MANIFOLD

This came to me in a dream: I saw space fractured into a zillion pieces. And not just random shards laying on the ground like a smashed crystal wine glass, no, they were ordered, all the diverse sizes and shapes imaginable intricately arranged.

The idea of a simplex is easy to visualize. They're dimensional. A 2-simplex can be thought of as a closed triangle with three vertices and three line segments -- or edges -- acting as its boundaries connecting the vertices. But, topologically, it can take any cell-shape, a cell being a generalization of a simplex. Each cell is homeomorphic -- maps invariants to invariants -- to a Euclidean space. In that sense, they resemble a manifold. Now, the term manifold is oftentimes generalized to mean any flat surface, but its significance lies in the fact that compared to a polyhedron, it is all of one piece. It has no sub-partitions joined together as does a polyhedron. In that manner, each manifold corresponds to a local flat Euclidean plane.

So we have simplex equals cell equals manifold. By the use of barycentric coordinates we can further sub-divide the cell into simple quotient spaces to create a polyhedron of space. With each factoring we create a composition series, further refining the cell's detail. The cell, it may be pointed out, is susceptible to continuous deformations. In the process its interior morphology rearranges its play of symmetries in a dynamic and self-organizing manner. And as the interior of a cell is sensivitve to its boundary shape, it is said to be shape-aware, analogous in regulatory capacity or methodology to the morphogenetic field of a biological cell. An important point that shouldn't be underestimated concerns the orientation -- spin -- imposed on a simplex, infusing it with a causal asymmetric tendency effecting ripples and contours. This feature eliminates random inconsistencies.

A change of basis takes place; we zoom in. Barycentric coordinates expand the vertices onto a larger scale, adding degrees of freedom, revealing underlying dimensions. As the space grows in complexity, each vertex, boundary and interior point transforms into a linear combination -- weighted by the barycentric coordinates acting as coefficients -- of the newly defined vertices. Essentially, the original coordinate system is relabeled. Also, any number of lines can be drawn through the interior of our cell, intersecting an edge at a point. For instance, for the triangle or 2-D simplex, a line can be drawn from each vertex to a point on its opposite side (edge), this point is then written in terms of the basis of three weighted vectors -- our barycentric-ized vertices:

point = a₁v₁ + a₂v₂ + a₃v₃.

For example, an electron can correspond to a factored network of quotient spaces in a fluctuating state, stuck in a loop due to the particular configuration, compressed into a bounded condition. Increasing and decreasing refinement within a certain range corresponds to its everchanging position and momentum. As the space of that cell increases in detail due to fluctuating factoring, a photon is absorbed. As it decreases, a photon is expelled. This represents a probability distribution as a result of, or corresponding to, the quantum partitioning. The electron can be localized by the barycentroid (or multiple barycentroids) of the cell, insensitive to noise and deformations.

Looking at our cell-complex from another perspective, continually varying spatial vortices collectively forming an ever-shifting strange attractor, an emergent phenomenon, manifests in the macro-world as spacetime, the fabric of it, to use a hackneyed mataphor. Piecewise, a strange attractor renders the overall appearance of the cell as it goes through its fluctuations, fractalizing space. A collection of these cells (simplex, node), interfacing at their boundaries (face, surface), on any required dimension, induces neighboring individual -- quantum level -- cells to arrange their interior structures in sympatico, either by means of extensions of partitioning lines from the source point crossing respective boundaries, or through resonation of sub-vortical frequencies.

Think of a cone made from a sheet of paper as you twist its pointed-end tighter and tighter. That is the space, or the simplicial complex, of gravity, barycentric coordinates standing for centers of mass [barycenter means center of mass or gravity]. Space, taffy-like, creates the surface appearance of curvature due to its contraction and stretching, to its folding and squeezing. The underlying cause may be attributed to the increasing refinement -- or factoring -- of the sections of simple quotient spaces -- factor groups of denser symmetry -- spontaneously generating, and nested in a hierarchy of sub-domains. Three-dimensionally, we can think of a tetrahedron as homeomorphic to a solid ball, an image that seems to be more agreeable for its natural symbol of centering.

Onion-skin slices superimposed on one another, interconnected and interrelating, subject to positive amplification and negative feedback constraints. Layers of spacetime embedded within each other -- concentric spheres -- yet separable along an imaginary axis. Viscosity thickening towards the bottom or center, bubbling up new micro-states from nano-moment to nano-moment, evolving fresh pseudo-stable surfaces through some kind of force convection.

Simple refers to non-divisibility. Each factor group -- each sub-domain of compacted space -- is a whole unto itself, locked in a preferred arrangement or quantized set of arrangements. The mesh size of any specific simplex net varies relatively, but must be no greater than a suitable Planck designation or limit based on the compact nature of the associated Lie group in Hilbert Space. An appropriate distance-metric can be imposed from without but understood only as an artifact for purposes of calculation. Being topological, the underlying field has no inherent metric; in a topological space distance has no meaning. Objects exist in and of themselves without reference to any background coordinate system. Furthermore, thinking of space as a fluid medium allows us to forgo the associated notions of rigidity and flatness of space inherent in our concept of manifold. It bends and twists and can reform into any conceivable topologically valid shape. Time and motion are thus seen as effects of this constant partitioning. Particles as constricted loops, trapped in a web of symmetrically bound energy. Forces as changes in the loop structure due to fluctuation of the spatial components -- the factors.

It's the energy of the Void itself that intiates this creative endeavor.

As John D. Barrow writes: "Space and time can be born and thus can die."

Adrian Dorn

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