http://arxiv.org/PS_cache/arxiv/pdf/0908/0908.0852v1.pdf ( from the above article )

" A long standing prediction of universality is the Efimov effect , which was presented in the early 1970's. It is an anamolous effect which occurs in neutral three-body systems when at least two of the underlying subsystems have resonant s-wave interactions , a diverging scattering length ( or at least much larger than the range of the physical force ). This corresponds to the subsystems being close to the thresholds for binding. Under these conditions an exceptional sequence of infinitely many geometrically spaced three body bound states occurs with an accumulation point at zero energy. More over , the ratio between the energies is a universal constant depending only on the mass ratios and the particle statistics. For three identical bosons the famous ratio is:

( Et^n ) / ( Et^n+1 ) = e ^ ( 2 * Pi / S(o) ) = 515.028

...it is emphasized that this specific scaling factor is independent of the mass and that the same effect in principle can be found at all physical scales. " ( end article quote )

.......an investigation showed the following:

The Efimov equation for quark resonance was described in the article above by this formula: ( e = natural log = 2.718281828 )

e ^ ( 2 * Pi / S(o) ) = 515.0280503

....S(o) equals the transcendental number ( 1.00624...). What was not explained in the article is the simpler meaning to the quark resonance phenomena. The natural log e form is equivalent to a dodecagon that has phi as the polygons edge:

e ^ ( 2 * Pi / S(o) ) = ( 10^4 ) / phi / 12

515.0280503 = 515.0283245

...phi has the unique ability to pull Pi out of the natural log e function , in ratio, to describe the resonance constant:

e ^ ( 2 * Pi / S(o) ) = ( 10^3 ) * phi / Pi

515.0280503 = 515.0362142

...this is related to the fact that ten times Pi divided by 12 equals phi squared:

phi^2 = 10 * Pi / 12

2.6180339875 2.617993878

....the relationship of the resonance constant ( 515 ) to phi is through one-half the dodecagon or hexagon(6-quarks):

515.0280503 * 6 = ( 10^4 ) * ( phi - 1 ) / 2

3090.168302 3090.169935

...in the presentation slide show, the resonance constant ( 515 ) is used in inverse form...1/515...If one makes one leg of a dodecagon polygon equal to one-tenth phi , then the perimeter is the resonance constant....1/51502834 / 10^3 :

( 10^-3 ) * Pi / phi = 1/515.02834 = phi / ( 10^4 ) * 12

1/515.0362142 1/515.0283245

...treating the perimeter like an approximate circle the radius equals: ( phi - 1 ) / 2

( 10^3 ) / 515.02834 / 2 / Pi = radius = ( phi - 1 ) / 2

.309021719 .309016993

....by placing the resonance constant as a leg on a dodecagon creates a polygonal perimeter of...( phi - 1 ) * 10^4... and a strange split of the circular radius of the dodecagon caused by the special qualities of phi...phi^2 = Phi +1:

515.02834 * 12 = perimeter = ( phi - 1 ) * ( 10^4 )

6180.34008 6180.3398758

...the dodecagon also illustrates the connection of 12 to phi and then to Pi. If the perimeter is 12 , leg equals 1 , then the radius equals a phi link:

12 and phi

1 * 12 / 2 / Pi = radius = 5 / ( phi^2 )

1.909859317 1.92983006

phi and Pi.....perimeter = Pi

( phi^2 ) / 10 * 12 / 2 / Pi = radius = .5

.500007659

...an interesting consequence of the resonance constant ( 515 ) is that by squaring the node ( 515.02834^2 ) , phi is eliminated from the equation:

515.03227 ^ 2 = ( 10^7 ) / 12 / Pi

...a new constant has appeared ( one erg = 10^7 = Giorgi constant ) and the leg of the dodecagon is now Pi ! Knowing the importance of Pi in the resonance constant , a series of Pi equalities can be shown: IN = reciprocal natural log e

Pi = IN( ( 10^4 ) * ( phi-1 ) / 12 ) * S(o) / 2 ................... S(o) = 1.006239915

Pi = IN( ( 10^3 ) * phi / Pi ) * S(o) / 2 ................... S(o) = 1.006237446

Pi = IN( sqrt(( 10^7 ) / Pi /12 )) * S(o) / 2 .................. S(o) = 1.00623868

Pi = IN( ( 10^4 ) / phi / 12 ) * S(o) / 2 ...................S(o) = 1.006239915

...What is the mysterious transcendental constant , S(o) ? Since the unit one is the non-decimal remainder of S(o) , it seems the more important number to investigate is the decimal remainder...( .006239915 ) that is derived from phi only equations. The inverse decimal remainder equals ( 1/.006239915 = 160.258593226 ). This is a very complex number , seemingly random , without a pattern , transcendental. This is the last puzzle of the Efimov resonance effect , directed by the resonance equation for the energy nodes of particles...e^(2*Pi)/S(o). To me to crack this puzzle and understand its meaning , one has to use the limited tools of the primal numbers ..37 , 37*18 , 137( fine-structure constant ) , Cheops constructs ( 2/Pi ) , etc. The first attempts were using S(o) as expressed in the equations as a power to 37 , the results were interesting:

37 ^ ( 1.006239915 ^ 22 ) = Pi * 20

62.83346517 62.83185307

...this result was fairly accurate with 22 being Kabbalic. The next attempt was using inverse S(o) as a power to 37:

( 37 ^ ( 1/1.006239915 )) - 30 = ( phi - 1 ) * 10

6.180699605 6.180339875

...again fairly accurate, odd to note that 37 was the only number up to 75 ( don't know about greater than 75 ) that when powered by inverse S(o) gave a good approximation for phi. The next primal number used as a power base to S(o) was 666 ( 37*18 ). This result was interesting because phi again was involved as well as the dodecagonal number 12:

666 / ( 1.006239915 ^ 12 ) = ( phi-1 ) * ( 10^3 )

618.0957263 618.03398758

...this result is very strange in that the number from the shield of Yale University's secret society , Skull and Bones , which is the integer , 322 , is related directly to phi by the dodecagon number 12:

322 ^ ( 1/12 ) = phi

1.618035289 1.6180339875

...this approximation is very accurate , 99.99992% of phi. However more accuracy is needed , and this was given by the constructs of the greatest structure on Earth, Cheops pyramid , 2/Pi. Its relationship to S(o) is stunning , but not surprising , and once this equality was shown , both the fine-structure constant , aem = 1/137.035999701 , and the strong nuclear force , a(s) = 14 , were able to derive S(o):

Efimov resonance formula : S(o) = 1.00624

e ^ ( 2 * Pi / S(o) ) = 515.0280503

..S(o) derivatives from 37 , 57 , a(s) and aem:

S(o1) = (( 10^ ( 2/Pi )) * 37 ) ^ -1 + 1 = 1.006239912554

S(o2) = ( sqrt137.035999701 * ( 3.7^2 )) ^ -1 + 1 = 1.006239922786

S(o3) = ( a(s) * ( 1/57 +1/37 ) / ( 10^2 )) ^ -1 + 1 = 1.006239924135

...so the secret of the decimal remainder of S(o) is primal. To check for accuracy plug S(o) into this formula for Pi:

Pi = IN(( 10^4 ) / phi / 12 ) * S(o) / 2

Pi(S(o)1 = 3.141592648 ...........99.9999998% of Pi

Pi(S(o)2 = 3.1415926804 .........99.9999992% of Pi

Pi(S(o)3 = 3.141592685 ............99.999999% of Pi

...one of the strangest coincidences of S(o) is with the actual dimensions of the Earth...radius = 3955 median miles. Since it has been shown that the Cheops constructs, 2/Pi , is a function of the Earth's radius:

2 / PI * ( 10^7 ) * 39.370078 = Earth's radius in inches = 3955.771795 miles

2 / Pi = Earth's radius / meter / ( 10^7 )

S(o) = (( 10 ^ ( Earth's radius / meter / ( 10^7 )) / 37 ) ^ -1 + 1 = 1.006239913

...thus it seems anything related to 2/Pi will derive S(o). A related phenomena is the Fermat form...e^ ( Pi +

...and the strong nuclear force a(s) = 14:

log( sqrt(( e ^ (( Pi +8 ) / 2 )) / a(s) )) = 1.999997681 / Pi

S(o)4 = (sqrt(( e ^ (( Pi + 8 )) / 2 )) / a(s) ) * 37 ) ^ -1 + 1 = 1.00623992316

Pi(S(o)4 = 3.141592682 .....99.9999991% of Pi

...S(o) has a link to the primal number 82944 or 288^2 , through the natural log e:

((( 10 ^ ( 2/Pi )) ^ ( S(o) ^ 6 )) + 82944 ) ^ ( 1/Pi ) = 100/e

36.7879441288 36.7879441171

...another equality involving the inverse decimal remainder of S(o):

S(o) = ( Pi / (( 2109 / 94 ) ^ 2 )) + 1 = 1.006240975

S(o) = ( 2109 / 94 / a(s) * ( 10^2 )) ^ -1 + 1 = 1.00623992413

....2109 / 94 is the inverse harmonic mean of the integers 37 and 57:

( 1/57 + 1/37 ) = 94 / 2109

...collective unconscious constant root .891891891... and S(o): .510998908 = electron energy

S(o) = ( 37 / 3030 * .510998908 ) + 1 = 1.006239920659

...the S(o) constant's flesh is the decimal remainder. But the kernel of the whole thing is the Cheops root ...10^(2/Pi) = 4.331315029

( 10 ^ ( 2/Pi )) * 18 = 10 ^ ( 70/37 ) ..............collective unconscious ..... ( 70.00001427 / 37 )

( 10 ^ ( 2/Pi )) / 18 = IN( sqrt phi ) ..............the golden number ........... ( 1.61810745 )

( 10 ^ ( 2/Pi )) * 37 = 1 / ( S(o) - 1 ) ..............quark resonance ................( 1.006239913 )

( 10 ^ ( 2/Pi )) /.37 = sqrt 137.035999701 ........electrical fine-structure .......( 137.0364491 )

J.Iuliano