (10, 1+0)

Addition of 4 (notice the symmetry):

1+4=5
5+4=9
9+4=4 (13, 1+3)
4+4=8
8+4=3 (12, 1+2)
3+4=7
7+4=2 (11, 1+1)
2+4=6
6+4=1 (10, 1+0)

Since the simplest 'all-inclusive' (1-9, multi-dimensional, cross-directional, and perfectly symmetrical) formula should be preferred (per the principle of parsimony), the following movement seems the best one yet for an infinity-loop.

1, 5, 9, 4, 8, 3, 7, 2, 6, [1], 6, 2, 7, 3, 8, 4, 9, 5, [1]...back and forth forever.

The bracketed 1's mark the turning points from addition of 4 to addition of 5 (and back).

In addition to the above, accounting for the prospect of moving in increments smaller than whole numbers (mainly in order to promote more symmetry and geometrical inclusiveness), I've marked out the halfway points between all of the numbers on the chart as follows:



Rotating around the outer perimeter clockwise in increments of .5, the movement by addition of 4 and 5 within the circle would result in an inverted nonagram stacked-up on top of the initial one.

This can be seen by moving from the close of the second axis at 1 to 1.5 ...and then running gamut inside the circle in 4's and 5's.

1.5+4=5.5
5.5+4=9.5
9.5+4=4.5 (13.5, 1+3.5)
4.5+4=8.5
8.5+4=3.5 (12.5, 1+2.5)
3.5+4=7.5
7.5+4=2.5 (11.5, 1+1.5)
2.5+4=6.5
6.5+4=1.5 (10.5, 1+0.5)

1.5+5=6.5
6.5+5=2.5 (11.5, 1+1.5)
2.5+5=7.5
7.5+5=3.5 (12.5, 1+2.5)
3.5+5=8.5
8.5+5=4.5 (13.5, 1+3.5)
4.5+5=9.5
9.5+5=5.5 (14.5, 1+4.5)
5.5+5=1.5 (10.5, 1+0.5)

Continuing along the outer perimeter in increments of .5, we'd then move from 1.5 to 2 and run the gamut again...

2+4=6
6+4=1 (10, 1+0)
1+4=5
5+4=9
9+4=4 (13, 1+3)
4+4=8
8+4=3 (12, 1+3)
3+4=7
7+4=2 (11, 1+1)

2+5=7
7+5=3 (12, 1+2)
3+5=8
8+5=4 (13, 1+3)
4+5=9
9+5=5 (14, 1+4)
5+5=1 (10, 1+0)
1+5=6
6+5=2 (11, 1+1)

...then on to 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9, 9.5, all the way back around to 1.

The resulting multi-layered geometry:



Notice how many more formations now exist within the circle.





Even this:



...among others.

Considering that such formations couldn't otherwise be formed within a closed system of 9 whole numbers, an argument could be made that moving in halves would be necessary in order to propagate those formations in the simplest way possible.