For those who have reached this far, it is time to cast off three dimensional thinking and look beyond four dimensional thinking, and see 'what is'.

Prior to constructing physics spaces using R-algebras, one could only tack on dimensions of space, time or some other with vector characteristics in some manner - 'ad hoc' - consistent with the lower dimension characteristics, and hope that a physics would be consistent with it that entities could exist in.

Construction of physics spaces via algebras changes all that!

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Just as the electromagnetic field may be constructed by the special four dimensional R-algebra using the weighted matrix product, an eight dimensional R-algebra may be constructed with base space consistent with this four dimensional R-algebra (physics space)!

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Thus, an eight dimensional physics space is constructed, with associated eight dimensional electromagnetic field. And when the upper four dimensional components vanish the field is the usual one we know and love.

Now, consider inert matter. Since its physics is well known, it is likely to have no components beyond the fourth dimension.

But, if any thing has an extension beyond the four dimensions (be it intuition, E.S.P, or whatever), the three dimensional existent we observe we would view as the tip of the iceberg. Ninety percent of an iceberg lies beneath the surface. Similarly, just how much of an entity lies in the dimensions above the usual four we cannot know from within the four.

Also, if the three dimensional extent of an object are twisted into the higher dimensions, what would be observed then within the lower four dimensions? Any entity restricted to the lower four dimensions is as vulnerable to higher dimension technology as ground troops are to air power.

Further, though the extent within the lower four dimensions may become inert, its extent beyond the lower four may not cease to exist. Indeed, assuming so would seem a rather blind presumption.

The signs on the products for the fourth dimensional space is clear, just use the cross product and the 0th crossed with any other is positive, non-zero crossed with the 0th would then be negative. How, then are the eight dimensional signs determined, self-consistent and consistent with the four dimensional?

The 4D vector diagram we start out with we know:

Consider the diagrams for 8D:

Note the full circulation all the way around the 4D space. The 8D space must have that property. The left 8D tries to emulate the circulation 34, 45, 53 (like the 12, 23, 31) and so on around. But some of the arrows are left ambiguously, so this is likely not the transcendant maxim.

 The right 8D has the full circulation property, and consistency with the 4D requires that 31 is positive, consistent with the property that: ij  is positive whenever:

These arrow diagrams may be drawn consistently for any positive integer dimension, but the bases forms a group and are chosen orthogonal which the Graham-Schmidt process insures. So, for each of the vectors in the lower 3 dimensions and 0th there must be a corresponding member for the next higher group, so that each product is unique So, the next higher must be:  2x4=2³=8 , and the next: 2x8=2⁴=16 , etc.

Thus, the groups of hyperspace-time "physics spaces" are of:  2^M dimension (M>0). 

Specifics of 2^M dimensional physics spaces and the particular structures of the eight and sixteen dimensional spaces are exhibited in the following ebook. This ebook is the basic course of study in a curriculum of 2^M dimensional technological development.