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.

L4Y3R C4K3 TH30RY

Proposal for the Unification of Quantum Mechanics and Relativity.

Accelerating a massive object to the speed of light, if it were possible, would be equivalent to sending said object through a type of inter-dimensional Young's slit experiment, in which all of its de Broglie wavelengths would interfere with one another to create a holographic image that extends throughout time as well as space.

The wave-function would be vibrating in phase with time and would therefore appear completely static in this dimension - a proposal which conforms to the Relativistic Principle of time contraction. This theory, which I have dubbed Layer Cake Theory, attempts to unify the concepts of quantum weirdness and relativistic principles in an intuitive and lateral kind of way.

LAYER CAKE THEORY

Experiments involving the passing of a single photon through a double-slit grating, and the interference patterns that resulted from this, were influential in the formulation of the particle-wave duality theory of matter. Matter, like that which composes your body, is said to be made out of waves of probability. On a small scale the position/velocity of a particle becomes uncertain. But when enough of these particles are combined together, renormalisation occurs leading to the collapse of the wave-functions into a stable amount of what we call 'matter'.

In one of his lectures, the famous physicist Richard Feynman made reference to an experiment in which a stream of photons were fired at a detector behind a solid steel plate. Every now and then the detector would register a hit. The researchers concluded that the photon had taken a trajectory allowing it to swerve around the metal plate and hit the detector. This means that when looking at a single sub-atomic particle that comprises your body, for instance, we may find that its probability wave-function is looping out towards the stars... Such is the weirdness of quantum mechanics.

When an amount of matter begins to accelerate, I propose that some of its particles accelerate out from it at speeds which are, in a sense, greater than that of light speed. This because these particles are accelerating in the dimension of time and not of space; looping out towards the future. The faster an object travels the more particles it emits. These particles are actually space-time manifolds, slices of the original object that travel into the future at tremendous speeds. Each of the manifolds are holographic (fractal) replicas of the original object. This means that as the object accelerates and more and more of these manifolds are ejected, its aspect (apparent size) begins to shrink in agreement with the Lorentz contraction.

Its over-all mass, however, is not depleted because the fractally encoded manifolds are representations of how the object appears at any one instance of time, and therefor each of them contain fractal information of the current state of the mass of the object in respect to its reference frame. Combined manifolds do not increase the mass of an object either, meaning that the mass is a fractal component of the entire structure. When an object, like a person, is at rest, they are the result of all of their space-time manifolds interfering with one another, to become a coherent image of themselves at any one moment.

The history, past, present and future of an object can be represented by an almost infinite number of space-time manifolds. The collective manifolds are fractal, because at any one time they all collapse to form one whole.

In the case of a large object, like the planet Earth, each separate instance of time in the life of this object is a separate space-time manifold. Due to the length and complex history of planet Earth the combination of this set of space-time manifolds at anyone instance is enough to warp the fabric of space itself.

I have already made clear that, in the case of an accelerating object, the loss of space-time manifolds does not decrease the mass of the entire object. So, I would appear to be contradicting myself by suggesting that the accruement of manifolds would lead to increased mass. But, again this is really not the case. It is true that larger objects of a similar density have more mass, but they also have a greater number of manifolds. This suggests that, on average, mass is equivalent to longevity. The more massive a body, the more manifolds it contains and the further these manifolds extend in 4-d space-time.


Again this is something that we also see on a more manageable scale with animals. For instance it has been suggested that elephants and mice have the same amount of heart beats (close to a billion) in their lifetime. But, in the case of the mouse, they are just happening at a far more rapid scale than when compared to the elephant. LCT would suggests that the reason for this is to do with the relationship between total body mass and longevity, although exceptions are sure to apply in the case of living animals, where so many variables exist.

Traditionally physicists have had a hard time marrying the concepts of Special and General Relativity with the weirdness of Quantum Mechanics. Layer Cake Theory (LCT) attempts to address this disparity and it does so in a very unexpected manner, by supposing that a massive object traveling at the speed of light is essentially equivalent to a particle that travels through a double slit grating. While the latter generates an energetic wave of probabilities stretching across space, the former is comprised of a highly energetic 4-dimensional probabilistic wave-form that stretches across time. A comparison between both Quantum Mechanics and Relativity, according to LTC is as follows;

Quantum Mechanics                                                   Relativity

---

Wave-particle duality                                                   Matter/mass accelerating towards
(as expressed by Young Slits experiment)                the speed of light.

---

the same particle/space being in two                       The same temporal instance being
or more different places at once                                being in two or more different instances

---

The accretion of matter/atoms creating                   The accretion of space-time manifolds
the renormalistion of quantum uncertainty              creating stable time

---

STELLAR OBSCURA

Black Hole Quantum Gravity Theorem

In order for a theory of quantum gravity to work in needs to be able to deal with gravitational singularities on the quantum scale. In my post, Hyperbolic Perspective and Quantum Gravity I outlined a theory which describes how gravity works on very small scales, but it is clear that something else is needed entirely for it to describe a gravitational singularity on the quantum scale.

The Inter-Dimensional Merry-go-Round

In the wake of his discoveries - regarding length contraction at speeds approaching that of light - the world renowned physicist Albert Einstein attempted to unify this concept with the gravitational spin of the Earth. He knew that the surface of the Earth rotated much faster than its core, and this meant that it must also undergo a more pronounced length contraction. The object he used to imagine this type of geometric contraction was the merry-go-round. He proposed that if a merry-go-round were to spin at very high-speeds, the outer-edge of the disk would undergo a length contraction forcing its edges to curve upward. From this he deduced that space itself was curving and this led to his Theory of General Relativity.

Just as a black hole warps space to an extreme degree we can imagine a merry-go-round that is spinning so fast that the resulting curvature warps it into a sphere. Shifting an object from one dimension into another higher dimension is an excercise in trans-dimensional geometry, which is exactly what we have done here, taking a flat 2-dimensional disk (merry-go-round) and turning it into a 3-dimensional sphere.

In the case of a super-massive star that is about to turn into a black hole, we need to imagine that we are taking a 3D sphere and transforming it into a 4D sphere.

As the star shrinks its surface is becoming more and more length contracted, which corresponds with an increase in the degree of space-time curvature and thusly in gravity. Although we can deduce from this that gravity is at its strongest with respect to the surface of an object, we also know that the force of gravity continues, as in the case of the Earth, down through the mantel to some degree until a null-point is reached at the core. It should be apparent, therefore, that the quantum singularity of a black hole is not 'the centre' of anything, but is in fact only that region of the gravitational field which corresponds to the surface of the star. This means that the entire surface of the super-massive star along with its gravitational field has turned in on itself, and its entire surface area (or circumference) has been reduced to zero. 

You might be wondering where the mass of the star has gone that produces these gravitational effects.When a star collapses it can be said that its entire mass is undergoing a length contraction, in all 3 dimensions. This is a Special Relativistic effect, which implies that the mass of the star is accelerating towards a central point, faster and faster. It reaches the speed of light (the event horizon) and then necessarily exceeds it. Once the star reaches twice the speed of light (at the plank scale) it turns itself inside out, resulting in the gravitational singularity. This effect is known as length expansion and is dealt with in the post titled Direct Relativity.

From then on the singularity can be described as an aperture into the 4th dimension. This is just like the aperture of a camera, or more accurately a camera obscura. In a camera obscura, light is allowed into a receptacle through only one point of entry (the size of a pin-hole). This has the effect of focusing the light at oblique angles creating an inverted (up-side down) image on the back of the black-out receptacle. A gravitational singularity has the exact same effect on the stars mass, only - by virtue of it being a 3-dimensional aperture - it turns the star inside out as well as upside down. In this way the singularity can be seen as a twist in the fabric of space-time, occurring at the Plank Scale.

Although a singularity cannot be said to have any sides, for the sake of argument (and common sense) lets imagine that an asteroid is being dragged towards the right-hand side of a black hole. From a holistic point of view i.e. one that spans all dimensions, the asteroid is actually being dragged towards the left-hand side of the star's mass, because the singularity has inverted the stars orientation. This type of inversion is what makes it reminiscent of the camera obscura.

In Hyperbolic Perspective and Quantum Gravity, I used a hyperbolic tiling to express how quantum particles, which are under a high degree of magnification, do not exhibit a noticeable degree of hyperbolic distortion (i.e. gravity). This same image could be used to describe a cross-section of the Earth's gravitational field; in which the bulk of space-time curvature is perceived at the edges and dead-space or null-point seen at the core. In order to create a singularity, all we need to do is invert the hyperbolic curvature. 

However, this still does not help us visualise where the star's mass has gone and what is taking place in the fourth dimension. In order to do that I will reduce the number of dimensions down and flatten out space into just 2 dimensions. The grey areas, which denote empty space are merely the result of flattening out the 3-4 D environment out into a 2D image. Such artifacts are regularly observed in map projections, when trying to reduce the curvature of the earth into a 2-d image. As in the case with the map projections no such areas of empty space really exist and motion from one area of space to another can be achieved without undue effort. This then is the fundamental domain of a black hole.

Here we see that the lines of force generated by the gravitational field of the star are being inverted through the singularity at the Plank Scale. What this tells us is that the boundary plain to the four dimensional universe is the 2D limit of the Plank Scale. To understand this same diagram from the point of view of the star in the 4th dimension we need to invert the image. I find this image to have striking correlation to a torus, or a flower. Marko Rodin would say that this is significant. I tend to agree, but I won't be dealing with exactly why this is so until a later date.

So you might be wondering, what is so new about this? Well, physicists generally assume that the mass of the star, while being compacted to an infinite density, remains in this dimension. Others (Marko Rodin for one) have supposed that the singularity may be a gate-way to another dimensions, whereupon the black hole shifts output to produce a white hole. I maintain that when the stars mass falls into the fourth dimension it remains in a static condition; for the obvious reason that there would not be enough mass to support the singularity otherwise.

The reason why the time does not impinge upon it is to do with relativistic laws, but more importantly, because in the 4th dimension time becomes space - and space becomes time. The only way that the star can gain or loose mass (not likely) is through its interaction with the 3D time-based universe. But if the black hole is frozen in time, does that not mean that it should be left behind, as time moves on in our Universe? It should, but because space acts like time in the 4th dimension the mass of the star is continuously dragged along into our time-frame of reference.

The fact that the fourth dimension is intrinsic to the geometry of singularities has been known about in the computer gaming industry for decades. This industry uses a 4-dimensional algebra set, known as the quaternions, to create smooth graphics and to allow for uninterrupted rotations around a 3-D environment. Without the use of the quaternions something known as Gimbal Lock occurs, which is basically defined as a singularity. This means that from a fourth dimensional perspective the singularity does not exist. Refer to the previous two diagrams.

If it has been known that quaternions, and therefor the fourth dimension, are intrinsic to the production and erasure of singularities, then why do physicists insist on thinking of the black hole as being something which exists in our dimension alone? Mathematicians have noted that quaternions/octonions are primarily spatial representations and therefore do not include time as a part of their framework. But I think that they are missing the point. There is no real fourth dimensional space, in my opinion, 4-d space looks exactly like 3-d space only we have swapped one of the spatial dimensions for a time dimension – see post; Within the Octonions for a fuller explanation of this.

As a final way to visualise what is taking place with the stars mass, take another look at the video of the rotating tesseract. Imagine that small cube at the centre of the structure is the uncollapsed star. Then, as the cube rotates, notice how it becomes distorted (length contracted) and finally turns itself inside out.