Painting a Picture of a Quantum Surface

Intro

In my last post I talked about spheres and taxi-cab geometry. Let me now put it all together and paint a multi-step picture that takes us from here is non-instancing multi-dimensional strings.

Step 1 – Exclusion Space

This is where I’m choosing to start. Exclusion space is based on the TPPT forming a space of octahedrons where they joins at points (and maybe much more rarely) at edges to form chains and clusters. This space is bounded dimensionally, so where two TPPT solutions share a point you get a join, but when there are three or more they exclude each other in such a way that the solutions all co-exist but not at the same time. If introduce the idea of historical precedence, then earlier octahedrons maintain their probability and later ones exhibit 1/n lower probabilities, ie. 1/2, 1/3, 1/4 etc

Step 2 – Active Clusters

Imagine now that there grows a primary cluster, maybe there is just one, maybe there are many (a multiverse). Let’s take one and ask the question, how big it is and are their any natural limits on it’s size and rate of growth, does it reach some limit and become stable.

For a start, we have already show that the density of the TPPT decreases with each generation. At some point per n (the rate of the counting number iterator) solutions become more rare and as a consequence octahedrons capable of joining the cluster are also less frequent. These ideas tend to increase the stability of the cluster. At some n = K, which is an inconceivably large number the cluster is at equilibrium with itself.

If we go with the historical precedence model (or another) we see probabilities 1/1, 1/2, 1/3 etc close to the center. As we go towards the outside the threads get seeded with combinatorial lower probabilities. At the surface a sparse matrix contains parts that are blinking in and out of existence with, maybe more off than on.

Step 3 – Patterns on the Surface

At the surface we can interpret the mountainous crags and caverns blinking on and off. But, interpret them as what? Let’s try and see if there are any patterns. Considering the surface as a whole (as there is no reason or limitation that forces us to consider only a subset) imagine the surface a sparse 2D matrix of bits. All the complexity of the inside structure and the combinatorial probabilities are now reflected on this matrix as the bits blink on and off.

Consider a set of bits that share the same pattern. Consider the entire surface analysed down into p distinguished patterns. I suggest we ignore the temptation to order them, as one pattern is really as good as any other, as long as a pattern is unique, the number of members is irrelevant.

Step 4 – Loops

Let’s start imposing our previously talked about constraints on solutions. That is, it’s time to look again for the signal in the noise. Frankly the above paragraphs describe a whole bunch of noise, so where is the signal. I propose that the signal is in the loops. Let’s do two chops in one go. Let’s ignore all the historical generations and focus on the generation boundaries. Within each generation focus on the loops and ignore all the tendrils. Loops have length, a critically important metric. I have suggested that loop length is akin to the spin in spin foam theory. This then leads on to theories of quantum gravity and the mechanism for the apparent creation of new space in the projection to MST.

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You’re not old enough to play with spheres

You have to admit pulling a sphere out of your ass is a pretty neat trick when you don’t even have real numbers to play with, so how did you get to spheres. Ok, the taxi-cab universe only goes so far, but this octahedral thing is based on spheres and close packing, isn’t that just a bit of a leap … even for you (or me).

This is the gap we have to cross. The whole point is that 3(3 – 2√2) is such a cute thing to play with, I mean its not even a counting number, it’s a ratio. So let’s start with the idea of the validity of the ratio in the first place. The ratio occurs in our minds, it doesn’t have to be real in the context of the counting numbers themselves. This is a meta analysis and I’m allowed to use tools that are too complex to express in the context of the thing being studied. So, what I’m saying is, I can derive a ratio between any two things, even counting numbers as long as the ration stays as an expression of analysis and does not play a role in the mechanics of the theory.

Here we have an analytical ratio, the “end game” ratio which would only be true at the end time that never comes. As the ratio of densities approaches this value what does it mean for universes derived from it. As an aside, let’s think of the ratio of densities as the 1st differential of the signal to noise ratio between generations. That is, its the rate of signal diminution over generations.

Now those bloody spheres, if the unit sphere represents the notional parent generation signal strength, and the small sphere the child generation signal strength then is this a model that contains any analytical meaning that could lead to the bringing forth of native spatiality.

When is a Sphere not a Sphere

I know some of you must be thinking, as I have, that the spherical radius ratio theory is flawed. First of all, why the radius, why not surface area or volume, as they sound more intuitive.

Here’s why: Taxicab Geometry

The only metric in Taxicab geometry is the radius, the rest is meaningless, as it is in the quantum world. There are no real spheres as these are just abstractions and constructions. Stuck here in MST we see spheres in space all the time. The planet is a sphere, so is the sun. Even the event horizon of a black hole is spherical, and yet its entropy is proportional to its surface area by virtue of holographic projection from a non 3D + time underlying quantum reality. See Leonard Susskind on The World As Hologram.

In our translation from the TPPT we only really claimed a relationship of ratio with the idea of octahedral close packing based on radius, not that the things had to related to spheres specifically. Like I said, spheres and surface area are constructions that require real numbers and we are a long way from these abstractions and illusions.

Thinking about octahedral close packing, 3D fractals and limits in the context of taxi cab geometry is not easy, but we will get there. Maybe there is some room for us to consider this mess in the context of yet another projection. There may be a role for Geometric (Clifford) Algebra here.

Finite Convergence and Entropy

Finite convergence is an interesting idea because it implies a contradiction in terms. Convergence is known to be infinite, but there is no infinity, and after a finite number of iterations, convergence is not complete.

Take the convergence of dimensional scaling factor of the Triples to 3(3 – 2√2). If there is no infinity it will never get there, right, and yet this convergence defines the nature of three dimensional spatiality. Take this conundrum also in the context of quantum cosmology, where the idea of infinite granularity in an infinite universe is a nonsense. Clearly the universe is not infinite, just big, and the quantum granularity implies a sense of non smoothness at the bottom of the abstraction stack.

The bottom of the abstraction stack for me is the counting numbers. If you don’t know why, go back and re-read the blog. It’s not smooth. The first abstraction is the Triples and after that we get the first idea of spatiality.There is lots of noise and a thin thread of signal. So now consider the non-infinite sequence that is the evolving convergence to 3(3 – 2√2) – I will coin this Russell’s number (why not it’s my number).

What does it mean for three dimensional spatiality as this sequence converges. Remember the number it’s converging to is the dimensional scaling factor of the octahedral graph. This is represented by the abstraction of the exploding unit sphere. As the sequence progresses what idea arises that map to the idea of the progression of the exploding unit sphere. The scaling factor (Russell’s number 3(3 – 2√2)) is the diameter of each of the six resultant spheres, so as the sequence progresses the unit sphere can be thought of as exploding into six slightly smaller spheres in terms of their diameter.

What does this mean, exploding into less than the perfect fitting six spheres of diameter 3(3 – 2√2). what positions do the spheres have if they do not fit perfectly, do they rattle around, are the positions given by an equation that expresses “uncertainty”. Is this the root of all uncertainty in the quantum universe and because the sequence is still running we still experience uncertainty. Does this imply that uncertainty will decrease over time, is this a parallel for entropy.

How does the dynamics of six trapped and expanding balls predict the structure of the early universe. Is this the source of the big bang, when after a time there is a fundamental change in the mode and nature of the positional uncertainty that MST becomes a possible abstraction.

Rethinking the Map

The mapping I have been using to date has a problem. My theory suggests that a mapping exists between the open and closed octahedral graph because their dimensional scaling factors are identical. The open graph was projected by taking the limit of the ratio of densities between generations. The closed graph was projected into 3D space and the dimensional scaling factor calculated using Euclidean geometry. Is there a dimensional difference in these target projections?

The open graph one may consider is a single dimensional projection as it relates to taking a limit between adjacent generations, or is it. Does it introduce a degree of freedom or remove one, that is, it has taken knowledge of a solution and generalised it into a comparative metric. In the closed case we go from zero to three, that introduces three degrees of freedom. How should this be represented in the mapping.

There is one more thing to consider, this is a mapping of solutions, and the TPPT does not represent all the solutions … only the primitive solutions. Typically the ternary tree has a scaling factor of 1/3 and a dimensionality of 3. From any one solution you can move to any one of three child solutions (or freedoms), or along a continuum to another parallel solution set. That’s four degrees of freedom.

There are also four paths from any one solution to any one of four connected solutions in the closed graph.

 

My Share of Rabbit Holes

To anyone who has read my maths adventure this far you will realise that maths is full of twists and turns and it’s fair share of rabbit holes. After some soul searching I have decided to leave this blog intact for posterity with all the rabbit holes in place.

The one conclusion I have come to after all this time is that my original result (the paper on Graph Scaling) still stands out for it’s core meaning. The implication is that the triples inherently define a three dimensional substructure of the counting numbers. That is, three dimensions are directly implied, the rest being noise. These dimensions are pure, they are NOT Minkowski space time, but the underlying spatiality of MST is core and doesn’t need complex and artificial construction (such as with the bubbles in quantum gravity) … it’s just there.

I have still to make any headway in terms of perturbation (nature of randomness) and instancing in this model. My older model seemed to have these features, but it was a rabbit hole!

I feel that the nature of the dimensional mapping is still important, and that modular forms may be in there somewhere giving rise to interference and perturbation. At any rate, the paper needs to be rewritten, i’s a first cut, but the significance of the result is the same.

It’s a Square World After All

An analysis of exclusion space and octahedral geometry shows us that probabilistic different versions of octahedrons that do not join the cluster are isolated and hence out of scope. That is, they place no further role in the universe. They are the noise. Only octahedrons that joins are signal and so we concentrate only on these.

In order to join they suffer a constraint. No longer able to take on all 54 possible configurations. The second to join is now constrained to only 18. As we add to the chain each new on has 18 possibilites but the inner ones are constrained to only 6. Finally as we move from a chain to a three way joint the possibilities fall to 2. The two-ness just won’t go away, we are stuck with it as this represents each of two mirror images or isomers. We can break this pattern by forming a cross-chain between two out-reached arms. Such a cross-chain sets in place one of the isomers on each side.

It is interesting to note that in octahedral geometry there are three planes that intersect orthogonally. This means that at a three way joint each new chain path is orthogonal to the others. This gives the overall impression of a set of right-angled square pathways, like a crazy jungle-gym in a kid’s park.

Think ahead now to the event surface at around K. The network is littered with loops and cross connections, probability constraints and exclusion have played their part in the network as it stands at n=K. Where now is the signal.

Let’s take a step back to better understand.

We are here because a2 + b2 = c2 has more than one solution.

There are multiple (a, b) and where there is coincidence in the TPPT we get octahedral definitions with matching edges. I would predict that as n gets larger, and as the width of a TPPT generation gets wider the possibilities for edge matching also increases. Hidden deep in the structure are parcels of certainty, and built on top of that uncertainly again outwards to the horizon.

According to some papers on spin foam they are looking for both a notion of spin and an evolution loop defined space over n to produce cells. The problem I have at this level is that what the hell is spin and why would loop space evolution over n be a source of signal. At some stage I would like to formulate a projection into 3-space without n, but I can’t see spin foam ideas at work here.

So, how do loops evolve into spaces? Unlike the spin foam and loop quantum gravity diagrams exclusion space is square, not loopy, there are no four ways vertices and all three way vertices (nodes) have their arms orthogonally. This means that a chain cannot be turned into a three way at any node because there are no free edges and uncertainly has already reduced leaving no option to rearrange the  entire chain. Only where an elbow exists is there a free edge. In general there is no preference for straight chains and elbows are just as likely, but this has yet to be proven. Maybe only a large scale simulation will shed some light on this. Nevertheless, the spin foam evolution idea that seems rooted in the simple line diagrams that appear in the papers don’t seem to map easily to a world more like a plumbing game on your phone.

At least we can assign some metric to the square loops. For example, they have a defined perimeter in terms of a node count and there is a notion of loop area and even loop volume. But it’s not signal.