4d DOMAIN CONNECTOR

In this post I will outline my idea of a 4-dimensional link-compliments that are not of the Lobachevskian kind, but could be better termed as O'Neill Space. I suggest that this concept has great applications to the field of cosmology and will outline said applications in this and later posts. To begin though we need to a bit about modular forms, link compliments and Lobachevskian space.

3D Fundamental Domains and Lobachevskian Hyperbolic Space

There is a video on Youtube, called Not Knot, that goes some way to explaining the concept of fundamental domains of link compliments. Link compliments are any closed loop of string (or strings) that consist of any number of crossings. In the past, mathematicians had difficulty identifying if one link compliment was the same or different to another. But, by breaking down the loops into discreet sections called fundamental domains they can avoid this confusion by building a picture that is specific to each knot.
Note; To quickly visualize what is meant by a fundamental domain, think of a cube. When you flatten out its sides you get 6 squares in the shape of a cross. This is the fundamental domain of a cube. According to the video, there is only one fundamental domain for each link compliment making them very useful for identification purposes.

In the case of the Borromean Rings (as seen in this video) the looped sections are at right angles to one another and the fundamental domain is that of a cube. The domain is copied outside itself infinitely until it tiles all of 3-d space. 

By taking the fundamental domain of the Borromean Rings and adding a further 90º degrees to each axis we can make a dodecahedron that tiles all of space in a similar manner. This is called Lobachevskian hyperbolic space.

Modular forms such as these were imperative in solving Fermat's Last Theorem, but some mathematicians (like researcher Jeffrey Weeks) think that they may go even further by explaining the exact size and shape of the Universe itself. If correct, it would mean that the Universe is  finite in size, consisting of a single dodecahedron-like structure that is mirrored across space and time to give the illusion of an infinite universe. Our galaxy and everything else that is inside this fundamental domain would also be mirrored across space, resulting in an infinite number of copies of the Earth, you and me.

4D Fundamental Domains and O'Neill Hyberbolic Space

It baffles me why mathematicians have been content to chart the fundamental domains of 3-dimensional knots, but have neglected to investigate four dimensional ones. It seems to me that if you want to accurately create a picture of 4-dimensional space-time you would need a 4D fundamental domain.

In an infinite universe (whether it is repeated or not) giving your fundamental domain any specific size seems arbitrary. A 4D domain gets around this because all of its domains are free-scale. 

To create a four-dimensional fundamental domain we could begin by taking the order-2 domain of the Borromean rings and following each of the steps in the first video until we have created our 3D grid pattern stretching off into infinity. Then we can rotate this pattern around the 4th dimensional axis of the original cube until all of the domains that lie outside of the original cube domain now also lie inside of it. Obviously this would have to be repeated with each cube in the matrix (an infinite number of them), and then within each cube within each of those cubes (producing a far greater infinity). This mirroring would continue ad infinitum in a fractal manner.

Furthermore, each time a new cube is mirrored, it would alter the pattern relative to another domain, meaning that it in turn would need to be mirrored over again. This process, which would start off slowly at first would quickly escalate, as the 4D domain becomes more and more connected with itself. (See the next post The Singularity Conjecture for more on how this pertains to reality and the future).

Next, we have to understand what an infinitely tiled hyperbolic grid would look like and how it would sit in relation to the rest of the ordinary 3D grid. To do this you start by making a 2D grid of 9 squares (left) and invert it so that the central square lies on the outside, and the other 8 squares sit in place of the original central square (see below). Next, you make grid with 5x5 grid and invert it, then a 7x7 one and so on. Once you have mastered this you can begin to progress to a 3d grid structure, the inversion of which would be a hyperbolic grid.

However, just because you have mastered this does not mean that you can create a 4-d fundamental domain with ease. There are still many thought-processes and trials which must be taken. The apparent shape of one of these domains is a rhombicuboctohedron (see below), and which, oddly enough, is related to the dodecahedron of Weeks domain proposal;

However this object must exist inside a our original cube domain, with lines leading to each vertex, shown below;

These lines join to create the following number of shapes; 6 cubes, 8 tetrahedrons, and 12 triangular prisms. The rhombicuboctahedron can be rendered in the following topological manner;

These are three different angles of the same hyperbolic domain. The original domain looks something like this (see below);

The net of this form looks like this;

And finally the 4D domain looks like this;

In the infinite grid of cubes, the central cube that is our original domain is surrounded by 26 cubes. This block of cubes, which is also a cube, is further surrounded by a nest of 98 cubes, which is surrounded by 218 and so on. If we imagine that our original cube domain has a volume of 1 metre cubed then the next set would have 26mˆ3, then 98mˆ3 and so on. When all of this is rotated around the fourth dimensional axis, it is clear that space is becoming larger the closer you get to the centre.

This doesn't make any sense, the space is becoming larger on smaller scales, which ought to be measurable in some way. The reason why it isn't is because these domains lie for the most part stacked in the fourth dimension and a considerable amount of their bulk lies hidden there, only accessible to us through line of sight perspective. However, it is clear that this space wants to find room for itself in 3D space, and this need, it appears, is driving the expansion of the Universe.

O'Neill Space is unpacking itself from the fourth dimension. The entire Universe is exhibiting this unpacking in the form of expansion just as an object that is slowing down from a tremendous sub-luminal speed undergoes a length expansion. 

As each nested cube expands, cubes within those cubes being to expand so the whole process is exponential. Furthermore, because the tiling of the fourth dimensional fundamental domain is infinite, this unpacking will continue indefinitely. 

Perhaps even more phenomenal, is that this theory suggests that the fourth dimension is not time, as Relativists believe it to be, but rather ordinary space. The process of its unpacking from one dimension to another, however, does take place over time, and this means that if you travel fast enough you will undergo a length contraction taking you against this unpacking flow rate. Obviously the further you gain access into the hyperbolic grid structure of O'Neill Space the slower the unpacking appears. This could mean that the boundary between both spaces is something like the event horizon of a black hole; a point of no return for these cubes of space. Time is nothing more than the perceived rate at which fourth dimensional O'Neill space is unpacking into ordinary 3D space, in this writer's opinion. 

Note; There are three basic topologies for the Universe; positively curved, negatively curved and flat. According to the main proponent of the finite universe theory, Jeffrey Weeks, the existence of dark matter does not favor a negatively curved universe. This is strange, however, because the dodecahedron tiled space that he envisages is a Lobachevskian Hyperbolic and therefore negatively curved topology. There must be some explanation for this obvious disparity, but I cannot think what it might be.

Note; While space-time is known to be a 4-dimensional vector plane of 3 space and one time, its topology is not explained by the 4-dimensional algebra equations of the quaternions. The reason for this is that quaternions are only spatial and do not deal with time. For this reason some scientists have concluded that the 4th dimension is not time, but an added spatial dimension interacting with ours over time. Time, they believe, must be an extraneous principle acting over all dimensions. I would agree with this concept.