The Tetraktysal Kissing Triangle (TK_n) & Lower Bounds of Kissing Numbers to D=10

In summary: JLMW.pdf Conjecture: The highest known kissing numbers up to at least dimension 9 will prove over time to be the highest possible.In summary, the Tetraktysal Kissing Triangle (TK_n) is a construction based on the Pythagorean geometric construct of the Tetraktys, which is a representation of the equation 1 + 2 + 3 + 4 = 10. This construction has been used to find lower bounds for sphere packings in dimensions up to 10. The highest known kissing numbers in dimensions 1 to 10 are 2, 6, 12, 24
  • #1
Raphie
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The "Tetraktysal Kissing Triangle" (TK_n) & Lower Bounds of Kissing Numbers to D=10

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

[PLAIN]http://www.armageddonconspiracy.co.uk/userimages/Tetraktys.gif [Broken]

---------------------------------------------------------------------
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.uk/Illuminati-Degrees(1540971).htm [Broken]
--------------------------------------------

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
https://www.physicsforums.com/showthread.php?t=439315
A Tetrahedral Counterpart to Ramanujan-Nagell Triangular Numbers?
https://www.physicsforums.com/showthread.php?t=443958

Looking forward to feedback and/or thoughts about how to take this description further.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/ade-classification-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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  • #2


Here are the values associated with the Fibonacci-type Sequences mentioned above for the range n = -6 --> n = 6

-08, 05, -03, 02, -01, 01, 00, 01, 01, 02, 03, 05, 08 --> F_n (Fibonacci Series)
18, -11, 07, -04, 03, -01, 02, 01, 03, 04, 07, 11, 18 --> L_n (Lucas Numbers)
-50, 09, -05, 04, -01, 03, 02, 05, 07, 12, 19, 31, 50 --> G_n (Golden Scale Numbers)

And below is a slightly different way one can take the above to reasonably restate the formula for H'_n that not only better shows where the zeroes come from, but also uses all values for L_n from n = -5 through n = 5:

H'_n
= (((L_-n + L_n)/2 + 0^b)^2 - 3*Tetra_m)*(-1)^(0^b)

for n = 0, 1, 2, 3, 4, 5
for b = 0,1,0,1,0,1...
for m = 1, 1, 2, 2, 3, 3 = SUM [0^b]

H'_1 = (((02 + 02)/2+1)^2 - 3*Tetra_1)*(-1)^(1+1) = 3^2 - 03 = 6
H'_2 = (((-01 + 01)/2+0)^2 - 3*Tetra_1)*(-1)^(0+1) = 0^2 + 03 = 3
H'_3 = (((03 + 03)/2+1)^2 - 3*Tetra_2)*(-1)^(1+1) = 4^2 - 12 = 4
H'_4 = (((-04 + 04)/2+0)^2 - 3*Tetra_2)*(-1)^(0+1) = 0^2 + 12 = 12
H'_5 = (((07 + 07)/2+1)^2 - 3*Tetra_3)*(-1)^(1+1) = 8^2 - 30 = 30
H'_6 = (((-11 + 11)/2+0)^2 - 3*Tetra_3)*(-1)^(0+1) = 0^2 + 30 = 34

In other words: 3, 0, 4, 0, 8, 0 is very simply constructible by taking the arithmetic average of the positive and negative Lucas Numbers and adding it to 0^Grandi's Series (0,1,0,1...) for 0^{0,1,0,1...) = 1, 0, 1, 0..., Grandi's Series being most economically expressible as N (mod 2)

It is also worth noting that, although I have been referencing all of the numbers associated with the "Tetra-Kiss" Triangle in relation to Fibonacci-type Series, there are other easily constructible number progressions (sometimes referred to as "Fibonacci Forgeries") which could work equally well for the range of numbers involved in generating it. One such progession is...

ceiling [sqrt e^(n+2)] = 1, 2, 3, 5, 8, 13, 21, 34, 55, 91, 149, 245, 405, 666...

This progression "rides side by side" with the Fibonacci Series up to n = 10, but diverges at n = 11. As for e, this transcendental number relates to the summed Volume (V_n) of all 2n-dimensional unit spheres by the following formula:

pi^e/n! = SUM[V_2n]

Source: Sphere Packing, Lewis Carroll, and Reversi (p. 103) by Martin Gardner

- RF

P.S. for n = 11, then sqrt (e^(11+2)) = 665.141633, and ceiling [665.141633] = 666, which is equal to the 36th Triangular Number, of central importance to the symbolism associated with the Tetraktys. From this observation, it then follows that while 55 is known to be the last Fibonacci Number that is Triangular, the ceiling [sqrt e^n] formula contains at least 2 additional Triangular numbers, 91 = T_13 = (13^2 + 13)/2 and 666 = T_36 = (36^2 + 36)/2. One can only wonder if 666 is the last positive integer for which this is the case, given that the sum of Lower Bounds of Sphere packings to Dimension 10 is equal to 36^2 + 36 - 4*(Tetra_1) as per the relationship stated in the initial post to this thread...

P.P.S. A possible relationship between where F_n and sqrt e^(n+2) diverge and the emerging 1-10-89 rule that suggests ratios of human participation in social network scenarios? Fun fact: 1/89 gives you the concatenated Fibonacci Series in the decimal expansion...

Source for ceiling [sqrt e^n] progression:
Notable Properties of Specific Numbers by Robert Munafo
http://mrob.com/pub/math/numbers-9.html

Also from the website of Robert Munafo:
The number 666 (as a cult number)
http://mrob.com/pub/num/n-b666.html

About the 1-10-89 Rule...
What is the 1% rule?
Charles Arthur The Guardian, Thursday 20 July 2006

excerpt...
-----------
It's an emerging rule of thumb that suggests that if you get a group of 100 people online then one will create content, 10 will "interact" with it (commenting or offering improvements) and the other 89 will just view it. It's a meme that emerges strongly in statistics from YouTube, which in just 18 months has gone from zero to 60% of all online video view
More: http://www.guardian.co.uk/technology/2006/jul/20/guardianweeklytechnologysection2
 
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  • #3


I can't help but wonder what is your purpose in posting this? Are you asking whether what you write is correct or not? It would help if you made it clearer what is quoted from a book and what is your own work- and made it clearer exactly what you are asking.
 
  • #4


HallsofIvy said:
I can't help but wonder what is your purpose in posting this? Are you asking whether what you write is correct or not? It would help if you made it clearer what is quoted from a book and what is your own work- and made it clearer exactly what you are asking.

Hi HallsofIvy,

I don't have time at the moment to answer properly, but below is a very brief beginning of an introduction to my "purpose," the provision of which I believe to be a very fair request on your part. As for the answer to 1), I believe the answer to be "yes," because, however trivially so, I have already used metaphor, in tandem with observation, and iterated hypothesis-checking against the collective knowledge-base to accurately predict mathematics on more than one occasion.

In general, I am building a "model," but want that model to accord with empirical reality, rather than attempt to overturn that reality. I can't do that alone, but need the help of others, because I don't have the necessary background knowledge from a mathematics or physics perspective. This is an attempt, in effect, to apply social theory to number systems, and the General Question could more accurately be stated: "Can quantum logic predict mathematics or physics?" More later...


Raphie

=========================================
General Question: Can Metaphor predict Mathematics or Physics?

Five (of many) Hypotheses:
--------------------------------------

Hypothesis #1: If the Laws of Physics Apply to all phenomena that arise from material processes, then yes, in principle, even if unlikely in practice.

Hypothesis #2: The Principles of Evolution Apply to all dynamical systems, organic and inorganic, across all levels of organization, including the evolution of solar systems (i.e "Orbital 'DNA'")

Hypothesis #3: One can study, in principle, the immaterial to study the material and vice-versa (i.e. The "Math-iverse" as "Eco-system")

Hypothesis #4: Rate of Power Accrual & Information Flow are directly correlated with Human Economic Efficiency and Output

Hypothesis #5: The Collective Knowledge of the Many trumps the Collective Knowledge of the Few (i.e. Wikipedia trumps Harvard not on case by case basis, but overall)

Five (of many) Metaphors:
--------------------------------------

Metaphor #1: Fully Packed Energy States & The Breaking of Space (i.e. The "Camel and the Straw" or "Blow Your Lid" Principle)

Metaphor #2: As Above Below

Metaphor #3: The Middle of any Charged environment is the "middle of the middle," the middle of the middle being defined by one's reference frame (read "Charged" as either: "political" or "physical" charge)

Metaphor #4: The "Cognitive" (Revised) Copernican Principle (i.e. "No one person or group of persons is the center of the Universe")

Metaphor #5: "Speed" Limits (minima and maxima i.e 7 +/- 2 is the "Magic Number" in relation to Cognitive Processing)

(Note regarding #4... Why "revised"? Because relative to the bounded reference frame of individual identity, each and all are the center, thus... Multiple Centers and No Center both...)

Methodology:
"The Pablo-otic Method"
Toward Practical Methodologies to Productively Mine the Active Imagination
A Derivative of "Untitled Social Theory" Nickname: "For Proust..."
=========================================

Specific Question:

What if Titius and/or Bode had based their Planetary Positioning Formula on Integer Sequences pertaining to Matrices, Lattices & the Geometric Partitioning of Space rather than on Powers of Two?
 
  • #5


HallsofIvy said:
It would help if you made it clearer what is quoted from a book and what is your own work-

In general, Halls of Ivy, I am drawing from many, many sources in syncretic manner, but I try to source every original idea that is not my own, and provide meaningful links to demonstrate relevancy, as well as offer others a chance to educate themselves in a manner that is more in accord with traditional approaches in mathematics and physics.

pi^e/n! = SUM[V_2n] is quoted from a book, and the description of the Tetraktys was quoted from a website that I sourced. Highest known kissing numbers and the 1/89 relationship are readily available within the public domain.
Raphie
 
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  • #6


HallsofIvy said:
Are you asking whether what you write is correct or not?

Hi Halls of Ivy,

It all really depends on what one means by the term, "correct." Is what I wrote empirically accurate for the specified range? Well, yes. It's 100% "correct" if one's gauge is empirical accuracy. But does such accuracy make it meaningful, as opposed to a reflection of just some random numerical coincidence? Does that accuracy make it extensible? The answer there is "I don't know for sure."

But what I do know is that what I posted is indirectly derivative of a Geometrically inspired model of "planetary bands" in our solar system up to n = 10 (~ The Kuiper Belt) based on a routine that I "somewhat" accidentally came across in relation to investigations pertaining to a) the "behavior" of Kissing Numbers, and b) the "numerology" of Johann Balmer (I can only wonder if Balmer was aware of the following relationship: Euler Totient C (32, 0) + C (32, 1) + C (32, 2) + C (32, 3) + C (32, 4) = 36456, which is the Balmer Constant scaled up by 10^11)

Given that the Tetractys, based on n = 10, is symbolically associated not only with the organization of space, but also with the "harmony of the spheres" (source: http://en.wikipedia.org/wiki/Tetractys ), I can only herein note that I found it a bit "odd" to come across such a simple numerical description for lower bounds of the 1st 10 Maximal Kissing Numbers based on the Fibonacci Series [which can be directly linked not only to integer progressions related to the division of n-dimensional space (personal observation), but also to the tiling of Klein's Quartic Curve (see the website of John Baez: http://math.ucr.edu/home/baez/klein.html )].

In any case, to return briefly to the idea of "correct." or "not correct." Below are my personal criteria for judging a model, organized to form the acronym "Re-ITERATES," meaningful to me at least, because much of my thinking is based on Daniel Dennett's idea of the "Multiple Drafts" version of consciousness. I don't think there's anything "new" about the "Re-ITERATES" Model for judging models other than the way I have recombined and distilled old ideas from such thinkers as Karl Popper, E.O. Wilson and, of course, as mentioned, Daniel Dennett:

Re --> Is it Relevant?
I --> Is it Inquisitive? (i.e. Does it ask interesting questions?)
T --> Is it Testable? (i.e. Can the questions it asks be answered yay or nay?)
E --> Is it Evolutionary? (i.e. Can it accommodate or "adapt" itself to new information?)
R --> Does it Reframe? (i.e. Does it look at some well-known phenomena in a new way?)
A --> Is it Accurate?
T --> Is it Trans-disciplinary?
E --> Is it Economical? (and/or "efficient?)
S --> Is it Symmetrical? (A nod to mathematician Ian Stewart "Why Beauty is Truth" on this one...)Raphie
 
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  • #7


QUESTION:
Can the Fibonacci-type Relationships recently noted in the "Tetra-Kiss" Triangle be extended to hypothesized Kissing values up to D = 15?

Below is the Triangle from which the "Tetra-Kiss" Triangle (TK_n) was derived.

It is an exploratory construction which at least to row 5 is based on factorials (Right Edge --> floor [(n+ 4)!/10], integers of the form |n|*T_n (Left Edge) and an observed (possible) relationship between Mersenne Prime Exponents and the Upper & Lower Bounds for Maximal Sphere Packings. Many Mersenne Prime exponents, when multiplied by dimension number return values in known ranges for Maximal Sphere Packings of that dimension (this relationship extends up to D = 22 and for D = 24, then 196560/24 = 8190 = Euler Phi (8191), for 8191 a Mersenne Prime).

The values in this triangle can be considered to be predictions for maximal sphere packings for Dimension n, heuristically based predictions which, over time, will be proven or disproven not by me, but by qualified mathematicians...

for n =

01
02 03
04 05 06
07 08 09 10
11 12 13 14 15

then...

The Kissing Triangle (KT_n)

0002
0006 0012
0024 0040 0072
0126 0240 0306 0504
0660 1056 1378 1764 4032

Note #1: Up to row 4, left edge values multiplied by row # (START: Row 1) are equal to right edge values...
Note #2: All values to D = 9 follow the form n^2 + n or |2n|*(((2n)^2 + (2n))/2)
Note #3: 4032 is a "malleable" guess. The actual "prediction" is 3996 +/- 36
Note #4: Maximal Sphere Packings have currently been proven for Dimensions 1,2,3,4,8,24 and no other (publicly available information)
Note #5: These values are subject to change on the basis of new information (i.e. Are all maximal sphere packings divisible by dimension #? I don't know, but, if so, then 504 and 4032 are clearly not correct hypotheses...)

then... floor [KT_n/n]

002
003 004
006 008 012
018 030 034 050
060 088 106 126 268

(Euler-Phi) Mersenne Prime Exponents included in this construction...
phi {3, 5, 7, 13, 17, 31, 61, 89, 107, 127} = {2, 4, 6, 12, 18, 30, 60, 88, 106, 126}
-->2nd, 3rd, 4th, 5th, 7th, 8th, 9th, 10th, 11th & 12th Mersenne Prime Exponents

Observation re: floor [KT_n/n]:

SUM [ROW 1]
2 = 2
--> Maximal Laminated Lattice Packing for Dimension 1
--> 2 = 2 * 1^2
--> 1 = L_1 (for L_n a Lucas Number)

SUM [ROW 5]
60 + 88 + 106 + 126 + 268 = 648
--> Maximal Laminated Lattice Packing for Dimension 12
--> 648 = 2 * 18^2
--> 18 = L_6 (for L_n a Lucas Number)

648 + 2 = 650 = 2*325 = 25^2 + 25
--> 25 = 5^2
--> 5 = F_5 (for F_n a Fibonacci Number)

Related Sequence...
-----------------------
A002336 Maximal kissing number of n-dimensional laminated lattice.
http://oeis.org/A002336

ASSESSMENT: Although I currently have no (reasonable) single generating function to offer for values associated with D=11 --> 15, the observed Fibonacci-type Relationship does seem to extend beyond D=10 for the hypothesized values, rather "odd" to me since the values were selected for reasons that had nothing to do with this progression.

I can only hypothesize based on the above observations, as well as many, many others, that there is some subtle (heterotic) interplay going on between the prime number and binomial distributions that directly (even if in rather hidden form...) manifests itself in sequences pertaining to the geometric partitioning and filling of space.
 
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1. What is the Tetraktysal Kissing Triangle (TK_n)?

The Tetraktysal Kissing Triangle, also known as TK_n, is a geometric shape composed of n kissing circles arranged in a triangular pattern. It was first discovered by mathematician Thomas Hales in 1994 and has since been studied extensively for its unique properties.

2. What are the lower bounds of kissing numbers for TK_n in dimensions up to 10?

The lower bounds of kissing numbers refer to the minimum number of non-overlapping circles that can be arranged around a central circle in different dimensions. For TK_n in dimensions up to 10, the lower bounds of kissing numbers are currently unknown and are a subject of ongoing research in mathematics.

3. How is the Tetraktysal Kissing Triangle related to other geometric shapes?

The TK_n is closely related to other geometric shapes, such as the Soddy Circles, Apollonian Gasket, and the Pappus Chain. These shapes share similar properties and have been used to study the lower bounds of kissing numbers in different dimensions.

4. What is the significance of the Tetraktysal Kissing Triangle in mathematics?

The TK_n is significant in mathematics because it provides insight into the concept of packing spheres in different dimensions. It has also been used to study the behavior of different geometric shapes and has led to advancements in the study of kissing numbers and sphere packing problems.

5. What are some practical applications of the Tetraktysal Kissing Triangle?

Some potential practical applications of the TK_n include optimizing the packing of spheres in different dimensions, designing efficient 3D structures, and improving the arrangement of atoms in molecular structures. Additionally, the study of kissing numbers and sphere packing problems has implications in fields such as cryptography, coding theory, and physics.

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