Monday, 17 December 2012

Tesseracts, Dimensions, and Reality

Okay, the past few days I've had a bit of an obsession on my mind. You see, mathematics of almost all types have had a special attraction for me my whole life and whenever someone brings up a subject like this to think about I go a little overboard.

I've been thinking about tesseracts. For those of you who do not know what a tesseract is, it is a 4 dimensional cube. That is to say it is a cube that is extended into a fourth dimension. It is rather difficult to visualize normally because we visualize things in 3 dimensions. But one way to help with that is to think about how we extrapolate a 2 dimensional square into 3 dimensions as a cube. The old imagery used is describing how we might try to describe a cube to a 2 dimensional world by unraveling it. The image is often represented as such:

That is basically how the tesseract is produced by “unfolding” the 4 dimensional cube so to speak into 3 dimensions by extending a cube(another “side” so to speak) from each of the 6 sides of a cube with one more extended that wraps around to the final side of the hypercube. One famous representation of the tesseract is Jesus Christ on a tesseract as a cross by Salvador Dali in 1954. It looks like this.

As I struggled to visualize this mathematical abstraction I began to argue with myself whether the 8th extended cube which represents the final side of the hypercube was actually enough. Who is to say that the visualization we apply to the 2nd dimension from the 3rd will work the same way from the 4th to 3rd? After all we can't actually see a 4-d cube nor visualize it so there is no way to really verify it by simply unraveling it like we do with the cube to squares. We can only abstract it and calculate it. But after thinking on it a while I realized a couple of things.

First I tried to see a pattern, something all mathematicians do to be able to abstract or calculate something further, and the more data points you have the more you can verify your calculation. So I need a little more data so I decided to apply what we did with the cube into a 2-d plane into a 1 dimensional plane, something mind you that is actually just as hard to visualize for us as a 4-d plane. But ultimately a 1d plane is simply a line, there is no height or width to it, only length. Though ultimately our conception of a line still has those technically, we can at least visualize that to a degree. When we unravel a square into the 1-d plane we just get a longer line, 4 times as long as one side of the square actually. But then I realized, the 1-d unraveled square has 2 times as many sides as there are dimensions of the original. A similar pattern is seen of the 2-d representation of a cube in that there are 6 sides to the cube, exactly 2 times the number of dimensions. This verifies the 8 cubes representing the tesseract, being 2 times the number of dimensions. We can then subsequently extrapolate to further dimensions requiring 10 tesseracts to represent a 5-d cube in a 4-d plane. 12 5-d cubes for the 6-d cube, ad infinitum. Its simply fun mathematics.

But what about the 0-dimension? Okay, yes now you think I'm crazy, but can you think about zero dimensionality? I know that if I were to try to unravel a line into the highly hypothetical 0-dimension I would think of it as two infinitesimally small points, not having length, width, or height, merely having two points that represent the ends of a line. Thats just how I would think about it. The question is the whether the reason I think of it that way is independent of my little formula of 2 times the dimensionality we're unraveling from. In this case 2 times 1 dimension becomes two points.

But think about it, does 0-dimensions really sound so off track to you when you think about black holes? Now I'm not saying I believe they exist because I disagree with the whole reason for postulating their existence, but those who do think they exist seem to be describing exactly what I'm postulating with my 0-dimension to me. Sure they'd argue semantics with me saying that in actuality there are dimensions, they're just so tightly wrapped infinitesimally small, kind of like the postulation of the big bang where the dimensions supposedly popped out of. Both of those are postulating pretty much what I'm describing here. Ultimately this is all still just idealistic mathematics.

But mathematics are not reality. They try to describe reality, but they are not reality. We could postulate various versions of universes that do not follow the laws of physics that we observe and still come up with completely legitimate mathematical equations to describe them. Thats why mathematics is so clean. It is idealistic, not realistic. And when we come to the question of dimensions I have the question the cohesion between this mathematical idealism and what the dimensions we observe and postulate are really like. Quantum physicists now postulate 10 dimensions of reality, or traditional 3 dimensions, a fourth we haven't really thought about much as a dimension in history as time, and 6 others that are supposedly tightly bound up with one another. The idea of the tesseract and its 3-d representation is based on the assumption that all of these dimensions are ultimately the same and we can unravel them all in such a manner so to speak. Thats what I take issue with, that assumption that all dimensions are basically the same in nature. Why do we assume this?

Take time for example. We recognize it as a dimension and in fact Einstein concluded he could not separate space and time. But in our normal 3 dimensions we can measure everything using the same measurements, such as meters. Whether it is up, down, forward, back, left, or right it can all be measured in a spatial manner. Time can not be. If we were to involve time as the 4th dimension in our tesseract we can visualize it as a further dimension. The longer the cube is in that dimension as we measure it, the longer that dimensions is. Ultimately though to make a cube all sides must be equal right? So then, how many seconds equals a meter? They are fundamentally different. Another thing is that we can somewhat freely move about in our 3 dimensions as I can choose which direction I move within the bounds of our physical laws such as gravity. However with time I can not choose a direction. It is unidirectional. And that in itself shows us its fundamentally different nature from the other 3 spatial dimensions. So why can't the other dimensions be fundamentally different as well? Is there any spatiality to them in the way we understand it? What if the other dimensions have nothing to do with moving through space or time, but rather measure the amount of energy of something or the electromagnetic orientation or such things as that? What if some of the dimensions are binary, rather than being able to move “infinitely” in either direction? Some of these questions have a few answers such as the nature of some or all of these dimensions I'm sure by physicists and I'm only asking questions. But I still think that the dimensionality is not quite so simply as unraveling the hyperdimensions in the same way we can from 3 dimensions to a 2 dimensional plane. Thats all I'm thinking.

Ultimately though dimensions are measurable and finite. How do we understand eternity in a time-bound reality? How do we understand an infinite God in a finite universe? Only through revelation. We are a 2-dimensional people being shown a cube outside the bounds of our universe and given glimpses of that larger reality through revelation. The revelation of Jesus Christ, being the Son of God, born as a man. He entered into our finite existence in order to show us and bring us into an infinite reality for which this reality is but a reflection and shadow. If you take it down to the Planck length of space and time, many have concluded that we ultimately live in a holographic universe. Something humbling and enlightening when you consider that Paul tells us the same thing when he says we see but a dim reflection. This reality is only a curtain. One day it will be pulled back entirely as we stand in the presence of a Holy God. When that happens, you'll either be ready for it or you won't. I suggest you be ready for it by getting to know what you can of Him now.

For His Glorious Name,

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