During this holiday season, I wish to leave the readers with some thought on the shape of existence. Probably, despite my intention to bring light upon this subject, I might leave you in doubt or confusion. But remember, it is upon the eyes of the beholder to make sense of what they see, and interpret to their utmost pleasure.
‘Geometry’, the mathematical science of the shape or curvature and arrangement of the internal structure of an object. It measures how the internal features of an object change and differ along its surface and volume. Topology in contrast, is a study of objects that might differ in geometry, but somehow appear to be similar, such that the union of objects under consideration are members of the same family. For example: A sphere and a cube are topologically similar, as they both can be inflated or deflated without addition of any external object or feature, in order to be transformed into each other. Whereas a sphere and a donut are not similar, because you cannot transform them into each other unless you carve a hole in the sphere or vanishing the hole in the donut, I.e without subtracting or adding something, they are not interchangeable. Though geometry may differ in all cases.
I suppose, the difference has been established above, as tersely as possible. I wish to move to its application on our existence and the universe. Now I will start with the topological aspect of things. There is a metric called chern class, developed by the mathematician S.S Chern. I will not go into the technical explanation, but for the purpose of my intended topic, I will try to simplify it in lay words to the best. It describes the number of dimensional objects that can fit into another particular dimensional object. It somehow tells us more about the nature of the bigger space that holds the smaller ones inside it, which will facilitate my elaboration of this math to the Universe itself.. For example: A one dimensional line can accommodate a zero dimensional point or the line itself, I.e zero or one degree of movement. Similarly a two dimensional sheet of paper can accommodate, a point, line and even a drawing of any two dimensional like a square or a circle I.e x-y axis movements.
Moving to our universe, it supports all the sub-objects or I would refer to them as sub-manifolds, from three to zero degrees of movement. The highest level of sub-manifolds it supports is three in accordance to its own physical dimensions of three. According to this geometrical observation in existence, we can understand that there always exists a manifold with dimensions that are equal to or higher than the highest dimensional sub-manifolds it can support. In short, if ‘n’ if the number of sub-manifold’s dimension then there always exists a space or more accurately a manifold of the dimensions ‘n+1’ to contain it. So if our universe supports 3-dimensional objects there has to be a space that is holding this 3-dimensional in a 4-dimensional arrangement. And a 5-dimensional arrangement holding that 4-dimensional space and so on, who knows this goes on till infinity.
Now geometry of space is the internal arrangement and shape of the universe. This geometry is defined by how it it is being stretched apart in that higher (4,5 or x-dimensional) space. As the universal space is stretched, it decides the shape of the space at any point and hence the shape of the object that is formed at that point. Remember it is always the vessel that defines, what shape the fluid that is filled in it, will take. So for believers of higher power in an objectified form or energy, that exists outside our universe, from this I wish to demonstrate that if that higher power is external species sitting outside our universe creating and affecting it, then they must be sitting in the 4,5, or x-dimensional space. Then as geometry and topology works, there has to be a space above encapsulating them. The most rudimentary understanding that can be gained from our existence and the space we inhabit is, that without a containing vessel, what could take shape and form. So, technically there might never be a God, as there always be a vessel of higher dimension containing above and above in layers, deeming the lower dimensional being, always lower in rank. Even the multiverses concept require a space of high enough dimensions to hold all the multiverses.
I do not intend to preach any idea, or offend anyone’s belief, it is only my intention to put light upon basic observation of existence and what it could mean for our understanding of it. Even if there is a higher power, we could never understand it, especially if it extends to infinity. And that true higher power might not have an objectified form.
The shape of things and the surface they exist on, are the determining factors for the physics that prevails in a system. Everything that may exist might be a geometry of infinite or unknown layers of dimension, something ever unfathomable by human mind.