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Klein four-group geometry

  1. Apr 30, 2012 #1
    I think this a map of the Klein four-group geometry. Comments?
    http://dl.dropbox.com/u/13155084/prime-%20square-klein%20four%20group.png [Broken]
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. May 1, 2012 #2
    I'm working on the explination.
    http://dl.dropbox.com/u/13155084/Given%20a%20divisor%20k.pdf [Broken]
    Last edited by a moderator: May 6, 2017
  4. May 1, 2012 #3
    A better visual.
    http://dl.dropbox.com/u/13155084/prime-%20square-klein%20four%20group.png [Broken]
    Last edited by a moderator: May 6, 2017
  5. May 4, 2012 #4
    Maybe its more than the Klein four group.

    "The divisors of 24 — namely, {1, 2, 3, 4, 6, 8, 12, 24} — are exactly those n for which every invertible element x of the commutative ring Z/nZ satisfies x^2 = 1.
    Thus the multiplicative group (Z/24Z)× = {±1, ±5, ±7, ±11} is isomorphic to the additive group (Z/2Z)3. This fact plays a role in monstrous moonshine."

    I checked. Only the divisors of 24 produce the specific symmetry such that:

    s+k =(n+k^2)/(2k)
    (s+k)^2 - s^2 = n

    (s+k)^2 = {1,0}
    s^2 = {0,1}
    n = {1,-1}

    (s+k)^2 = {1,0}
    s^2 = {0,1}
    n = {1,2}

    (s+k)^2 = {1,0}
    s^2 = {0,1}
    n = {1,3}

    (s+k)^2 = {1,0}
    s^2 = {0,1}
    n = {1,5}

    (s+k)^2 = {1,4,1,0}
    s^2 = {0,1,4,1}
    n = {1,3,5,7}

    (s+k)^2 = {1,9,4,0}
    s^2 = {0,4,9,1}
    n = {1,5,7,11}

    (s+k)^2 = {01,09,16,12,01,09,04,00}
    s^2 = {00,04,09,01,12,16,09,01}
    n = {01,05,07,11,13,17,19,23}

    (s+k)^2 + s^2 = 1 (mod 12) ** only modulo 12. ???? additive inversion of the identity element to its multiplication group Z/12Z*????
    ((s+k)^2 - s^2)^2 = 1 (mod 12)

    In the Klein Four-Group the quadratic residues of (s+k)^2 and s^2 are permutations of the symmetric group of order 4 (S4).
  6. May 4, 2012 #5

    It seems to be that only, or perhaps mainly, you know what you're talking about: "Klein four group geometry"? "A group geometry"?

    If time passes by and you get no answer, perhaps it's time to give more background, books, papers for people to know what you mean. It may be that

    what you call "some group geometry" is known by another name to someone else...

  7. May 9, 2012 #6
    Thank you for your reply DonAntonio.

    Let me preface my responses with the fact that I am self taught so I may be describing things incorrectly or may be over looking something obvious, but that is why I am here, to learn. Also, I quote wiki a lot. Sorry for that.

    Geometric group theory

    "the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces)."

    The geometry I speak of relates to the points in Euclidean space defined by (x,y,z) where

    I had hoped that this would provide more detail:
    http://dl.dropbox.com/u/13155084/Given%20a%20divisor%20k.pdf [Broken]

    The reason I say "Klein four group geometry" is because that group seems to fit best or "mainly" as you stated.
    Last edited by a moderator: May 6, 2017
  8. May 9, 2012 #7

    Well, it still looks a little odd: geometric group theory is a well-known, pretty advanced subject within group theory. I studied

    a little of it no less than with Prof. Iliyah Ripps while in graduate school. Undoubtedly this could be a rather tough subject for

    a non-mathematician.

    What you're talking about, though, seems to be something else, related, as above, to group theory, geometry, number theory, etc., but in

    a different way, apparently...

    The link you wrote looks interesting but if you can I'd like to see books, papers, etc. about that in order to decide whether it is

    something I can mess with (interesting, level, etc.) or not.

    Last edited by a moderator: May 6, 2017
  9. May 9, 2012 #8
    I'll see if I can find any real reference material for you.

    It seems to me to be related to Pythagorean Triples. Here is something I wrote quite a while ago when I started down this path. It's very armature I know but it was my first attempt to make since of the relations I was noticing. It might help give a little background I guess.

    http://dl.dropbox.com/u/13155084/Pythagorean%20lattice.pdf [Broken]
    Another visual:
    http://dl.dropbox.com/u/13155084/CircleRecusion.png [Broken]
    Last edited by a moderator: May 6, 2017
  10. May 9, 2012 #9
    As far as number theory goes, one relation is the Divisor Summatory Function equivalence.

    [itex]D(n)=\sum_{k=1}^{\lfloor\sqrt{n}\rfloor}\left(2 \cdot \left\lfloor \frac{n-k^2}{k}\right\rfloor+1\right) [/itex]


    also the Coupon collector's problem



    http://dl.dropbox.com/u/13155084/divisor%20semmetry.png [Broken]

    http://dl.dropbox.com/u/13155084/DSUMv2.htm [Broken]

    http://dl.dropbox.com/u/13155084/prime.png [Broken]

    Last edited by a moderator: May 6, 2017
  11. May 9, 2012 #10
    For whatever it's worth Don Antonio, I used Jeremy Ebert as an example of potentially undiscovered genius in a paper submitted last fall to two professors at Columbia University (Sociology Department).

    He has already rediscovered the Dirichlet Divisor Sum geometrically, and there is no telling how far he could go with proper support from experts. It's a tough trick to pull to go backwards from simply "getting it" to understanding how it is that you "got it."
  12. May 9, 2012 #11

    Uneducated intelligence can easily be wasted in vain, in particular in realms such as mathematics where so often a hefty base is

    needed to build upon it.

    If he wants support then he'd rather go to some Maths Depts. in some university and approach some people there.

  13. May 10, 2012 #12
    I've shown this to one of my employees who is also a Math teacher at a local College. I'm really interested in what your take on this is Don Antonio.
  14. May 10, 2012 #13

    A related number theory paper:

    Note the geometry of Gauss circle problem.

    Last edited by a moderator: May 6, 2017
  15. May 11, 2012 #14
    A natural question to ask:

    "What was the response of the math teacher?"

    Hopefully he or she even knew what it was, but, based on anecdotal experience, I would not bank on it, tho', in principle, the basic formulation for D(n) is junior high school, if not elementary school, level. It's just a quotient table when you get down to it, albeit a quotient table that currently confounds all modern techniques available with regards to our ability to fully understand it...

    - AC
    Last edited: May 11, 2012
  16. May 11, 2012 #15


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    Hey JeremyEbert.

    I'm a little lost as to what you are trying to ask. Do you want specific kinds of comments on your diagrams and your PDF or do you have some specific question you had in mind?

    You have put in a lot of effort and I don't want to see it go to waste by asking undirected questions. Are you trying to solve something in particular? Are you trying to investigate something for a particular reason? Are you looking for comments on a particular issue?

    If the above is the case then this would help us give a more directed answer and know exactly what to focus our attention on to initiate the conversation.
    Last edited by a moderator: May 6, 2017
  17. May 11, 2012 #16
    Radio silence thus far.
  18. May 11, 2012 #17
    Thanks for your interest. I see you've had some post that deal with quantum. I've noticed that the geometry of this equation produces a type of parabolic coordinate system.

    http://dl.dropbox.com/u/13155084/CircleRecusion.png [Broken]


    This coordinate system seems to be of importance in the quantum world when it comes to angular momentum.

    With the similarities between the Riemann zeros and the quantum energy levels of classically chaotic systems and the parabolic coordinates created by this geometry, it seems like this might be another connection between primes and the quantum word. What do you think?
    Last edited by a moderator: May 6, 2017
  19. May 11, 2012 #18


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    Science Advisor

    Well oddly enough, there has been shown to be a link between the Riemann Zeta function and Quantum Field Theory.

    I remember reading about this ages ago from work associated with Alain Connes, but I don't have any deep knowledge, only superficial awareness of the fact. I would continue your work if this is what you are working on, because if you want ways to solve the Zeta problems and connect it with general quantum phenomena (discrete structures, diophantine systems and anything involving some kind of discrete system or finite-field) then that would be extroadinarily useful.

    Here are two results from a google search involving the Zeta function and quantum field theory:


    Last edited by a moderator: May 6, 2017
  20. May 12, 2012 #19

    Give me any 6 Lucas Sequences with P's and Q's constructed from 4 integer variables (r_0, r_1, r_2, r_3) in toto and if the ratio between successive terms of those six sequences at the limit as n approaches infinity is an integer, then those same P's and Q's can be used in a very simple, regular and consistent manner to construct a quartic and its resolvent cubic that will have integer roots.

    P = (r_x + r_y)
    Q = (r_x * r_y)
    D = P^2 - 4Q (Discriminant)

    Minimal Start Terms:
    U-Type Lucas Sequence = 0, 1
    V-Type Lucas Sequence = 2, P

    Recursive Rule:
    -Q*a(n) + P*a(n+1) = a(n+2)

    In other words, seems to me not only that there is a very clear linkage between the mathematics of the quartic (and thereby V4, the Klein Four-Group...) and Diophantine systems, but that this linkage may help explain how Jeremy is uncovering what he is uncovering.

    If you are unclear as to what I am referencing, I can show you how to map those P's and Q's to the quartic. It took me all of about a day to work out the "translation" once Jeremy got me thinking about it.

    - AC
    Last edited: May 12, 2012
  21. May 12, 2012 #20


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    I'm not sure what a Lucas sequence is. I only know the basics of number theory, but I think if you show me something like an external web-page or something that clarifies your ideas I can take a look at it.
  22. May 13, 2012 #21
    Don Antonio,
    Here is a link with some useful information regarding this although this site seems to be political in nature.
    Last edited: May 13, 2012
  23. May 13, 2012 #22
    Nice AC! I'll have to play with this!
  24. May 13, 2012 #23
    with the equivalence relation

    given the divisors of 12 have a different property than the divisors of 24, is there a link to equal temperament?


    Last edited: May 13, 2012
  25. May 14, 2012 #24
  26. May 14, 2012 #25
    Consider the form:

    C(a(n+2), j) (Binomial Coefficient)
    a(n + 2) = x*a(n) + y*a(n+1) + z

    The Lucas Sequences are just a special case of the above where:
    j = 1
    z = 0
    x = r_0 * r_1 = Q
    y = r_0 + r_1 = P

    Take, for example, r_0 = phi and r_1 = -1/phi
    r_0 * r_1 = -1 = Q
    r_0 + r_1 = 1 = P

    Start terms: 0, 1 (U-Type Lucas Sequence) gives the Fibonacci Series
    Start terms: 2, 1 (V-Type Lucas Sequence) gives the Lucas Series

    The recursive rule is -(-1)*a(n) + 1*a(n+1) = a(n + 2)

    (sqrt (P^2 - 4Q) +/- 1)/2 = (sqrt(1^2 - 4(-1)) +/- 1)/2 = phi, 1/phi

    You can read more about Lucas Sequences on Wolfram MathWorld or on Wikipedia. All kinds of identities follow from the maths and it's pretty simple stuff actually. http://mathworld.wolfram.com/LucasSequence.html

    Lucas Sequences are also related to Carmichael's Theorem which involves the introduction of new prime factors into integers associated with recursively based sequences. Thus, for example, Mersenne Numbers (2^x - 1) [which are a Lucas Sequence] can only be prime where x is prime.

    - AC
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