# Klein four-group geometry

## Main Question or Discussion Point

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]

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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]
I'm working on the explination.
http://dl.dropbox.com/u/13155084/Given%20a%20divisor%20k.pdf [Broken]

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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]
A better visual.
http://dl.dropbox.com/u/13155084/prime-%20square-klein%20four%20group.png [Broken]

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Maybe its more than the Klein four group.

Wiki:
"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=(n-k^2)/(2k)
s+k =(n+k^2)/(2k)
(s+k)^2 - s^2 = n

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

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

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

Z/6Z
(s+k)^2 = {1,0}
s^2 = {0,1}
n = {1,5}

Z/8Z
(s+k)^2 = {1,4,1,0}
s^2 = {0,1,4,1}
n = {1,3,5,7}

Z/12Z
(s+k)^2 = {1,9,4,0}
s^2 = {0,4,9,1}
n = {1,5,7,11}

Z/24Z
(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).

Maybe its more than the Klein four group.

Wiki:
"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=(n-k^2)/(2k)
s+k =(n+k^2)/(2k)
(s+k)^2 - s^2 = n

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

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

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

Z/6Z
(s+k)^2 = {1,0}
s^2 = {0,1}
n = {1,5}

Z/8Z
(s+k)^2 = {1,4,1,0}
s^2 = {0,1,4,1}
n = {1,3,5,7}

Z/12Z
(s+k)^2 = {1,9,4,0}
s^2 = {0,4,9,1}
n = {1,5,7,11}

Z/24Z
(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).

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...

DonAntonio

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...

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.

wiki:
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
x=s=(n-k^2)/(2k)
y=sqrt(n)
z=(s+k)

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.

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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.

wiki:
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
x=s=(n-k^2)/(2k)
y=sqrt(n)
z=(s+k)

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.

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.

DonAntonio

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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.

DonAntonio
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]

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As far as number theory goes, one relation is the Divisor Summatory Function equivalence.

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

http://en.wikipedia.org/wiki/Divisor_summatory_function

also the Coupon collector's problem

http://en.wikipedia.org/wiki/Coupon_collector's_problem

related:

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]

http://math.stackexchange.com/questions/5574/a-triangular-representation-for-the-divisor-summatory-function-dx

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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."

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."

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.

DonAntonio

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.

DonAntonio
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.

As far as number theory goes, one relation is the Divisor Summatory Function equivalence.

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

http://en.wikipedia.org/wiki/Divisor_summatory_function

also the Coupon collector's problem

http://en.wikipedia.org/wiki/Coupon_collector's_problem

related:

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]

http://math.stackexchange.com/questions/5574/a-triangular-representation-for-the-divisor-summatory-function-dx

A related number theory paper:

Note the geometry of Gauss circle problem.

http://www.homepages.ucl.ac.uk/~ucahipe/Lfunctions.pdf

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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.

"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

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chiro
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]
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.

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"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

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.
Chiro,
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]

http://mathworld.wolfram.com/ParabolicCoordinates.html

This coordinate system seems to be of importance in the quantum world when it comes to angular momentum.
http://www.ejournal.unam.mx/rmf/no546/RMF005400609.pdf

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?

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chiro
Chiro,
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]

http://mathworld.wolfram.com/ParabolicCoordinates.html

This coordinate system seems to be of importance in the quantum world when it comes to angular momentum.
http://www.ejournal.unam.mx/rmf/no546/RMF005400609.pdf

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?
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:

http://arxiv.org/pdf/hep--th/0701284.pdf

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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.
Chiro,

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

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chiro
Chiro,

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_0 + r_1)
Q = (r_0 * r_1)
D = (r_0 - r_1)^2 = P^2 - 4Q

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
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.

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.

DonAntonio
Don Antonio,
Here is a link with some useful information regarding this although this site seems to be political in nature.
http://wlym.com/~animations/ceres/InterimII/Arithmetic/Reciprocity/Reciprocity.html

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Chiro,

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
Nice AC! I'll have to play with this!

Maybe its more than the Klein four group.

Wiki:
"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=(n-k^2)/(2k)
s+k =(n+k^2)/(2k)
(s+k)^2 - s^2 = n

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

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

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

Z/6Z
(s+k)^2 = {1,0}
s^2 = {0,1}
n = {1,5}

Z/8Z
(s+k)^2 = {1,4,1,0}
s^2 = {0,1,4,1}
n = {1,3,5,7}

Z/12Z
(s+k)^2 = {1,9,4,0}
s^2 = {0,4,9,1}
n = {1,5,7,11}

Z/24Z
(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).
with the equivalence relation
s=(n-k^2)/(2k)
k=versine
sqrt(n)=sine
s+k=cosine

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

http://en.wikipedia.org/wiki/Equal_temperament

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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.
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