Dimensions of Space

Einstein’s Theory of General Relativity and Quantum Mechanics are two of the most significant theories in physics, yet they are fundamentally incompatible. Despite this, both theories excel in predicting the phenomena relevant to their respective domains. Their primary disagreement revolves around the concept of gravity. Professor Carlo Rovelli has formulated a theory called Loop Quantum Gravity to articulate the nature of quantum gravity. Rovelli posits that:

> Space manifests as a spin network where its nodes symbolize elementary particles, while the connections depict their spatial relationships. Space-time arises from processes where these spin networks morph into one another, described by sums over spinfoams. A spinfoam embodies the history of a spin network, resulting in a granular space-time characterized by the merging and separation of the graph's nodes.

> … What constitutes the essence of the world? The answer is straightforward: particles are quantized fields; light is a product of quantized fields; space is merely a field also composed of quanta; and time emerges from the activities of this same field. In essence, the universe is entirely formed from quantum fields.

This series suggests that the structure of space is composed of Platonic solids, with their vertices and edges paralleling the nodes and connections of Rovelli’s Loop Quantum Gravity.

Tarski’s findings indicate that Elementary Euclidean Geometry serves as a comprehensive axiomatic framework within AdS space, independent of the dimensions of the Platonic solids, meaning various sizes of Platonic solids can coexist within the same space.

Anti-de Sitter Space

A notable characteristic of AdS space is its potential infinitude, balanced by a boundary that encapsulates all elements of AdS space in one lesser dimension. Essentially, AdS space possesses a finite boundary, which acts as a holographic representation of a potentially infinite AdS expanse. A visual analogy for such a boundary can be found in M.C. Escher's wood carving, "Circle Limit 111."

The original AdS space is comprised of Platonic solids of varying sizes. The boundary within AdS space serves to delineate all content (shapes) as a hologram, represented in fewer dimensions. Each shape can be characterized in the hologram through its length and number of sides—using only two rather than three vectors. Furthermore, the boundary in AdS space implies that only a finite number of shapes can depict the space infinitely. This feature can be illustrated in the hologram through the arrangement of shapes by increasing size, employing fractal descriptions, i.e., self-similar shapes. The reduction of an infinite sequence to a finite assortment of shapes parallels the mathematical principle of renormalization. As Wikipedia states: “Renormalization is a technique … in the theory of self-similar geometric structures … [that is] used to address infinities arising in calculated quantities by modifying these values to counteract the effects of their self-interactions.”

Shapes within the boundary could be arranged by fractals of descending size. To differentiate between the forms in AdS space before and beyond the boundary, the shapes representing those beyond could mirror the original shapes but inverted.

In geometric terms, a pseudosphere is defined as a surface with constant negative curvature. Mathematicians have established that the surface area of a pseudosphere is finite, despite its infinite extent along the rotational axis. One may envision AdS space as a three-dimensional pseudosphere.

The edge of the pseudosphere resembles a boundary within AdS space; a boundary that contains data regarding all Platonic solids ‘beyond’ its rim. Mathematically, these Platonic shapes can ‘overlay’ other forms. As elaborated in Appendix D of the book Physics from Finance, fibre bundles are mathematical frameworks utilized in quantum mechanics. These bundles assist in describing the product of two spaces, illustrating how one space can be mathematically mapped onto another. A Platonic solid may function as a specific type of fibre.

A point from one Platonic solid could connect with a point on another Platonic solid within the same space. While many of these connections may be random or trivial, some bundles may exhibit a significant global structure. A Mobius strip, for instance, is formed by attaching a line to every point on a circle with a twist in the fibers encircling the circle. Another noteworthy example is a Klein bottle. In a space resembling a Klein bottle, Platonic solids become inverted. In summary, the negative curvature of AdS space filled with Platonic solids naturally leads to the emergence of complex descriptions of the spatial fabric.

AdS space is isotropic, maintaining uniformity in all orientations. It also exhibits spatial homogeneity, being identical everywhere at a given moment. Since the boundary contains a hologram of all shapes within the AdS space, these shapes are integral to the content of the entirety of AdS space, composed of both right-side-up and inverted shapes. The inclusion of both kinds of shapes aligns with the topology of a Klein bottle; locally, the surface displays negative curvature, while globally, the topology reflects that of a Klein bottle.

Inside a Klein Bottle

Each Platonic solid has a corresponding dual, a solid whose vertices are located at the midpoints of the original solid's faces (in two dimensions, a Platonic solid and its dual can be expressed using the same mathematical notation). Rotating a two-dimensional Platonic figure reveals its three-dimensional form. Mathematically, both a Platonic solid and its dual can be represented in one less dimension when combined with another parameter, valued at 0 or 1, where 0 denotes one form of the solid and 1 signifies its dual. Thus, a three-dimensional Platonic solid or its dual can be mathematically portrayed by a two-dimensional shape paired with a (0, 1) bit of information.

The duals of the Platonic solids include: - The cube and the octahedron are duals of each other. - The icosahedron and the dodecahedron are duals of one another. - The tetrahedron is self-dual.

George Spencer-Brown’s Laws of Form outlines the mathematics that could govern the movement of information into different regions, such as the event horizon of a black hole. Information contained within a black hole represents a Distinction, a ‘mark’ in a region that is separate from the original AdS space. A more detailed discussion of Spencer-Brown’s Laws of Form will appear in a subsequent article. The key takeaway is that entering and exiting a black hole can be interpreted through Spencer-Brown’s mathematical framework.

According to Wikipedia:

> In mathematics, an abstract polytope is an algebraic partially ordered set or poset that encapsulates the combinatorial properties of a conventional polytope without specifying purely geometric attributes like angles or edge lengths. A polytope generalizes polygons and polyhedra across any number of dimensions.

As stated by wiki.org:

> The duality of a pair of abstract polyhedra represents a specific relationship between two partially ordered sets, each signifying the elements (faces, edges, etc.) of a polyhedron. Such a ‘poset’ may be depicted in a Hasse diagram. The diagram of the dual polyhedron is derived by inverting the original diagram.

Topology is a branch of geometry devoid of dimensions or one that considers geometries exceeding three dimensions. Often referred to as ‘rubber sheet geometry,’ topology investigates forms that can be ‘stretched and manipulated.’ The only stipulation is that all primary surface characteristics must remain intact during the transformation or ‘mapping’ from one form to another. When a form is ‘stretched’ into an additional dimension, it retains characteristics or ‘memory’ of its original configuration. Three-dimensional Platonic solids ‘stretched’ into a fourth ‘time-like’ dimension would still preserve the memory of their three-dimensional origins.

The topology of a Klein bottle can be equated to a three-dimensional shape possessing a fourth dimension. When the topology of AdS space resembles a shape akin to a Klein bottle, abstract polyhedra like Platonic solids, on the surface of that space, may revert to their original state, i.e., emerge from a boundary, but in an inverted position, thus appearing as their own duals. The Klein bottle’s topology aligns with an AdS space that establishes a Distinction, where this Distinction is encoded as a (0, 1) bit of information. Every element within such an AdS space holds information about the topology of the entire expanse.

Spencer-Brown asserts that the creation of a Distinction leads to the emergence of an additional dimension, a time-like dimension. In summary, when a three-dimensional AdS space adopts the topology of a Klein bottle, that space transforms into four dimensions with the inclusion of a time-like dimension. As discussed in Article 4, Could there be a logical explanation for our universe, a time-like dimension in AdS space might be represented by the Fibonacci series.

Klein Bottle and Mathematical Singularity

In conclusion, an AdS space consisting of Platonic solids can be mathematically characterized by the topology of a Klein bottle, which inherently generates a time-like dimension. Each component of that space contains information about the topology of the whole. Future articles will expand on the notion of AdS space-time adopting the topology of a Klein bottle to elucidate how the content within that space-time acquires quantum-like attributes.

The central question posed in this article is:

Could the origin of our universe have a rational explanation?

To acquire a copy of the book Orbiting Stars, which encompasses the initial drafts of all these articles, please visit https://www.amazon.com

To view the headings of all the articles scheduled for publication in this series, please click on https://readmedium.com/orbiting-stars-and-origin-of-our-universe-338906930f51