user737 wrote: ↑Wed Feb 19, 2020 8:25 pm

'Trace an anneagram with the nine on top, then trace the

**shapes of motion**, these are the shapes obtained by multiplication (yes, not additive, multiplicative just as in RS) and for example 2*7 = 14 and 1+4=5. Once the shapes are traced find the contradiction and resolve it (doing this in yourself is also part of the work). Notice how the shapes of motion are a close match to Larson's basics motions, and how the material (physical) and cosmic (etheric) sector are evident, having 9 as the symmetry center. One can also see the ethical sector (astral plane of consciousness) in

**3-6-9**, the prime mover. Notice the x9 shape, the

**all seeing eye** on top of the

**pyramid**.'

I am sure it relates to what I am after in some way but it is not immediately obvious to me, so please forgive my shortsightedness: what should I be looking at?

I wonder if it has anything to do with what follows concerning π.

I have been trying to figure out exactly what the relationship is between π and Φ,

only to realize the former can be derived by the latter:

Beginning with a circle whose diameter is √5,

place two unit squares inside of the circle side-by-side

(either horizontally or vertically, the latter is shown above)

and find √5 as any diagonal between two opposite corners (AB as shown)

(this diagonal thus being

*equal* to the diameter of the circle).

Add 1 unit distance to this diagonal (√5 + 1) and find the mid-point (/2).

Note: this midpoint coincides with the circle whose diameter is 2r = 1

by way of rotating the √5 diagonal about the origin of the circle we began with,

and thus

*"kisses"* the square four times

*equidistantly*, thus a more precise π

can be begotten directly from Φ

*expressed as a ratio of Φ*.

**This ratio is 4/√Φ = π and/or 16/Φ = π²**.

For reference, here is the triangle used to construct the Giza pyramid:

And an overlay of the geometry of 4/√Φ as it concerns the circle/square of 2r/1:

Concerning:

https://en.wikipedia.org/wiki/Kepler_triangle
I am unsure as to the motive for the above, but the construction of the geometry

is conceptually unsound: concerning the same Kepler triangle, a circle that circumscribes it

completely ignores the imperative need for the diameter of the circle to be √5, the same to which

the Kepler triangle must apply. If the circle is not √5, there is no valid coupling and π remains 'transcendental'

however if coupled to the √5 diameter circle (as shown), the coupled relation permits a geometry

whose co-operation between Φ and π are respectively reflected in the following:

f(x)=x²-x-1

f(x)=x

^{4}+16x

^{2}-256

graphed with an overlay of the same geometry:

With golden spirals (for visual aid) and geometric solution utilizing 4x golden rectangles:

Images such as the following can easily be constructed with a single golden spiral model

and one 'bulge' function (paint.net) to magnify what is directly in the center (otherwise hidden).

I have a general inclining towards the notion that the natural geometry of the universe

can be captured as some particular relationship between only π (as 4/√Φ) and Φ, recalling:

user737 wrote: ↑Wed Jan 29, 2020 2:46 pm

ckiit wrote: ↑Wed Jan 29, 2020 3:55 am

What are the odds of a single polynomial whose +/- root solutions are +1 and +phi^3, thus scale invariant?

Would that not be an equation which describes the physical universe?

+Φ

^{3} = Φ

^{2} + Φ = Φ

^{2} / (Φ-1) = (Φ+1) / (Φ-1) = -Φ × Φ

^{2}= -Φ

^{3}
(1/Φ)

^{2} + 1/Φ = +1

where Φ = (1 + √5)/2 -- the golden ratio

but only if the relationship between the two is known

(ie. π is

not autonomous, it is naturally coupled to Φ).

Finding the exactitude of π can never be done with polygons (as

*approximated* by Archimedes)

thus by using the

*natural* curve of the Φ spiral the circumference of the circle can be known.

EDIT this relationship alone generates the same pattern...

as seen on the recently engineered Ferocell devices:

EDIT2 if one takes the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21 etc. and continues indefinitely,

one may notice two important points:

i. the last digit of every 60 numbers completes a recurring pattern, wherein

ii. every 5th number of this pattern is divisible by 5

The first is important because it establishes the precedent needed to derive a circle (or folded, as in infinity ∞), and

the second is important because it reveals the "fold" in/of the same circle induced by the √5 operation of Φ. Thus,

the golden ratio (Φ) and the circle (π) are scale-independent linear/circular counter-parts to one another at 60 "cycles".