Vesica Piscis and Sacred Geometry

When finding simple arithmetic relationships to represent the mean orbit distance of planets and other objects in the Solar System (see Lay On MacDuff at A Page In The Life)  it's especially exciting to see that p/4 (or (4/p)^{^2} ) since this intuitively means we are dealing with solid angles...wait while I check because I don't remember exactly why I thought this to be so exciting...

I even remember thinking seeing this (4/ p)^{^2}   to be elegant at one point....

...not for lingam or yoni reasons;

Icthus

The vesica piscis (Latin for ‘bladder of a fish’ _see ichthus + Early Christian Symbol) is the shape formed at the intersection of 2 Circles of equal size. The term is used for the shape of any symmetric lens and is also called a mandorla (Italian for ‘almond’).

The ratio of the width of the vesica piscis to its height is  Ö3~ 1.7320…best whole number approximations are:

Subject to mystical speculation by Kabbalists, New Agers, and Freemasons, it serves as an important symbol within Freemasonry (and guess who had their hands all over the construction of the Washington Monument).

While New Age thinkers interpret the vesica piscis as a yonic symbol (roots in Hindu ideology, the traditional symbol given female genitals), it serves as a proportioning system in Architecture, especially Gothic Architecture (ie., ‘the rule of the German Architects”).

...but because it has to do with machines and the Reuleaux Triangle...

  http://mathworld.wolfram.com/ReuleauxTriangle.html,

...which I think of as a drill bit (how boring) and some in my sector are more familiar with as a Wankel engine.

...and not because I am trying to break into a Masonic Lodge, or get clubbed over the head, or fed an elixir that will cause kidney failure, or make me want to leap from tall buildings or do anything else with (more) fatal/painful consequences (I guess my secret admirer ran out of ammo)....

Woodcut 3 from Rosarium Philosophorum
Sacred Geometry, Freemasonry

Depicted above is Woodcut 3 from Rosarium Philosophorum showing how the Reuleaux Triangle is embedded in the ‘sacred (3-Circle) geometry’ formed by the King and Queen, The Sun and Moon, and The Dove (which reminds me…Where’s Malcolm?!).

The Dove is associated with the feminine and is also connected to Water (The Mother Goddess). To Christians, The Dove symbolizes the Holy Spirit, while to early pagans it symbolized the ‘yoni’ or female sexual organs (see vesica piscis).

In this example, the Reuleaux Triangle is formed by the 3 Circles of equal radius centered at the second chakras of the King and Queen and the midpoint of the Dove’s body. The dashed lines are the axes of the vesica piscis. The rectangle of Ö3 has vertical sides which correspond to the spines or center lines of the King and Queen.

{Swadhisthana, the second chakra (in Sanskrit chakra means 'wheel') symbol includes the Sanskrit intonations:

ba + bah + ma + ya + ra + la
and corresponds to the Muslim Sufi 'nafs' (Ar. soul/consciousness/breath/self), refer to navel/nave/naval + cow/ship/nifs/nafs/nauta/ships/cloud/cumulous/heaven/nouf_nose_beak from the Fr. + transepts(church)/cardinal directions/story/upper part of the nave/large choir/chancel_lattice/hub_(of a wheel_by extension any center)/umbillic/navel/end of a scroll roller...}.

...but because it brings to mind a homework assignment from 2009

A form that looks like a modified 'bean curve', 
a 4th order planar curve having arithmetic genus=0,
and a singularity at the origin, an ordinary triple point.
(See Limaçon, Inverse Curve of Hyperbola)

....and how this all ties in with the Heavens being 'musghari' or 'geared' ....

Sign 2:164 refers to clouds being 'harnassed' (الْمُسَخِّرِ musghari==is geared) between the earth and the Sky ..word for harnessed also has to do with 'gears' and we understand gears to transmit motion from one body to another as in toothed gears, they serve to 'engage' an activity/equipment like when putting something into gear, and gears also transmit power and/or direction (like in a vehicle transmission)--a term closely related to equipment and machines.

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....just remembered why the fascination with (4/p)^{^2}  (added on Feb 23, 2012)

1 sphere == 8 octants == 2 hemispheres ==12.57 Steradians== 4p sr==All of Subtended Space

and:

(4/p) ^{^2}  Steradians  (sr) ~1.621 sr~1.618~(1/2 + (Ö5)/2)~ j:

It is equivalent to:
~ 0.129 spheres ;(1 sphere ~12.57 sr)
~ 0.52 x (3.142 sr) == p  the solid angle subtended by 1 face of a regular tetrahedron from its center
~ 0.77 x (2p/3 sr) == the solid angle subtended by 1 face of a cube from its center
~ 1 x  (p/2 sr)==the solid angle subtended by 1 face of a regular octahedron from its center

and its reciprocal (p/4)^{^2}  Steradians (sr) ~ 0.6169 sr~ 1/j:

which is equivalent to:
~0.04909 spheres
~0.59 x (p/3 sr)== the solid angle subtended by 1 face of a regular dodecahedron from its center
~0.98 x (p/5 sr  )==the solid angle subtended by 1 face of a regular icosahedron from its center
~9100 x (6.8 x 10^{^-5} sr)== the average solid angle subtended by the Sun from the Earth

Some interesting math equivalents to 16/p^{^2}  include:

While in terms of the math it gives us a lot to think about (i.e, it relates simply to Zeta of (2) and the Pythagorean triple constant for the hypotenuse, the Gamma function, and the complete elliptical integral of both the 1st and 2nd kind, etc) , since this study began with a look at the Solar System the fact that Saturn's Sidereal (9) vs its Synodic (256) periods (according to the Babylonians; I don't know what it is according to NASA--they lost a Space Shuttle (or 2), Remember?--See ? WE All make mistakes, so Why Not Me?) are also related to this ratio is of some interest  and worthy of consideration:

understood interms of:
(4/p)^{^2} =(Saturn's Synodic Period/Saturn's Sidereal Period) * f(t) = (T_syn/T_sid) * f(t)

or

(4/p)^{^2} = (16/3)^{^2*}f(t)____________-Equation 1

where     f(t) = [1/ (sqrt(3)+32 integral_0^(1/4) sqrt(-(-1+t) t) dt)^2)]
and let T ' ^{^2} = (16/3)^{^2}


Equation 1 can then be written in terms relating (4/p)^{^2}  to Saturn's Synodic v Sidereal Periods and some function of time f(t) as:

(4/p)^{^2} = T ' ^{^2 *}f(t)  _____________Equation 2

For planets orbiting the Sun, having period T and average radial distance R, the ratio T^{^2}/R^{^3}=k,
a constant independent of the planet's (or object's) mass.

Where k = 4p^{^2}/(G*M_Sun);
and
G=6.673 x 10^-11 N.m^2/kg^2  is the Gravitational Constant, 
M_Sun =1.98892 x 10^30 kg is the mass of the Sun.

it's worth emphasizing that k works out to roughly (3/10)p^{^2} x 10^{^18}

(Yeah, I know--You all Knew this already. It's okay for me to be a laggard, according to my 'brothers' I'm about 10 years away from figuring all this out, which is so not fair considering how I already got there and The Nameless One threw the game in everyone else's favor but mine for no reason that I can figure!--Excuse me, but was that You fluttering Your Falcon Wings at Me in the Skies over Malibu yesterday (Feb. 21, 2012)?--Such a Flirt! Don't be doing that after I saw You with another woman--You Know, nothing makes a man less attractive more quickly, to me anyway, than to see him keep company with another woman. Oh, a Eureka! moment here--I think I finally figured out why He ditched me--told You I was slow...And not only does He ditch me but doesn't want me running around with the rest of the pack either---the most dangerous weapon in His arsenal--He wants me to be bored to double-death.)

--actually I think my brothers were very optimistic with the 10 year estimate---having no idea how difficult this simple math is for me and how frustrating i find that---plugging right along:

Rearranging;

T^{^2}/R^{^3}=k
and

k=4p^{^2} /G*M_Sun

to

4p^{^2} = T^{^2} *f(r)__________Equation 3  

where f(r) =[G*M_Sun /R^{^3}]

expressing the relationship between the orbital period and some function of radial distance f(r):
(2p)^{^2} = T^{^2} *f(r)__________Equation 3'
(4/p)^{^2} = T ' ^{^2 *}f(t)__________Equation 2

dividing Eq. 3 by Eq. 2 gives us:

(p⁴/4) = (T/T ')^{^2} * ( f(r)/f(t) )___________Equation 4

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