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Old 24-06-2011, 06:04 PM   #2
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Exclamation dimensional equivalence & the isomorphic omniform

To a certain extent, as long as we maintain the total number of dimensions, we can "convert" a temporal dimension into a spatial one; and vice versa.

This depends on the observer's "frame of reference" somewhat. If we take the example of a "Flatlander" (2Dspace+1Dtime) we can extract an analogy for arbitrary dimensional equivalence with the following "":

Let's say we have a static (abstract, and therefore fully "spatial") Sphere in 3Dspace and attempt to describe this to the 2Dspace of the Flatlander, then their experience must be augmented somehow.

To do this, we can pass the (3Dspace+0Dtime) Sphere through the Flatlander's 2Dspace+1Dtime world sheet. The Flatlander would observe a single point in 2 dimensions, which evolves in the additional time dimension, into a circle that increases in diameter, reaches a maximum and then shrinks back to a point. Notice that the Flatlander and the Sphere now have the same total number of dimensions:

Flatlander = 2Dspace+1Dtime = 3DIM
Sphere = 3Dspace+0Dtime = 3DIM

The Flatlander is able to observe the static Sphere by extending their observation into an extra time dimension - animating the Sphere by slicing it into layers and observing them as a sequence of events.

So, by this scenario, we can surmise that any completed spatial dimension can be "converted" into a temporal dimension in an arbitrary fashion, eventhough the observer's frame of reference may be fixed. Eventhough the 2 spatial dimensions of the Flatlander seem to limit their observation, they can still extrapolate into the 3DIM of the Sphere, and then recognise this fully spatially completed form as existing - albeit outside of their plane of direct observation.

For example, if 11D M-Theory has 11 spatial dimensions, then each one of these can be split into temporally equivalent dimensions - retaining the 11DIM constraint on the total number of dimensions:

11Dspace + 0Dtime = 11DIM
(Ultimate Frame)
10Dspace + 1Dtime = 11DIM
9Dspace + 2Dtime = 11DIM
8Dspace + 3Dtime = 11DIM
7Dspace + 4Dtime = 11DIM
6Dspace + 5Dtime = 11DIM
5Dspace + 6Dtime = 11DIM
4Dspace + 7Dtime = 11DIM
3Dspace + 8Dtime = 11DIM (Normally Apparent Frame)
2Dspace + 9Dtime = 11DIM (Flatlander)
1Dspace + 10Dtime = 11DIM
0Dspace + 11Dtime = 11DIM

The difference between each ratio being that of the observer's frame of reference - eg, the Flatlander would need an extra 9 dimensions of time to be able to recognise M-Theory in their 2 dimensions of space.

In this way, we can say that a spatial dimension is a completed time dimension, because once the 3D Sphere has passed through the 2D Flatland fully, its description is complete.

11DIM of space would then be the highest level of observation in the hierachy of reference frames - the Ultimate Frame. Theoretically, we can arbitrarily switch between these reference frames, eventhough we seem to be limited to 3Dspace, we can utilise an extra 8 dimensions of time to make up the difference to 11DIM in total!!!

The question then is, how do we change our frame of reference???
(Even if we can, would we want to?)


PS - Is the Flatlander at a disadvantage seeing as they only have 2 spatial dimensions? (We saw how they could still recognise a Sphere by adding an extra temporal dimension, eventhough they are stuck inside the 2D Flatland.)

PPS - I just realised that this has to do with entropy, in that the extra temporal dimension required - to recognise a higher spatial dimension - removes information from the given reference frame. For example, what if the Sphere was more egg-shaped - they'd have to wait until they could filter the observed form out of the category of "Spherical". The other thing to notice is that this approach only deals with abstracted "map", it does not describe the "territory" - unless with M-Theory we have the finest level of discernment, so that we have effectively found the "" map/territory omniform.
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