Quote:
Originally Posted by riseball  This is something that has fascinated me, I noticed that if you embed a square in a square like a fractal as it gets bigger the perimeter and the areas follow the DOUBLING CIRCIUT. The original square in the middle is a 1X1 square and gets progressively bigger...  |
There is something interesting about "√2", which is why Stan Tenen had great issue with Dan Winter ripping off and misrepresenting his work as related to "Phi".
The phenomena in question is this:
√2 = 1.4142135623730950488016887242097... = R
(R - 1) = 0.4142135623730950488016887242097... = E
(1/E) - 2 = E
This is related to a "geo-synchronous orbit" or "escape velocity" in a gravitational system.
It's also the origin of the spiral shape under the Wadjet (or ""):
It is not a "phi" related spiral. It is a "√2" related spiral.
I find it strange that a reciprocal (1/x) relationship is equivalent to adding or subtracting the number 2. Let's investigate further.
This analysis seems to link √2 to a "doubling" series (or at least a factor progression of 2 = 0,
2,
4, 6,
8, 10, 12, 14,
16, 18, 20, ...etc.)
NB - note that the number "1" is missing from the series... perhaps this echoes the "1/64" missing fraction of the Wadjet:
(1) = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 (+ 1/64 missing from the eye)
The "Golden (Phi) Spiral" and related spirals are called a "", whereas the one involving the √2 reciprocal is called an "".
The former tend to be "static" and the latter "dynamic" when found in Nature.
I found something relating these
Spirals to Fluid Dynamics, Gravity and
Coriolis Forces (involved in some of Nassim's
Physics papers):
r = a + b θ
r/r= [a + b θ]/r
1 = a/r + (b θ)/r
eg,
a = 4
r = 2
b θ = (-2)
where:
r is the radius (distance from locus),
a is the increment of radius (for a given increment in angular displacement),
b is the proportional factor of the angular tracing (θ) of the spiral around the locus (with respect to r)
Then we can say,
(1/E) - 2 = E
(1/E) = E + 2
1 = E^2 + 2E
1 = 1/(E^2 + 2E)
1 = (1/E^2) + (1/2E)
(1/E^2) + (1/2E) => (a/r) + [(b θ)/r]
Then, solve to find the required parameters for this particular Archimedean Spiral (E = √2 - 1).
More on the Logarithmic/Archimedian Spiral discussion from
Stan Tenen can be found:
Golden Mean SpiralGolden Mean Spiral - AddendumAsymmetric Spiral (r θ = 1)
sadukan.
PS If we take Stan Tenen's suggestion (r θ = 1) we end up with,
θ = (1/r)
Now substitute into our previous analysis:
(1/E^2) + (1/2E) => (a/r) + [(b θ)/r]
1 = (1/2E) + (1/E^2)
1 = (a/r) + [(b/r)/r]
r = a + (b/r)
Identical to the form of the original equation:
E = (1/E) - 2
r = a + b θ
so,
a = (-2)
b θ = 1/E
thence,
E = √2 - 1 => r = a + b θ
where,
θ = √2 + 1
√2 = θ - 1
√2 = E + 1
1 = [(θ - 1)/√2] + [(E + 1)/√2]
1 = (1/√2).[(θ - 1) + (E + 1)]
√2 = (θ - 1) + (E + 1)
√2 = θ + E
θ = √2 - E = 1
(r θ = 1)
(E θ = 1)
1 = E√2 - E.E
1 = r√2 - r.E
(E√2 - E.E) = (r√2 - r.E)
(√2 - E) = [r√2)/E] - r
(√2 - E)/r = [(√2)/E] - 1 = 1 + √2
r = √2 + 1
(√2 - E)/(√2 + 1) = 2 + √2
1/(√2 + 1) = E
iff:
r = 1/E
r = E+2
(radius (r) is a factor 2 series of E)
E = √2 - 1
E = r - 2
(r - 2) = [1/(r - 2)] - 2
r(n) = 1/(n - 2)
Wolfram|Alpha gives
this graphical representation,
which gives us the required spiral:
eg,
0 = 1/(-2) = (-0.5) (electron, ie zero radius???)
1 = 1/(-1) = (-1) (whitehole???)
2 = 1/(0) = singularity (blackhole???)
3 = 1/1 = 1
4 = 1/2 = 0.5
5 = 1/3 = 0.333...
6 = 1/4 = 0.25
7 = 1/5 = 0.2
8 = 1/6 = 0.166...
9 = 1/7 = 0.142857...
10 = 1/8 = 0.125
etc...
NB - the plot crosses the axes at two points: 2 and 0.5 (doubling/halving).
PPS I intentionally left off the factor of pi for the part of the angle, to make it easier to follow the argument.
I didn't really want to get into trigonometry, as I think that would just complicate things further.
It's a simple ratio relationship anyway:
PPPS Also, since I haven't done anything like this for a while, for those of you who are able to follow, you might like to check the validity of the foregoing analysis.
PPPPS The above sequence is known as a divergent with associated .
It then also links in to Riemann's - ζ(1) for the Harmonic Series - which is related to
Prime Numbers; implying the unsolved .
"
over it is 19"[74:30]
(
DiYu - 18 Levels of Hell)
Philip LeMarchand
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Esto es un cuadro/anteproyecto dado a nosotros por el Di-s de Israel de que después de los 6.000 años de tiempo (el final del Olam Hazé), Él entrará en la plenitud del matrimonio con Su pueblo, cuando el Mesías (Mashíaj) judío Yeshúa/Jesús gobernará y reinará con Su Novia durante la Edad Mesiánica (Atid Lavó). En el pensamiento tradicional judío (casa de Judá), la Fiesta de las Trompetas (Rosh HaShaná) es el día de la plenitud del matrimonio del Di-s de Israel con Su pueblo.