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Raphie
#1
Nov6-10, 11:47 PM
P: 153
The Tetraktysal Kissing Triangle (TK_n)
Nickname: "The TetraKiss Triangle"
A Fibonacci, Lucas, Tetrahedral Convolution Construction for Lower Bounds of Sphere Packings to Dimension 10
Based Upon the Pythagorean Geometric Construct of the Tetraktys


Introduction to Kissing Numbers:
Kissing number problem
http://en.wikipedia.org/wiki/Kissing_number_problem

Highest known Kissing Numbers to Dimension 10 --> 2, 6, 12, 24, 40, 72, 126, 240, 306, 500



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SYMBOLIC BACKGROUND OF THE TETRAKTYS
... for Pythagoras, the number 10 was divine. 1, 2, 3 and 4 were also revered because they add up to 10 and they form the divine triangle - the Tetraktys - which symbolized the four elements (earth, air, fire, and water) and, in its totality, also the mystical fifth essence, the Quintessence.

The Tetraktys is an equilateral triangle composed of dots in four rows, a visual representation of the equation: 1 + 2 + 3 + 4 = 10. The Tetraktys contains a hexagon and a three-dimensional cube, as shown in the picture. It is a truly extraordinary figure. It also symbolises key musical intervals: 4:3 (the fourth), 3:2 (the fifth) and 2:1 (the octave). If the Tetraktys is extended by adding new rows, up to a total of 36, the 36th "triangular" number is 666: the Number of the Beast in the Christian Book of Revelation. The number 36 has a crucial significance for the Illuminati, as does the Tetraktys extended to order 36.

Image & Text from Armageddonconspiracy.co.uk
http://www.armageddonconspiracy.co.u...s(1540971).htm
--------------------------------------------

LEGEND
K_n --> Maximal known Kissing Number in n-dimensional space
L_n --> Lucas Number = phi^n - phi^-n(-1)^n
F_n --> Fibonacci Number = (L_(n - 1) + L_(n + 1))/5
G_n --> Golden Scale Number = F_(n-2) + F_(n-1) + F_(n) + F_(n+1) + F_(n+2)
T_n --> Triangular Number = (n^2 + n)/2!
Tetra_n --> Tetrahedral Number = ((n + 1)^3 - (n + 1))/3!

Point + Triangle + Hexagon
--> P' + T' + H'
--> 1 + 3 + 6
-->10

P'_n Union T'_n
= 2*F_(L_n + 1)^2


P_1 = 2*(F_(2 + 1))^2 = 2*(2)^2 = 8 --> F_6
T_1 = 2*(F_(1 + 1))^2 = 2*(1)^2 = 2 --> sqrt (L_0*G_0)
T_2 = 2*(F_(3 + 1))^2 = 2*(3)^2 = 18 --> L_6
T_3 = 2*(F_(4 + 1))^2 = 2*(5)^2 = 50 --> G_6

n = 0, 1, 2, 3

H'_n
= ((L_n + 1)*a) - 3*Tetra_m)*(-1)^(a+1)


H_1 = (02 + 1)*1)^2 - 3*Tetra_1)*(-1)^(1+1) = 3^2 - 03 = 6
H_2 = (01 + 1)*0)^2 - 3*Tetra_1)*(-1)^(0+1) = 0^2 + 03 = 3
H_3 = (03 + 1)*1)^2 - 3*Tetra_2)*(-1)^(1+1) = 4^2 - 12 = 4
H_4 = (04 + 1)*0)^2 - 3*Tetra_2)*(-1)^(0+1) = 0^2 + 12 = 12
H_5 = (07 + 1)*1)^2 - 3*Tetra_3)*(-1)^(1+1) = 6^2 - 30 = 30
H_6 = (11 + 1)*0)^2 - 3*Tetra_3)*(-1)^(0+1) = 0^2 + 30 = 34

n = 0, 1, 2, 3, 4, 5
m = ((2*n + 3) + (-1)^n)/4 = 1,1,2,2,3,3

{P} = 08
{T} = 02, 18, 50
{H} = 06, 03, 04, 12, 30, 34


Arrange as an ordered set...

02
03 04
06 08 12
18 30 34 50

Multiply by n...

01
02 03
04 05 06
07 08 09 10

=

002
006 012
024 040 072
126 240 306 500


=

K_01
K_02 K_03
K_04 K_05 K_06
K_07 K_08 K_09 K_10



Triangle Sums by Row
--------------------------------
000 000 000 002 --> 0002 = 2*(01^2 + 01) - 0 = 2*T_01 - 0
000 000 006 012 --> 0020 = 2*(04^2 + 04) - 0 = 2*T_04 - 0
000 024 040 072 --> 0156 = 2*(12^2 + 12) - 0 = 2*T_12 - 0
126 240 306 500 --> 1328 = 2*(36^2 + 36) - 4 = 2*T_36 - 4

for 0, 0, 0, 4 --> 4*Tetra_(n-2)

Where...

01 - 00 = 01 = |0^2 - 1| --> |F_0^2 - 1|
04 - 01 = 03 = |2^2 - 1| --> |F_3^2 - 1|
12 - 04 = 08 = |3^2 - 1| --> |F_4^2 - 1|
36 - 12 = 24 = |5^2 - 1| --> |F_5^2 - 1|

And where... 0,1,4 & 12 --> F_(2n + 1) - 1 for n = 1,2,3,4

0 = F_1 - 1
1 = F_3 - 1
4 = F_5 - 1
12 = F_7 - 1

Or, alternatively, where p_n denotes nth prime number...

Euler Phi (p_01) = 1
Euler Phi (p_03) = 4
Euler Phi (p_06) = 12
Euler Phi (p_12) = 36


For 1, 3, 6 and 12 are the number of vectors associated with Kissing Numbers of Dimension 1,2,3 & 4 (2, 6, 12 & 24)

Please note that some pretty interesting things happen when you sum the parts of this construction in an appropriate manner. More on that another time...

Related Threads:
One Alternate Look at the Periodic Table
http://www.physicsforums.com/showthread.php?t=439315
A Tetrahedral Counterpart to Ramanujan-Nagell Triangular Numbers?
http://www.physicsforums.com/showthread.php?t=443958

Looking forward to feedback and/or thoughts about how to take this description further.

Best,
Raphie


BACKGROUND

The above construction is a by-product of recent (and excellent) feedback given me by CRGreathouse whom, in general, I would like to thank for the time he has taken reviewing and responding to my past postings. In relation to his feedback, I was thinking about how to reasonably justify the manner in which I was presenting certain formulas and this is what I ended up with rather accidentally. I have a number of other exploratory constructions for Sphere-Packings based upon prime numbers, factorials, the binomial theorem, Pronic & Pentagonal Pyramid Numbers etc., but find this one to be not only the most elegant, but also the one that best ties in with other aspects of the long term project I am working on: Organic Symmetry: Explorations in Linking Lattices & Matrices to the Unification of Social & Physical Spaces

Please do keep in mind, the above is a "construction" aka a "model" and insofar as this is the case, it is not necessarily "wrong" or "right," but an observationally-based description that I believe is best judged in terms of simplicity, accuracy and economy. That said, what the above suggests to me, not in isolation, but heuristically, in tandem with multiple other observations not herein presented, is that... CONJECTURE: The highest known Kissing Numbers up to at least Dimension 9 will prove over time to be, in fact, the highest possible.

Highest known Kissing Numbers to Dimension 10 --> 2, 6, 12, 24, 40, 72, 126, 240, 306, 500

Related Papers:
Seven Staggering Sequences (pages 10 & 11)
www2.research.att.com/~njas/doc/g4g7.pdf
Kissing Numbers, Sphere Packings, and Some Unexpected Proofs
www.ams.org/notices/200408/fea-pfender.pdf

Also relevant...
ADE classification, McKay correspondence, and string theory
http://motls.blogspot.com/2006/05/ad...ion-mckay.html

It should be mentioned that my longstanding guess for highest possible sphere packing for Dimension 10, not justified within the context of this thread, has been that it will be found to be, not 500, but 504 --> floor [7!/10]. Other guesses for Highest Kissing Numbers to Dimension 15 ( --> 660, 1056, 1378, 1764, 3996 +/- 36 = 11*60, 12*88, 13*106, 14*126, 3996 +/- 36 = 660*6 = 3960 or 63^2 + 63 = 4032) are based on observed relationships between Factorials and the Distribution of the Mersenne Prime Exponents which I believe to be, more or less, in some manner yet to be understood, a Kissing Number equivalent to the "Ley Line" (or, perhaps more apropos in this case, "Lie Line") of Riemann Hypothesis fame.

From a value perspective, it should also be noted that If there were anything to the above from the standpoint of higher mathematics, the relationships presented above could suggest new approaches to the study of the Geometry of Lie Groups, not in isolation, but in relation to the manner in whichsuch groups in higher dimensions geometrically interact with one another.
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