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#73
May1111, 04:36 PM

P: 205

very interesting patterns if I use a little recursion. Its in 3D best viewed in 1900*1200 resolution. Keep the mouse off of the page while it loads and it will plot in 2d first. source code is available. http://www.tubeglow.com/test/PL3D/P_Lattice_3D.html



#74
May1111, 05:57 PM

P: 153

Keep in mind, ala Marcus du Sautoy, that every time you add in a new prime you are, in effect, adding in another variable that will create "waves" and "ripples" interacting with all the other waves and ripples created by the other primes, meaning that the shapes you model will morph endlessly as you go further and further down the number line, or, rather, further out from the origin along the surface of your Minkowskiesque "prime lattice light cone." But, just because you know that it will "morph," this in no way excludes the possibility, even perhaps likelihood, of regularities in relation to the manner of "timing" by which new variables are introduced, because each and every new variable is recursively made possible only by the "multiplicative failure" of the primes that preceded it to fully "cover" "number space". Also keep in mind that variables equate with dimensions: Variable == Parameter == Dimension Best, RF 


#75
May1211, 08:46 PM

P: 205

edit: I attached a better image and its projection. this is a kind of root system right? http://en.wikipedia.org/wiki/Root_system http://upload.wikimedia.org/wikipedi...ystems.svg.png 


#76
May1211, 10:28 PM

P: 205

More "timing" links. Natural squares and partition numbers maybe? this is based on 12:
http://1.bp.blogspot.com/_u66d4_gsS...1600/roots.PNG 


#77
May1311, 03:41 AM

P: 153

And in relation to the number 12... n(n+ceiling(2^n/12)) http://oeis.org/A029929 The first 8, but only 8 numbers in this series are proven (lattice) kissing numbers.  RF RELATED LINK Leech lattice http://en.wikipedia.org/wiki/Leech_lattice The Leech lattice is also a 12dimensional lattice over the Eisenstein integers. This is known as the complex Leech lattice, and is isomorphic to the 24dimensional real Leech lattice... 


#78
May1311, 09:55 AM

P: 153

e.g. 40  24 = 4^2 40 + 24 = 4^3 One can use this mathematical fact to easily obtain integer solutions to the following: Period^2 = 4*pi^2/GM * Distance^3 (Kepler's 3rd Law) e.g. (40 + 24)^2 = (40  24)^3 = 4^6 = 4096 (6 + 2)^2 = (6  2)^3 = 2^6 = 64 (= sqrt 4096) The Pentagonal Pyramid numbers, of course, are the summation of the Pentagonal numbers, which are already wellknown to be related to the "timing" and/or "tuning" of the primes. p^2  1 == 1 mod (24) for all p > 3 (p^2  1)/24 is Pentagonal for all p > 3. And, also, as I mentioned previously, 24 s^2 is the Period^2 one obtains if one replaces L/g in the formula for a pendulum with zeta(2)^2 = (pi^2/6)^2, where (the reciprocal of) zeta(2) gives the probability of two randomly selected integers being relatively prime.  RF Note: Pentagonal Pyramid Numbers have a very easy to remember formula n*T_n = (+) Pentagonal Pyramid # and n*T_n = () Pentagonal Pyramid #, for T_n a Triangular Number. 


#79
May1411, 12:42 PM

P: 205

Raphie,
Thanks! I'm working on some projections.I definitly see an Eisenstein integer connection. edit: I'm working on the pendulum 


#80
May1411, 08:31 PM

P: 153

A060967 Number of prime squares <= 2^n. http://oeis.org/A060967 0, 0, 1, 1, 2, 3, 4, 5, 6, 8, 11, 14, 18, 24... RULE: SUBTRACT 1 1, 1, 0, 0, 1, 2, 3, 4, 5, 9, 10, 13, 17, 23... RULE: ITERATE INTO THE "QRIME" NUMBER SEQUENCE {0, 1 U Primes} indexed from 1; p'_(n1) 0, 0, 1, 1, 2, 3, 5, 7, 11, 17, 29, 41, 59, 83... RULE: ADD 1 1, 1, 2, 2, 3, 04, 06, 08, 12, 018, 030, 042, 060, 084... RULE: MULTIPLY BY (n2) 2, 1, 0, 2, 6, 12, 24, 40, 72, 126, 240, 378, 600, 924... In formula form... K_(n2) = (n2) * (1 + p'_(1 + COUNT[Number of prime squares <= 2^n])) for n = 2 > 10 The 3rd through 11th values are the (proven lattice) Kissing Numbers up to Dimension 8, the very same ones you get by inserting n into the formula: n(n+ceiling(2^n/12)). And the 2nd and 12th values? T_1^3 = 1 and T_3^3 = 378 (& 600 = 2*T_24, while 924 is a Central Binomial Coefficient, the sum of proper divisors of which = 1764 == 42^2) 378  totient (107) = 272 = K_9; 107 = p'_28 = p'_(T_7) = p'_(T_(Lucas_4) 001  totient (002) = 000 = K_0; 002 = p'_01 = p'_(T_1) = p'_(T_(Lucas_1) 2 is the 1st Mersenne Prime Exponent, and 107 the 11th (1 = Lucas_1, 11 = Lucas_5, and 1 and 28 are both kPerfect Numbers). These two numbers also have the property, that, when triangulated, you get a Lucas Number. There are only 3 Triangular Lucas Numbers: 1, 3 and 5778. (See: Lucas Number http://mathworld.wolfram.com/LucasNumber.html ) 0001 = T_001 = Lucas_01 = Lucas_(Lucas_1) = Lucas_(Lucas_(T_1)) = Lucas_(1*T_1) 5778 = T_107 = Lucas_18 = Lucas_(Lucas_6) = Lucas_(Lucas_(T_3)) = Lucas_(2*T_2) (1 and 6 are kPerfect Numbers, 1,6 & 18 are the first 3 Pentagonal Pyramid Numbers) Also, see.... k*Lucas_n + 1 is a prime of Lucas Number Index http://www.physicsforums.com/showthread.php?t=497766  RF 


#81
May1411, 10:15 PM

P: 153

A POSSIBLY RELATED SEQUENCE Suppose the sum of the digits of prime(n) and prime(n+1) divides prime(n) + prime(n+1). Sequence gives prime(n). http://oeis.org/A127272 2, 3, 5, 7, 11, 17, 29, 41, 43, 71, 79, 97, 101, 107... e.g. (2 + 3)/(2+3) = 1 (3+5)/(3+5) = 1 (5+7)/(5+7) = 1 (7+11)/(7+(1+1)) = 2 (11+13/((1+1)+(1+3)) = 4 (17+19/((1+7)+(1+9)) = 2 (29+31/((2+9)+(3+1)) = 4 (41+43/((4+1) + (4+3)) = 7 (43+47/((4+3)+(4+7)) = 5 (71+73)/((7+1)+(7+3)) = 8 (79+83)/((7+9)+(9+7)) = 5 (97+101)/((9+7)+(1+0+1)) = 11 (101+103)/((1+0+1) + (1+0+3) = 34 (107+109)/((1+0+7)+(1+0+9) = 12 ALSO... Numbers n such that 1 plus the sum of the first n primes is divisible by n+1. http://oeis.org/A158682 2, 6, 224, 486, 734, 50046, 142834, 170208, 249654, 316585342, 374788042, 2460457826, 2803329304, 6860334656, 65397031524, 78658228038 002  002 = 000 = K_00 012  006 = 006 = K_02 (Max) 600  224 = 336 = K_10 (Lattice Max known) 924  486 = 438 = K_11 (Lattice Max known) 6/(5+1) = 1 42/(6+1) = 6 143100/(224+1) = 636 775304/(486+1) = 1592 Like I said, especially given that these two progressions are ones I came across in the process of writing that last post to you, "hmmmm..." RELATED PROGRESSIONS Integer averages of first n noncomposites for some n. http://oeis.org/A179860 1, 2, 6, 636, 1592, 2574, 292656, 917042, 1108972, 1678508, 3334890730, 3981285760, 28567166356, 32739591796, 83332116034 a(n) is the sum of the first A179859(n) noncomposites. http://oeis.org/A179861 1, 6, 42, 143100, 775304, 1891890, 14646554832, 130985694070, 188757015148, 419047914740, 1055777525624570390, 1492138298614167680, 70288308055831268412, 91779857115464381780, 571686203669195590338 Numbers n that divide the sum of the first n noncomposites. http://oeis.org/A179859 1, 3, 7, 225, 487, 735, 50047, 142835, 170209, 249655, 316585343, 374788043, 2460457827, 2803329305, 6860334657 This number, in particular, I find interesting... 142835 = 5*7^2*11*53 = (142857  par_8) = (142857  22) vs. 1/7 = .142857 (repeating) Indexing from 0, 142857 is the 24th Kaprekar Number 1, 3, 7 and 225, the 1st 4 terms in that last sequence above == (2^1  1)^1, (2^2  1)^1, (2^3  1)^1, (2^4  1)^2.  RF 


#82
May1411, 11:18 PM

P: 153

Jeremy, as an FYI, and by way of giving another example, if one desires to mathematically derive, say, the Dimension 10 Lattice Kissing Number from a convolution of primes and partition numbers, a far simpler way to do it is as follows:
p'_((par_n  1) * p'_(par_(n1)  1) p'_(11) 1 = 0 p'_(11) 1 = 0 p'_(21) 1 = 1 p'_(31) 1 = 2 p'_(51) 1 = 6 p'_(71) 1 = 12 p'_(111) 1 = 28 0*0 = 0 = K_0 0*1 = 0 = K_0 1*2 = 2 = K_1 2*6 = 12 = K_3 6*12 = 72 = K_6 12*28 = 336 = K_10 In order, that formula returns maximal (proven except for Dimension 10) lattice sphere packings for Dimensions equal to 6 consecutive Triangular Numbers: T_1, T_0, T_1, T_2, T_3, T_4 On the other hand, if you simply add 1 to the first 7 partition numbers, and multiply by n... (1+1)*0 = 0 = K_0 (1+1)*1 = 2 = K_1 (2+1)*2 = 6 = K_2 (3+1)*3 = 12 = K_3 (5+1)*4 = 24 = K_4 (7+1)*5 = 40 = K_5 (11+1)*6 = 72 = K_6 ... then you get Maximal (proven) Lattice Sphere packings to dimension 6. Best, RF 


#83
May1511, 10:48 AM

P: 205

I'm going through you posts now. I reworked the visual a little. Click any where on the page after it loads the black back ground then press:
1 = normal growth of the equation. after it builds for a while you can notice the pattern and the timing. Seems to be timed like a pendulum. or 2 = normal "inverse growth". or 3 = fractal pattern generation up/down arrrow = zoom in out left/right arrow = fractal limit increase/decrease. d = 3d on/off http://www.tubeglow.com/test/PL3D2/P_Lattice_3D_2.html 


#84
May1611, 10:17 AM

P: 153

Jeremy, firstly, the page you linked to doesn't seem to work with my system.
Secondly, I wouldn't read too much in to any single example I might give. It's all of the examples, taken together, and the picture they are seeming to paint (or the tune they are seeming to play) that I find most interesting. Thirdly, a critic would reasonably note that the indices of the prime numbers I am giving are all quite small. And that's a fair point. But then one has to explain away as "coincidence" relationships such as the following: For 1, 2, 3, 4 and 6 the solutions to the Crystallographic Restriction Theorem, then consider lattices in the following Dimensions: (1  1)^2 + 1  totient (1) = 0 (2  1)^2 + 1  totient (2) = 1 (3  1)^2 + 1  totient (3) = 3 (4  1)^2 + 1  totient (4) = 8 (6  1)^2 + 1  totient (6) = 24 Dimensions {0 & 24} Union {1, 3, 8}, the dimensions associated with the Standard Model of Physics = SU(3)×SU(2)×U(1) Then, for F_n a Fibonacci Number and T_n a Triangular Number... And for... 2, 4, 6, 10, 22 == totient (1st 5 safe "qrimes") == 2 * (1, 2, 3, 5, 11) where... 01 = p'_(1  1) = par_1 02 = p'_(2  1) = par_2 03 = p'_(3  1) = par_3 05 = p'_(4  1) = par_4 11 = p'_(6  1) = par_6 Then... p_00001  p_01 = p_F_02  p_((F_0)*(T_(pi(pi(01) + 1))) + 1) = 000002  002 = 000002 = K_0 p_00003  p_02 = p_F_04  p_((F_1)*(T_(pi(pi(02) + 1))) + 1) = 000005  003 = 000002 = K_1 p_00008  p_04 = p_F_06  p_((F_2)*(T_(pi(pi(03) + 1))) + 1) = 000019  007 = 000012 = K_3 p_00055  p_07 = p_F_10  p_((F_3)*(T_(pi(pi(05) + 1))) + 1) = 000257  017 = 000240 = K_8 p_17711  p_31 = p_F_22  p_((F_4)*(T_(pi(pi(13) + 1))) + 1) = 196687  127 = 196560 = K_24 Note: 1, 2, 3, 5 & 13 are the Prime Numbers  (2^n  1) is Twice Triangular (aka "The RamanujanNagell Pronic Numbers"). And 2, 3, 5, 17 and 257 are all Fermat Primes, while 2, 3, 7, 17 and 127 (and also 19) are all Mersenne Prime Exponents, the 1st, 2nd, 4th, 6th and 12th (19 is the 7th). p'_1  1 = 02  1 = 1 p'_2  1 = 03  1 = 2 p'_3  1 = 05  1 = 4 p'_4  1 = 07  1 = 6 p'_6  1 = 13  1 = 12 The condensed way to state the above is as follows:  for... K_n = nth Kissing Number p'_(n1) = nth n in N  1 < d(n) < 3 > {0,1,2,3,5,7,11,13...} c_(n1) = nth n in N  1 < totient(n) < 3 > {0,1,2,3,4,6} E_n = nth Mersenne Prime Exponent F_n = nth Fibonacci Number then for range n = 0 > 4... FORMULA K_((c  1)^2 + 1  totient (c)) = (p'_(F_(2(p'_(c  1)))))  (E_(p'_c  1))  2, 3, 5, 7, 13 [= {p_c} == {n in N  d(p_c  1) = c}], as well as being the first 5 Mersenne Prime Exponents, are also the unique prime divisors of the Leech Lattice: K_24 = 196560 And, as I believe you may already know, this particular set of primes has been associated with anomaly cancellations in 26 Dimensional Bosonic String Theory by Frampton and Kephart: Mersenne Primes, Polygonal Anomalies and String Theory Classification http://arxiv.org/abs/hepth/9904212 Best, RF ============================================ Also... 00 = p'_1 = (1  1)  01 = p'_(p'_1) = (2  1) 02 = p'_(p'_(p'_1)) = (3  1) 03 = p'_(p'_(p'_(p'_1))) = (4  1) 05 = p'_(p'_(p'_(p'_(p'_1)))) = (6  1) 11 = p'_(p'_(p'_(p'_(p'_(p'_1))))) = (12  1) for 1, 2, 3, 4, 6, 12 > the divisors of 12 And also... (01 * 0) + 3  d(01) = 02 = p'_01 > 01st Mersenne Prime Exponent (02 * 2) + 3  d(02) = 05 = p'_03 > 03rd Mersenne Prime Exponent (03 * 4) + 3  d(03) = 13 = p'_06 > 05th Mersenne Prime Exponent (05 * 6) + 3  d(05) = 31 = p'_11 > 08th Mersenne Prime Exponent (11 * 8) + 3  d(11) = 89 = p'_24 > 10th Mersenne Prime Exponent for 1, 3, 5, 8, 10 > Sum of Divisors (SUM d(n)) for n = 1 through 5 Here are the 1st 14 Mersenne Prime Exponents (inclusive of 1)... 1, 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521 (Range = Lucas_1 > Lucas_13) As you may or may not have noticed, in the last few posts I've referenced every one of these excepting 61 and 521 (= Lucas_13 = Lucas_(p'_(sigma_5)), indexing from 0, the 13th Mersenne Prime Exponent). ((611)*11) = totient (p'_11^2) = 660 = (T_36  sqrt (36)), by the way, is a simple group that, musically speaking, is oneperfect 5th above A440 and its my prediction for the maximal Kissing Number in 11 dimensions (+ or  12). "Coincidentally," 660  12 = 648, is the maximal known lattice sphere packing in 12 dimensions (= 2*18^2 = Lucas_0*Lucas_(sigma_(5))^2 = p'_(Lucas_5)^2  p'_(sigma_5)), while 36 (=2*18) is the totient of the 12th prime number, 37. Finally, in the interests of clarity, note that the below are all just absurdly longwinded, even if contextually relevant, ways of stating: 0, 1, 2, 2, 4: (pi(pi(01) + 1)) == d(pi (01)); 01 > (11)th Mersenne Prime Exp == pi (pi (02)); 02 > 1st Mersenne Prime Exp = 0 (pi(pi(02) + 1)) == d(pi (02)); 02 > (21)th Mersenne Prime Exp == pi (pi (03)); 03 > 2nd Mersenne Prime Exp = 1 (pi(pi(03) + 1)) == d(pi (03)); 03 > (31)th Mersenne Prime Exp == pi (pi (05)); 05 > 3rd Mersenne Prime Exp = 2 (pi(pi(05) + 1)) == d(pi (05)); 05 > (41)th Mersenne Prime Exp == pi (pi (07)); 07 > 4th Mersenne Prime Exp = 2 (pi(pi(13) + 1)) == d(pi (13)); 13 > (61)th Mersenne Prime Exp == pi (pi (17)); 17 > 6th Mersenne Prime Exp = 4 And finally, finally... a statement such as p_((F_4)*(T_(pi(pi(13) + 1))) + 1) is an even more longwinded way of stating: p_31, which is the 12th Mersenne Prime Exponent and/or the iterated 8th Mersenne Prime Exponent. As such, I went back in the post and included the condensed formula... 


#85
May1611, 08:07 PM

P: 153

for... c_(n1) > {0,1,2,3,4,6} == pi {1, 2, 3, 5, 7, 13} > Divisors of Prime Divisors of Leech Lattice > Integers with Totient < 3 > n in N  d(p_c  1) = c > Solutions to 2*cos (2*pi/(n + 1  sgn(n)) is in N E_(n1) > {0,1,2,3,5,7,13,17,19...} > 0, 1 Union Mersenne Prime Exponents then... K_((c_n  1)^2 + 1  totient (c_n)) = (2^E_(c_n  1))  2)*((E_(c_n  1)  1)*totient (c_(n1))) Expansion: K_00 = (2^01  2) * ((01  1)*totient (0)) = 0000 * 00 = 0 K_01 = (2^02  2) * ((02  1)*totient (1)) = 0002 * 01 = 2 K_03 = (2^03  2) * ((03  1)*totient (2)) = 0006 * 02 = 12 K_08 = (2^05  2) * ((05  1)*totient (3)) = 0030 * 08 = 240 K_24 = (2^13  2) * ((13  1)*totient (4)) = 8190 * 24 = 196560  POSSIBLY RELATED PROGRESSION y such that y^2=C(x,0)+C(x,1)+C(x,2)+C(x,3) is soluble 0, 1, 2, 8, 24, 260, 8672 R. K. Guy, Unsolved Problems in Number Theory, Section D3. http://oeis.org/A047695  c_n & E_(c_n  1)  1) can be linked in the following manner: Denote iphi(x)_n as the nth integers with a totient of x (The "Inverted Totient Function") Denote s(x) as the number of Solutions to iphi(x) Denote J_n as the y solutions to 2^y  1 is Triangular (RamanujanNagell Triangular Numbers) Then... iphi(J) >  iphi(00) > 00;  Solutions = 1 (Mathematica Definition) iphi(01) > 01 02;  Solutions = 2 iphi(02) > 03 04 06;  Solutions = 3 iphi(04) > 05 08 10 12;  Solutions = 4 iphi(12) > 13 21 26 28 36 42;  Solutions = 6 00 = 2T_pi(01) = 2T_d(011) = 2T_0 02 = 2T_pi(02) = 2T_d(021) = 2T_1 06 = 2T_pi(03) = 2T_d(031) = 2T_2 12 = 2T_pi(05) = 2T_d(051) = 2T_3 42 = 2T_pi(13) = 2T_d(131) = 2T_6 And right there you've got yourself, potentially, a nice clean bridge between (just for starters...), RamanujanNagell, the Solutions to the Crystallographic Restriction Theorem and the Divisors of the Leech Lattice/FramptonKephart Primes. s(J) = c iphi(J)_1 = (J1) + totient (c) iphi(J)_c = 2T_(d(J)) iphi(J)_c = 2T_(pi(J+1)) and... Delta (iphi(J)_1, iphi(J)_c) = p'_(J  2) = 0, 1, 3, 7, 29 Thus, for instance... K_(totient(s(J))J) = (2^J + 1)*(totient(s(J))J) K_00 = (2^(00+1)  2) * (00*1) = 0 K_01 = (2^(01+1)  2) * (01*1) = 2 K_04 = (2^(02+1)  2) * (02*2) = 24 K_08 = (2^(04+1)  2) * (04*2) = 240 K_24 = (2^(12+1)  2) * (12*2) = 196560 Hard to get more simple than that. With the nice little bonus that... pi (2^01) = 0001 = 2^00 + 0 = 2^(T_1) + 0 pi (2^02) = 0002 = 2^00 + 1 = 2^(T_0) + 1 pi (2^03) = 0004 = 2^01 + 2 = 2^(T_1) + 2 pi (2^05) = 0011 = 2^03 + 3 = 2^(T_2) + 3 pi (2^13) = 1028 = 2^10 + 4 = 2^(T_4) + 4 Thus... pi (2^J+1) = 2^(T_(s(J)  2)) + n If that and the other relationships presented in this post don't make you and/or anyone else who comes across this go "hmmmm," then I really don't know what will. Go back to the beginning of this post and you'll see you can make both formulas relating to the previous post far, far simpler by substitution. But it still doesn't change the nature of the relationships. Rather, all that changes is the apparent simplicity of the relationships. e.g. K_(n(mod 5)+0) = (s(J)_(n1) + 0*6)(n+0)^0 = 00, 02, 06, 012, 024 K_(n(mod 5)+4) = (s(J)_(n1) + 1*6)(n+4)^1 = 24, 40, 72, 126, 240 Best, RF 


#86
May1611, 10:36 PM

P: 153

In relation to the Richard Guy sequence posted above, ...
0^2, 1^2, 2^2, 8^2, 24^2, 260^2, 8672^2 = 0, 1, 4, 64, 576, 67600, 75203584 These squares correspond with Cake Numbers of index 1, 0, 2, 7, 15, 74 & 767 Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3)+n+1. http://oeis.org/A000125 [0], 1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, 299, 378, 470, 576, 697, 834, 988, 1160, 1351, 1562, 1794, 2048, 2325, 2626, 2952, 3304, 3683, 4090, 4526, 4992, 5489, 6018, 6580, 7176, 7807, 8474, 9178, 9920, 10701, 11522, 12384, 13288, 14235, 15226... I know this set of numbers well... e.g. Product [Cake_n  0!/Cake_n  1!] = 1, 1, 2, 8, 64, 960, 24960, 1048320, 67092480... 00000000!/00000000  1! = 0000^2  d(0000) + 1; 0000 = M_00 00000001!/00000001  1! = 0001^2  d(0001) + 1; 0001 = M_01 00000008!/00000008  1! = 0003^2  d(0003) + 1; 0003 = M_02 00000960!/00000960  1! = 0031^2  d(0031) + 1; 0031 = M_05 67092480!/67092480  1! = 8191^2  d(8191) + 1; 8191 = M_13  RF 


#87
May1711, 12:27 PM

P: 153

Jeremy, I just mentioned the number 67092480. I mention it because you are looking at pendulums in relation to the primes and you're interested in gravity. 67092480 = (90^2 + 90)^2 + 2*(90^2 + 90) [follows the form of a parabola] is one of the key terms in the below formula which involves just one sign change to go from A to B. If only to protect myself from charges of "numerology," think of it as "sudoku," not physics, because the numbers are all based on lattices (and primes) and on the very "what if" proposition of "what if fractals were to guide the evolution of all dynamical systems, organic and inorganic alike, across all levels of organization from the very, very small to the very, very large (i.e. what if it were possible to develop an atomic model of the solar system as opposed to a planetary model of the atom?...)
For... s(1 + 2 + 3 + 5 + 13) = s(24) = 10 Solutions iphi(24) = 35, 39, 45, 52, 56, 70, 72, 78, 84, 90 Range: 55 = T_10 = F_10 = ceiling [e^((102)/2)] A) The Gravitational Constant ((((4*pi^2)^(1(1)^0)*pi)/0.007297352570631)((2*10^34/10^(12 + (9*0)))*(35 + sqrt (35)/10^4))/(((67092480  1)^(2+0)/sqrt (67092480 + 1)^(2+0))*(299800649  sqrt (67092480 + 1))))^(1/1) = 6.67428281 * 10^11 B) The ~ Planetary Positioning Ratio @ n = 10 ((((4*pi^2)^(1(1)^1)*pi)/0.007297352570631)((2*10^34/10^(12 + (9*1)))*(35 + sqrt (35)/10^4))/(((67092480  1)^(2+1)/sqrt (67092480 + 1)^(2+1))*(299800649  sqrt (67092480 + 1))))^(1/2) = 1.68845301 EMPIRICAL PLANETARY POSITIONING RATIO @ n = 10 Source for Values: Wikipedia Planet Pages (with Asteroid Belt set at 2.816 AU, which yields minimal possible value) ((394.8165/301.0366) + (301.0366/192.2941) + (192.2941/ 95.8202) + (95.8202/52.0427) + (52.0427/28.16) + (28.16/15.2368) + (15.2368/10) + (10/7.2333) + (7.2333/3.8710))/ 9 ~ 1.68845075 39.48165 + 30.10366 + 19.22941 + 9.58202 + 5.20427 + 2.816 + 1.52368 + 10 + .72333 + .38710 = 110.05112 floor[110.05112] = 2*T_10 = 2*55 The numbers in that equation are not at all "random." For instance... 299800649 = (G_(240) * L_(24+0))/10^2 ~ (G_(246) * L_(24+6))/10^2 = (289154*103682)/100 ~ (16114 * 1860498)/10^2 for G the Golden Scale (sum of 5 consecutive Fibonacci Numbers) and L the Lucas Series Delta ((G_(n0) * L_(n+0)), (G_(n6) * L_(n+6))) = 256 == totient(p_55) == totient(p_(T_10)) 6 == K_2 == L for pendulum set to Zeta(2)^1 = L/g 24 ==K_4 == T^2 for pendulum set to Zeta(2)^1 = L/g (67092480  1)/sqrt (67092480 + 1) = 8190.99976... (10^34*(9.10938215*10^31))/(10^34*(6.626067758602965*10^34/(2*pi)))^2 = 8190.99976... floor [299800649  8190.99976] = 299792458 Best, RF Note: 7245 = iphi(24)_7  iphi(24)_3 = 27. 27 and 45 are the roots of E_6 56+70 = iphi(24)_5 + iphi(24)_6 = 126 = K_7. These are the roots of E_7... ADE Classification http://en.wikipedia.org/wiki/ADE_classification RELATED LINK A Cute Formula for Pi http://www.physicsforums.com/showthread.php?t=475539 


#88
May1711, 03:15 PM

P: 205

Raphie,
Your “what if” fractal scenario is exactly what I’ve been thinking for a while. To me, your posts have shown some amazing connections, I only wish I had the knowledge and insight to give you meaningful feedback other than, “ Holy S@#! Yea, I see the connection now!” I feel like I’m in a crash course in number theory and I love it, I only wish I could contribute more. I’m still reviewing the depth of your latest posts. I think the link I posted was down for a while. It seems to be working on several different systems now. I would really like you to take a look at it if it’s working for you. As regards the “pendulums in relation to the primes” and my interest in gravity, I see some interesting results I think. After the page loads the black background, press “1” and let that run for 10 sec or so, a pendulum type motion starts to become apparent. Now press “2”. It “flips” the equation so it’s contracting instead of expanding. It resembles, to me, a “ball in a cone” . Other functions again: 1 = Expansion 2 = Contraction 3 = Recursion (Reload page first) UP/DOWN = ZOOM IN/OUT LEFT/RIGHT = RECURSION LIMIT. Default limit of 2 is loaded. Increasing the limit yields some familiar patterns. d = 3D ON/OFF http://www.tubeglow.com/test/PL3D2/P_Lattice_3D_2.html 


#89
May1911, 03:35 PM

P: 153

As for crash courses in "number theory." I wouldn't really call it that. If anything, you're getting a crash course in applied social theory. The little ditty that guides many of my mathematical explorations? Where e and F ride side by side, the orbits may not collide. Where they diverge clear order may not emerge. But the source of that ditty isn't my interest in physics or mathematics, but rather my interest in human cognition. Recent research indicates that both e and the Golden Ratio seem to play a role. Combine that with the rule of 7 +/ 2 (Miller, 1956) as well as Sir Roger Penrose's hypothesis that the mathematics of quasicrystallization can be applied to human brain plasticity, and then ask yourself the question: Is humankind distinct from nature, or just one small, however amazing, sliver of nature? If not distinct, then did nature apply one set of rules to us and another to the rest of nature? If no, then in principle, humankind and all of it's products, material and immaterial alike, from buildings to social networks, properly become (hard) scientific objects of study. And the rules that apply to the physical world, may also, to varying degrees of efficacy, be applied to the immaterial world and viceversa. This isn't a new idea. Durkheim, the Sociology equivalent of Albert Einstein in many respects, was saying the same thing, more or less, about a hundred years ago: Man is not an empire within an empire  Emile Durkheim (Elementary Forms of Religious Life) And E.O .Wilson, evolutionary biologist and author of "Consilience" has been saying the same thing for years now to anyone who will listen. Rather insanely, and more than a little baffling to me, I have been censored on more than one occasion by scientists (quite well educated and well meaning ones at that...) to whom such thoughts are heresy, a criminal offense tantamount to suggesting that the earth is round (back in the days before we became "enlightened"). This, even as higher maths are being used to demonstrate the evolutionary basis of Cooperation... Nice Guys Finish First by David Brooks May 16, 2011 The New York Times http://www.nytimes.com/2011/05/17/opinion/17brooks.html Oh, and the metaphor I employ? E = PI^2 It's my evolutionary adaptation of e = mc^2. Just substitute "Power" for m and "Information" for c. As a hypothetical, exploratory construct, P maps to Power Laws (related to e) and I maps to optimal flow of information (related to phi). Where P or I equals 0, then E, effectively, equals 0 (there's actually another part of e = mc^2 most people aren't familiar with, which is why a photon has no mass, but does have energy...), leading to the following statement most activists are quite familiar with: SILENCE = DEATH  RF RELATED PAPER Period Concatenation Underlies Interactions between Gamma and Beta Rhythms in Neocortex Roopun, Kramer et al. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2525927/ RECOMMENDED "The Blank Slate: The Modern Denial of Human Nature " by Steven Pinker. http://www.amazon.com/BlankSlateMo.../dp/0142003344 A few other thinkers of interest: Carl Jung, Sigmund Freud, Daniel Dennett, Richard Dawkins, Albert Laszlo Barabasi, Clay Shirky, Duncan Watts... 


#90
May2011, 07:01 PM

P: 153

Tying together a few seemingly disparate concepts for you Jeremy...
PROBABILITY  x^2 + 2*(a*x) + a^2 = 1 Heads or tails... .5^2 + 2*(.5*.5) + .5^2 = 1 Pareto's Law... .8^2 + 2*(.8*.2) + .2^2 = 1 PALINDROMIC FORM  sqrt (10^2*7^2 + 10^1*2*(1*7) + 10^0*(1)^2) = 71 sqrt (10^0*7^2 + 10^1*2*(1*7) + 10^2*(1)^2) = 17 17 + 71 = 88 = 2*T_9  2 sqrt (10^0*9^2 + 10^1*2*(1*9) + 10^2*(1)^2) = 19 = sqrt (0081 + 180 + 100) sqrt (10^2*9^2 + 10^1*2*(1*9) + 10^0*(2)^2) = 91 = sqrt (8100 + 180 + 001) 19 + 91 = 110 = 2*T_10 PALINDROMIC FORM w/ "Interference"  sqrt (10^0*11^2 + 10^1*2*(1*11) + 10^2*(1)^2) = 021 = sqrt (00121 + 220 + 100) sqrt (10^2*11^2 + 10^1*2*(1*11) + 10^0*(1)^2) = 111 = sqrt (12100 + 220 + 001) 21 + 111 = 132 = 2*T_11 sqrt (10^0*13^2 + 10^1*2*(1*13) + 10^2*(1)^2) = 023 sqrt (10^2*13^2 + 10^1*2*(1*13) + 10^1*(1)^2) = 131 23 + 131 = 154 = 2*T_12  2 form: (square root of...) Parabolic Cyllinder + (square root of...) Parabolic Cyllinder Sum of Central terms for 10,1 & 11,1 = 180 + 180 = 360 (> Fundamental Domain of the Crystallographic Restriction Theorem) = 220 + 220 = 440 (> "Arbitrary" Reference frame for the Western Musical Scale = A440 = 8*T_10 = 8*F_10 = 8*Ceiling [sqrt e^8]) (x + y)^2  (9^2 + 2*(1*9) + 1^2) = 100 (1^2 + 2*(9*1) + 9^2) = 100 etc... THE INVENTOR OF THE PENDULUM CLOCK & EXPECTED VALUE  Via Wikipedia... Excerpt # 1 The idea of the expected value originated in the middle of the 17th century from the study of the socalled problem of points. This problem is: how to divide the stakes in a fair way between two players who have to end their game before it's properly finished? Excerpt # 2 Three years later, in 1657, a Dutch mathematician Christiaan Huygens, who had just visited Paris, published a treatise (see Huygens (1657)) "De ratiociniis in ludo aleæ" on probability theory. In this book he considered the problem of points and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens also extended the concept of expectation by adding rules for how to calculate expectations in more complicated situations than the original problem (e.g., for three or more players). In this sense this book can be seen as the first successful attempt of laying down the foundations of the theory of probability. http://en.wikipedia.org/wiki/Expected_value Christiaan Huygens Christiaan Huygens... was a prominent Dutch mathematician, astronomer, physicist and horologist. His work included early telescopic studies elucidating the nature of the rings of Saturn and the discovery of its moon Titan, the invention of the pendulum clock and other investigations in timekeeping, and studies of both optics and the centrifugal force. http://en.wikipedia.org/wiki/Christiaan_Huygens Huygens achieved note for his argument that light consists of waves,[1] now known as the Huygens–Fresnel principle, which two centuries later became instrumental in the understanding of waveparticle duality.  RF 


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