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.