A times B equals A minus B

[AntonioLao]

The field of rational numbers besides being closed under addition, subtraction, multiplication, and division (division by zero is not allowed) has the property that for any two rational numbers A and B related by the equivalent rational functions: A=B/(1-B) and B=A/(1+A), AB=A-B is always true. For example, if A=1 then B=1/2 and for A=-2 then B=2. For the graph of A=B/(1-B), the function is undefined at the point (1,-1). For the graph of B=A/(1+A), the function is undefined at the point (-1,1). That is to say that these points are not the solutions for AB=A-B for their corresponding rational functions. The split domains of the first graph are (-∞,1) and (1,∞) corresponding to the split ranges: (-1,∞) and (∞,-1). The split domains of the second graph are (-∞,-1) and (-1,∞) corresponding to the split ranges: (1,∞) and (-∞,1). For the rational functions A=-B/(1-B) and B=-A/(1+A), their product is reduced to the same AB=A-B. However, the two points of discontinuity are respectively (-1,-1) and (1,1). The split domains for the first are (-∞,-1) and (-1,∞) corresponding to the same split ranges: (-∞,-1) and (-1,∞). The split domains for the second are (-∞,1) and (1,∞) corresponding to the same split ranges: (-∞,1) and (1,∞).

All these functional descriptions suggest that A times B equals A minus B hides hyperbolic discontinuities at four points: (1,-1), (-1,1), (-1,-1), and (1,1). If these become row matrices then four fundamental 2 by 2 square symmetric Hadamard matrices can be constructed. The first has row 1 given by (1,-1) and row 2 by (-1,1). The second has row 1 given by (-1,1) and row 2 given by (1,-1). The third has row 1 given by (-1,-1) and row 2 also given by (-1,-1). The 4th has row 1 given by (1,1) and row 2 also given by (1,1). The matrix product of the first or the second with the third or the 4th gives the zero matrix. These demonstrably imply that Hadamard matrices are suitable for representing a quantum theory of the spacetime continuum and also provide formulations for both the origin of mass and charge.

[SteveA]

Consider viewing a sequence of objects in time.

If there are no repetitions, this would give the appearance of an infinite sequence.

On the other hand, if there were repetitions, this would imply closed/looping motions with set(s).

Now consider this in the context of viewing a finite/closed set of objects, for example, 3 vertices of a triangle. How do we determine that this is a closed set of objects? I believe it arises from detections of repetition/identities, which requires at least one duplicate observation in order to determine that a sequence is closed/finite.

If we were to scan a collection of elements and began with A, B, C, ... we, as of yet, cannot determine the size of the set. If we observe the next element to be A, then we can determine the set is closed and contains the elements A, B and C, but this requires that make observations from a "present" state containing 4 observations, the final one of which is a duplicate of a previous state.

In a sense we could compare this in terms of viewing a ratio of 3/4, as we would be observing 3 spaces/objects from a position of 4 memories. Notice that if we had 4 objects and 4 memories to retain observations, we could not detect a repetition of these 4 objects and hence the properties of 4/4 could appear to be infinite as it would not be observed as an "enclosable"/less than unity ratio in memory (basically, identity can appear to possess infinite qualities).

Anyway, your example of B=A/(1+A) could match that form of B representing the observation of A objects from a position of (1+A) memories, so there could be correlations here with observations of physical quantities.

In that case, if we rewrote A=B/(1-B) as -A=B/(B-1) this would be similar to trying to reverse/invert this perspective and view a set of B from a space/memory of B-1, (which might contain underdetermined/statistical observations as neither a contained set, nor a 1 to 1 match would exist and if we assumed both the observations of these objects as well as the filling of memory were synchronized in time (which we should expect), then B and B-1 are relatively prime and would construct a repetitive wavelength of observation with a period B*(B-1), with predictions made from the B-1 set being perfectly uncorrelated (at least for observations over a period as a multiple of B-1. Notice that idealized randomness is not actually random at all and non-idealized, or irregular distributions imply an unobserved structure remains and would not be truly random either - randomness in the sense of being uncaused/uncorrelated would appear to be impossible - these are unknowns instead (memory and perceptions aren't sufficient to encapsulate them - which ironically could also imply an identity in terms of size ... quite interesting)).

As a sidenote, we get some chaotic relationships when "viewing" ratios with shared common factors such as (X+2)/X, when X is even, as this appears to construct two objects with the perfectly uniform statistical properties of (B+1)/B, but embedded as two symmetrical objects of size X/2 (or half the full scale of view).

In your other thread you mentioned squaring uncertainty. I believe reality is asymmetrical (all things are unique) - unknowns construct an appearance of symmetries (unknown asymmetries - notice that any number of unknowns can appear identical to the influence of one).

I don't believe squaring is a fundamental operation (it's non-linear and destroys information, for example, x^2=(-x)^2, and in this case f(x)=f(-x), which destroys the sign/polarity information of x when being viewed by the result of f. This would not be a structure that could be physically constructed in a deterministic manner as observing, for example, a result of 4 would imply two possible causes - in an informationally conserving system only one of these causes should be real).

As a better example, imagine that we have a sequence of observations measuring some quantity of events, A. In order that a variable density of such events in time could be detected, there must exist at least a single other event, B as well as some memory, "window" or period of time over which these events are observed.

Fundamentally the sequence would contain binary information (we'll assume we have no reference for time outside this) and for a memory of length n, we would have 2^n possible observable states.

Now if we don't, or can't, retain the specific sequence of over time but instead only count the occurances of A within this window, then every observation would be of a quantity of A from 0 to n.

If n is 1, then we can square these quantities and there is no information lost (0^2=0, 1^2=1) and this value of 0 and 1 would describe precisely, at every moment of time, which observation, A or B was occuring. We have 100% data retention at this point.

Now let's say that we accumulate A into a quantity of an energy instead and allow, for example, n=2.

Now at any moment in time we have between 0 and 2 elements of A visible in our window. Let's see how this compressed representation masks temporal information:

BB=0
AB=1
BA=1
AA=2

If we detect a quantity of 0 or 2 As, then no information is lost and we can reconstruct the observation, but if the observation is of either AB or BA, we lose information regarding the asymmetry in temporal ordering.

Notice that any "laws" derived from such a 1-D quantitative description would be symmetrical in both directions in time - if we reverse these elements within this window, it wouldn't alter the quantity of that 'energy'.

Notice also that one component of energy is an entropy ... something to consider is whether or not it is a true entropy or that the representation as a scalar quantity is underdetermined as to the specifics of an observation. If we were able to retain all components of an observation, there should be no such entropy present, but instead observations of (a much larger) preexisting quantity of information. (I believe this is why some physical theories utilizing concepts of multiple time dimensions (which are constructed from spacial perceptions - the "dimensions" of space can be correlated with properties of conscious perceptions) have met with some success in that they retain a larger quantity of information regarding events).

[SteveA]

Notice the again a series of progressive qualities of these ratios for n/m:

n/m<1
a) if n=m-1 we have a (completely closed and static/non-moving field (i.e. space))
b) if n<m-1
1) if gcd(n,m)=1 then we have a uniformly moving object of size n moving in space.
2) if gcd(n,m)>1 then we have identical objects of size n/gcd(n,m) moving synchronously together as a single object (for example, an atomic nuclei)

n/m=1 (static infinite)

n/m>1
a) if gcd(n,m)=1 then we have a uniform perfectly distributed randomness)
b) if gcd(n,m)=p then we have a p way symmetrically distributed statistical "wavefunction" (for example, some forms of atomic orbital shells))

Notice also that a recursive observation of observations would construct a sequence of viewing n via n+1 and then viewing n+1 via n+2 etc.

I've posted some comments regarding the ability of the gcd function to mirror many qualitative properties of physical observations (for example, we have holographic 3-D properties). Some example images are here: http://www.toequest.com/forum/your-t...utation-9.html

[AntonioLao]

Steve, I have a question: Using your combinatorial or computational binary randomization approach how do we describe the 8 directional invariance properties? These are left-front-top, left-front-down, left-back-top, left-back-down and their complements right-front-top, right-front-down, right-back-top, right-back-down. Moreover, these are not representable by numbers but by directions of tri-vectors. They are also multi-dimensionally orientational invariance.

[SteveA]

Notice that non-orthogonal rotations in Euclidean space can construct irrational ratios of distances. An irrational quantity would either have to be a dynamic ratio constructed in time via. two interleaved processes one for distance as well as one for the viewing scale and space would then become a dynamic structure itself.



I generated these images using techniques I've mentioned here and on that other thread I posted the link to. In these images, the vanishing points and horizon lines arises from ratios near unity, those the gcd can be seen as a nested and recursive/ fractal computation because:

gcd(n,m)= { 
   if n>m : gcd(n-m,m) ;
   if n=m : n ;
   if n<m : gcd(n,m-n) ;
}

Hence, it's a fractal (actually chaotic - I define the difference to be whether or not different n-way symmetries are constructed and contained within a closed space, this appears to be the most predominant feature of objects that are viewed as chaotic structures).

Notice that this can also be viewed as a sorting and selection of objects in a 1 dimensional space (vortices within a conserved volume of space):

If we place n and m on a single number line, they become indistinguishable and exchangeable except as two quantities, though if they are superimposed as a single point, we have a result of the gcd. (Notice that this is the way the mind observers positions of objects in space - two otherwise indistinguishable objects appear to possess distances (which could be vectors) relative to an observational origin, and a detection of a single object twice at a specific location appears to be a single/stationary object).

So if we first extract all n=m and determine these to describe a space of stationary objects (which is the final result of a gcd computation), we're still left with n != m values, which can be "presorted" into n<m by swapping spaces of n and m (a closed pathway/orbital in space).

If we can then determine that n<m, an iteration of the gcd computation becomes gcd(n,m)=gcd(n,m-n) and we have no information lost if we then redescribe this as a new space (within a new context, such as a new perceptual quality - for example, the objects have now changed color).

Notice that we do not see a true 3 dimensional + time space. Vision is 2-D + time (a stereo pair of 2-D, limited field observations) - if we could really see 3-D + time, then we should be able to see the interiors of objects (but this would make space appear static and stationary if all information regarding physical events was then observable as no entropy over time would be available).

Now a fixed object should arise from making an observation of a quantity less than that the width of memory of the observational perspective, such that the entirety of the object can fit within perceptual memory, otherwise not all of the object would be visible and it would surpass perceptual capabilities.

So we have objects being observed from ratios (constructed analogous to wavelengths/pathways of space) of n/m, as seeing n contained within m and hence n<m, if n is to appear as closed/finite - such a sorting is implicitly present in the qualities of perception - if an object is finite, then n<m and if an object is both finite and moving, then n<m-1. If an object exists as a symmetric collection of closed objects, then n=p<m-1, where gcd(n,m-1)=q, where q>1 and describes the quantity of such objects being detected.

There are two forms of vanishing points/horizon lines - one is chaotic/irrational (for example, seeing blackbody radiation believed originated from an unknown source) and the other is coherent - such as the vanishing point constructed via the surface of a solid object (a trajectory between two previously existing points).

In these cases we can construct an example of one of the simplest forms of chaotic/irrational vanishing points as a recursive phi/fibonacci sequence where the gcd approaches 1 between n and m and n~=phi*m. For example, looking along the fibonacci sequence, we can select an adjacent pair of 1,1,2,3,5,8,13,...

and, for example:

gcd(8,13)=gcd(8,13-=gcd(8,5)=gcd(8-5,5)=gcd(3,5)=gcd(3,5-3)=gcd(3,2)=...=gcd(2,1)=gcd(2-1,1)=gcd(1,1)=1

Notice that every iteration of this in terms of gcd(n,m), constructs the relative order in space of:

n(0) < m(0)
n(1) > m(1)
n(2) < m(2)
n(3) > m(3)
...

And we have these positions orbiting in space, spiralling inward. (converting an entropy in terms of their relative primeness into an entropy describing their motions in space):



We also have another form of vanishing point and this arises when n=m, though in this case we could not directly observe such a relationship in time (it would appear "uncontainable" in perceptual memory and with "infinite", though static, properties in that respect - a timeless perceptual context of all temporally dynamic qualities of objects n<m).

Though we can see that there exist fractal (re)representations of this relationship in terms of n-way symmetrical features of objects contained within a perceptual space - the most significant is the binary duality, which we could represent with an observation of n=m/2 (or (m-1)/2 might be the largest stationary, finite and perceptually comprehensible analogy we could observe - for large m, that could take a while to construct via. recursive self-observations, though with a bit of interactive growth, such as a binary selection of possible motions this could be sub-linear (potentially logarithmic in that specific case, though a factorial selection would be faster, it's interesting to see that using something like a classical statical "energy" term would be slower than linear and potentially require at least polynomial, if not an exponentially longer time ... could be intentional).

Though basically these vanishing points are similar to those constructed by the various angular ratios of observation within a 3-D (though actually 2-D) "grid"/matrix:



If we then extend an object to higher dimensional properties, such as a computation of gcd (n,m,p), we now have a 2-D array of vanishing points, relative to a spacial "scaling" of p, and it would appear required that n+m<p, unless the system of n and m is statistical, in which case we could have n+m>p.

Anyway, this constructs an array of vanishing points of various qualities over the plane of n and m, relative to the scaling (viewing distance) of p.

I guess to get back to your point, if we then interacted within such a space, using these vanishing points as references, these would become conserved properties within that space.

[AntonioLao]

Thank you and thank you. Your images look as impressive as ones found in James Gleick's book: CHAOS, the making of a new science. Aside from their randomized chaotic patterns there are also patterns of self-similarity (e.g. Julia sets and Koch curve) and strange attractor (e.g. Butterfly Effect). Others like the Sierpinski carpet or Menger sponge has infinite surface area, yet zero volume. I realized fractal structures are generated by pseudo-random number generators and making them to have fractional dimensions. Truly random patterns, I think, would be completely featureless. Could this be the spacetime continuum???

[SteveA]

Thank you and thank you. Your images look as impressive as ones found in James Gleick's book: CHAOS, the making of a new science. Aside from their randomized chaotic patterns there are also patterns of self-similarity (e.g. Julia sets and Koch curve) and strange attractor (e.g. Butterfly Effect). Others like the Sierpinski carpet or Menger sponge has infinite surface area, yet zero volume. I realized fractal structures are generated by pseudo-random number generators and making them to have fractional dimensions.

Thank you for seeing the many cross associations.

We can additionally find structures similar to continued fractions (a+1/(c+1/+...) lie along trajectories and that these can describe objects similar to nested orbital structures (wheels within wheels).

Truly random patterns, I think, would be completely featureless.

Yes, I gave up trying to understand randomness and went the other way ... if you had to make it work, what properties would it have to possess. It appears it's got to logically be a hierarchy of properties with a singular "prime mover" ... on the other hand, such a structure does not move either and time is simply a creative unknown, though we could potentially map out some possible "directions" in time (possible an inverse form of space existing as the ratios n/m>1)

Notice that we could also look at those images in comparison to Earth - solid objects would be n<m (we have water approaching the horizon which could be analogized with ratios that approach 1 from below and then ironically 1 itself is not visible, but a unity - in effect the distant horizon comes back to the viewing origin as gcd(n,n)=n, yet gcd(n,m+1)=1 and so we have a discontinuous jump from 1 to "infinity" (n would represent the observational "size" and n appears static and unencapsulated and non-repeating relative to itself and hence appears infinite), so in effect, the obstruction of the viewed horizon line is only a single quantum unit distant) and then as we move to ratios above 1 to 1 then we have more creative and energetic components - air and all the way to a vertical view of the "Big Bang" with, down to a quantum unit at the bottom.

I believe we can even find some close analogies with beauty and intelligence on this scale (beauty and intelligence are complimentary views on this scale - much like the left/right brain division. Form and change, noun and verb)

Could this be the spacetime continuum???

It appears to at least be a naturally resonant and information conserving structure. I believe most everything science is working on is related to this structure ... though there are still loose pieces (that appears to always be the famous last words of any theory ... just before the picture gets even bigger and more beautiful!). It's probably more representative of a natural field of formatting for information upon which intelligence operates.

[SteveA]

We should also have some immediate mathematical correlations with physical concepts of motion such as within/beyond/through/by/over/outside/inside/under etc. These could then be describing various properties of a ratio between object and observation.

I believe we can also show that most all the Millenia Problems are based upon this same structure as well as the ideas of a holographic universe (in which case atoms would, yes, be very much "entangled" because they're constructed by a smaller quantity of conscious information as a collective structure - though apparently there is a creative ability to alter these and that's where it gets a bit tougher to figure things out).

[AntonioLao]

Thank you. I'm working on a quantum theory of the spacetime continuum. If it is truly featureless then it would be impossible to quantize it. But my trust in mathematics allowed me to think that this featurelessness can be described by square symmetric Hadamard matrices and the fact that they are embedded in a sieve of Diophantus makes it more personally convincing. The proof I think would come from the successful implementation of cold fusion for deuterons.

[AntonioLao]

These could then be describing various properties of a ratio between object and observation.

I have to agree that ratio has became the yardstick to gauge everything in the universe. But as a dimensionless quantity each ratio suggests a scalar multiple of of something as the greatest common divisor or denominator. On the other hand, if the dimensions of the numerator and the denominator are different, for example, distance for numerator and time for denominator then the ratio becomes rate and physically rate simply certain change and motion becoming dynamic instead of static values like ratio.