infinity of infinity

[AntonioLao]

It is physically logical to deduce that there is an infinity of direction to choose from. Moreover, for each chosen direction there is an infinity of other direction pointing in the same chosen direction. This idea suggests that physical reality is really made from an infinity of infinity of equally orientable direction comprising the totality of the space-time continuum. Fortunately, out of this double infinity only two possible groups of relatively local directional symmetry can be created. One is the group of inward directions and the other is the group of outward directions. If these are in a dynamic equilibrium then the result is simply perceived physically as the complete manifestation of the space-time continuum. Since this double infinity of directions have equal absolute magnitudes, each value can be set to unity. Consequently, the minimum number is three unique directions in order to form a local group of magnitude difference, mathematically known as the triangular inequality.

There are three distinct definitions. First is the triangular inequality from complex imaginary numbers. If ��₁ and ��₂ are complex numbers then complex modulus |��₁+��₂|, |��₁|, and |��₂| are related by the triangular inequality: |��₁+��₂| ≤ |��₁| + |��₂|. Second is the triangle bounded by three points A, B, and C such that the lengths of all three sides are |AB|, |AC|, and |BC| and the three triangular inequalities are |AB| ≤ |AC| + |BC|, |AC| ≤ |AB| + |BC|, and |BC| ≤ |AC| + |AB|. The third is the triangular inequality for three dimensional vector additions. Let |��| denote the absolute magnitude of vector ��, |��| the absolute magnitude of vector ��, and |��+��| the absolute magnitude of vector ��+��. The vector triangular inequality is then given by |��+��| ≤ |��| + |��|. These can be extended to the Cauchy-Schwarz inequality for integrals. If ��(��) and ��(��) are two real functions then {∫[��(��)��(��)]����}² ≤ {∫[��(��)]²����}{∫[��(��)]²����} if and only if all these integrals exist. This existence is equivalent to their convergent infinite series expansions if and only if their corresponding limits also exist. Together, the generality of these defined inequalities forms the infinity of triangular infinity.

[mkirkpatrick]

From any point does infinity stretch endless in absolute extreme,but what of the point itself? Does the point transcend that which is infinite?


regards michael.

[AntonioLao]

That's a good question. But if each space-time point is associated with a particular direction then it is associated with an infinity of other different directions. Two points of different directions are then associated with a double infinity of directions. This two points configuration would then form a quantum of space-time and two distinct configurations are the H-pluses and the H-minuses.

[SteveA]

I tend to think a fundamental unit of time is the largest physical quantity available and this quantity may grow unbounded (it seems a logical paradox to assume anything within time can stop or start it, so our observations of finite limits does not mean that these cannot be increased).

Other structures in time should then be, similar to the growth of branches of a tree, interleavings of this fundamental quantity in various dimensions.

So, if I took your comments and placed them into this context, we could have infinitely many directions of infinite depth by taking the largest quantity required to construct all these - the volume and then spiral it around in a growing number of dimensions (we could not immediately move in an infinite number of dimensions because no such reference would be available, hence the first motion would simply be other/else/out/different/beyond/next etc. instead of same/here/now/this/self etc.)

The second step would then be more specific in the possible directions it could be as it could be neither of the previous two qualities and we can continue to do this creating orthogonal forms of motion relative to the previous qualities and construct new dimensions. Along with that, if we wanted to append quantities to these dimensional qualities, such as something is not simply "different', but varies by 1/3 or 3 times the variance we could then, for example combine previous qualitative dimensions into quantities and construct sets of qualities.

Such a construction of sets is yet another form of motion within this space and could be created in a rigid/deterministic manner, for example, by continually counting in a discrete representation where the base/extend/digital range of representation for each dimension was equal to the current number of dimensions and then adding a new dimension and counting through the next set of extended bases (excluding the previously constructed hypercube).

So we'd have a volume of growth occurring with layers constructing growing volumes at each addition of a dimension of:

(0^0?) 1^1, 2^2, 3^3, 4^4 etc. and the volume of each successive "shell" would be "n^n-(n-1)^(n-1)".

(I think there's a more natural manner in which this should grow that has a more point, line, triangle, tetrahedron ... form instead of the sequence, point, line, square, cube, hypercube, but I don't know what the name of that form of construction is).

There should be nothing larger than infinity (in which case we could even define it as the largest natural number ... which we do not specifically know, hence it becomes a single variable, n, capable of describing all unknowns in a system as its selection is unbounded and determining a single integer, n could require more information than the entire universe supplies).

[SteveA]

I made a couple mistakes in the previous post, if define n as being THE single deterministic source of fundamental units of time, then it's largest and there would not then exist n^n, but instead, for example a m^m such than m^m<=n, and m would be the dimensionality and maximum distance possible in that form of construction dependent upon n.

Also, I think the sequence I mentioned, point, line, triangle, tetrahedron (hypertriangle? hypertetrahedron?) would be indicative of factorial growth, m!

I say this appears more natural for a few reasons 1) we can't move in space without moving toward something - everything is a physical endpoint - you can't navigate past some distant star except via. inertia or selecting a new object "beyond" it, but notice that making continual selections beyond something potentially leads in a circle and does not guarantee a "straight" motion occurs (whatever that might be).

So unless there's a way to navigate orthogonally "away" from everything, the space would appear to be constrained by endpoints describing the limits of its dimensions (which should effectively be endpoints in memory - creativity might arise from something similar to adding new references to a space) and so for 3 points, we have a triangle, for 4 we have a tetrahedron etc.

Notice also that in terms of information/energy conservation we could consider the maximum information in a set of m elements as existing when each element is distinct. In this case we have a maximum number of permutations they can be reordered as being m! and hence, once again there are some close correlations.

The quantity m! appears to be more of a virtual/mental construction made by assumptions of independence and orthogonality (which may be fundamentally true of objects, though that would not appear something deterministically verifiable)

Here's a very interesting thing to look at. Consider the difference between log(m^m) and log(m!).

log(m^m)=log(m)*m
log(m!)=log(m)+log(m-1)+log(m-2)+...+log(1)

Now we can do some interesting operations on the factorial version and find many fractal structures as well as complex linear/logarithmic relationships between them. For example, if we assumed m is a power of 2, we can recursively construct:

log(m!)=log(m!)-log((m/2)!)+log((m/2)!)
log(m!)=log(m!/(m/2)!)+log((m/2)!)
log(m!)=log(m!/(m/2)!)+log((m/2)!/(m/4)!)+...+log((2)!/(1)!)

This creates a fractal sequence of self similar structures of the form log(p!/(p/2)!) and we can multiple them by log_2(m) to approximate this value.

Another interesting operation is to break these factors into alternating odd/even sequences:

log(m*(m-2)*(m-4)*(m-6)*...*2)~=log((m-1)*(m-3)*(m-5)*...*1)

There are some close approximations for the growth of a factorial and they have some very interesting possible correlations to a chaotic physical space. Here's Sterlings approximation http://en.wikipedia.org/wiki/Stirling%27s_approximation

m!~=(2*pi*m)^(1/2) * (m/e)^m

We can rewrite this as:

m!~=m^m * ((2*pi*m)^(1/2) * (e^-m)

And I see some correlations here between an energy following a chaotic/gaussian/brownian diffusion in a hypercube that's an m^m volume, and we have a 1/e decay in correlation across each dimensional "layer".

There's an even closer approximation I've seen that appears to add something like a 1/6th cycle rotate to the mix (which could be similar to compensation for a 3 dimensional space where 3!=3*2*1=6 possible permutations of orientation in 3 dimensions and then we might have chaotic/fractal/prime properties creating the "1/e decay" of information arising from statistical measurements that do not retain the information regarding the permutations of events over time)

Yes, my comments are very abstract but I know you tend to be able to follow these lines of thought and understand some of the general properties as well. (I admit I quickly get lost but there's still enough of a figurative "ledge" to hold on to and the appearance of pathways that I know are connected together ... the question to me is whether or not that's how all chaotic structures are formed or if there are some additional qualities that could be added in the mix that I'm missing. The physical limits to these structures appear to depend upon the availability of distinct and orthogonal properties of experience - basically whatever unique forms of conscious experience form the "perimeter" of that space, the equivalent fundamental "letters"/unique qualities of conscious experience).

Well, as usual, my thoughts as still a work in progress trying to tie the pieces together and see what the limits might be (until yet something else pops up and adds another layer of obfuscation to it! )

[AntonioLao]

There should be nothing larger than infinity

Dealing with the physical or abstract attribute of direction we need not answer the question of 'which is larger or which is smaller' since each direction has no magnitude. Direction can be the same as a vector quantity of zero length. It is properly defined as a spinor by Elie Cartan when he developed the theory of spinors in the early years of the 20th century.

[Profpat]

From any point does infinity stretch endless in absolute extreme,but what of the point itself? Does the point transcend that which is infinite?


regards michael.

Hi Michael;

The dimensionless point is the only thing (VOID, NOthing) that is infinite (immeasurable), anyTHING else is only potentially infinite.

Best,

Pat

P.S. How is the retirement going?