Webster says: “Dimension (in the general sense): an element represented in an equation by a symbol that functions analogously to symbols representing the three spatial dimensions and regarded as a constituent of a geometric space”
Whew! That’s a mouthful. To be truthful, I thought that I’d try to come up with a definition of my own, but I think that this serves the purpose, because it is both accurate and confusing. First, it defines dimension as relating to the ‘three spatial dimensions’ without describing specifically what they are. Secondly, it doesn’t require that a dimension be any sort of specific quantity or physical attribute except that it must be ‘regarded as a constituent of a geometric space’. How convenient for String Theory and its 10 to 23 ‘dimensions’ that must be hiding somewhere. I guess that ‘happiness’ or ‘paranoia’ could be dimensions too because they definitely influence how one perceives the geometry of space.
In the past, for physical space, the dimensions have been defined as having to do with direction and the amount of space described. A line is considered to have one dimension both because it can only describe first order equations and because it represents only one type of measurement; length.
If you add another line, and customarily it is oriented perpendicular to the first, you get two dimensions, which engender second order equations and area. But there is not really any new feature to the measurement gained by the addition, since we’re still utilizing lines; we have just added another direction and ‘area’ into the mix. But of course, we still have not added a variable for rotation, and any off axis direction must be defined by an association with both of the linear axes, requiring second order equations to calculate length.
If you add a third line, which is perpendicular to the other two in the most common construction, you now have three dimensions and space, rather than just area. Viola! 3-D! Put on your red and green glasses.
But because each of these elements (axes) are of the same nature and composition, and distinguishable only by their direction and their relationship to the other two, they are interchangeable, and as discussed above, codependent.
What if we could come up with a system, a geometry that had dimensions that could stand alone, were instantly distinguishable and did not rely on each other for definition? Wouldn’t that be, as they say, more robust?
What if we defined dimension as follows:
Dimension – a physical attribute of a velocity that describes a unique aspect of its relationship to spacetime.
At least it’s a little easier to understand. And as we shall see in the next chapter, it makes a lot more sense in the long run.
These last three chapters really form the basis for the geometry that is coming in the next section. It is somewhat surprising, though it really shouldn’t be, that we are already using a velocity to define our distances. Of course astronomers had to use light years because the distances to even the closest neighboring stars are so great that defining them in terms of even the largest Earth-based measurements (miles? Leagues?) would be absurd and make interpreting the relative distances between celestial entities much too difficult to allow them to be properly compared. None of us are very good with extremely large numbers and few even know what the next grouping above ‘trillion’ is even called.(How many zeros are there in a jillion?)
The definition of ‘one’ as the center of the numerical universe is both enlightening, and if I may say so myself, profound. This topic will likely be the subject of a later essay. It has tremendous philosophical implications that have only been lightly touched upon here.
But the definition that has been presented for dimensionality is one of the real strengths of this new geometry, because, as we’ll soon see, each one of the primary velocities carries with it the characteristics that allow one to define each dimension not by some particular arbitrary orientation, but by the form, the nature of the velocity itself. It is details such as these, that evolved after the original concepts for the geometry were created, that help to illustrate the soundness, the applicability and the appropriateness of those ideas.
This is not some happy accident; it is because we have chosen to define the components of our universe using its own most prevalent forms, and as we shall see later on, this choice can enable us to provide better, more accurate expressions for the physical forces and energies that we observe.
– O. Penurmind