There exists a deeper understanding of what constitutes $3$-space. Each orthogonal two-dimensional plane is most commonly defined as two lengths (or axes) perpendicular to one another, sure, but what does that really mean? The concept of orthogonality brings the concept of relative direction (or orientation) into the picture. This structure of relative orientation, when combined with length, is the defining characteristic of a two-dimensional plane.

Because of the symmetry that actually exists regarding how direction and distance can be represented mathematically, there is an alternative way to portray $3$-space. This method uses direction as the indices (or “metric”) rather than length. In the same manner in which the normal methods can produce a unitless direction from three orthogonal lengths ($x,y,z$), there are alternate methods in which a unitless length can be produced from three orthogonal directions ($\upsilon x, \upsilon y, \upsilon z$). The variables $\upsilon x, \upsilon y, \upsilon z$ are values or quantities which can be seen as amounts represented by the area under the normalized two-dimensional curve associated with each angle.  The two-dimensional curve for each $\angle\upsilon$ is expressed using the criteria provided in the model.

Once again, it is a smooth function that approaches a sine curve when $\upsilon\to0$, and that approaches a hyperbola when $\upsilon\to\frac{\pi}{2}$.

If this system is worked out there will exist a completely different set of geometric relationships that occur in Euclidean $3$-space. This allows the development of a coordinate system that contains bizarro vectors that’s conceptually and mathematically similar to the one that is used normally. Quantities that are vectors in normal parlance become scalar quantities and, conversely, scalar quantities become something akin to a vector. By this we mean that lengths become un-oriented or directionless in the same way that direction or orientation has no length in the normal coordinate system. The radius or diameter of a sphere is this type of un-oriented length.

This alternative system expresses Euclidean $3$-space in a way that affects the mathematical treatment of spacetime. In flat or Newtonian spacetime the effect is indistinguishable from the normal perception. It’s only when we look at this model in relativistic spacetime that the ramifications appear. This is due the difference in ranges for the quantities that are being combined (length, direction, and time).

When distance ($l$ength) and $t$ime are combined in spacetime the result is bounded by $-c$ and $+c\::$

$-c\leq lt \leq +c$

When direction ($\tau$urns) and $t$ime are combined in spacetime the result is bounded by zero and infinity:

$0\leq \tau t\leq \infty$

Note that when structured this way, the sign is actually associated with length rather than direction. This has an effect when spacetime is curved. A subtle anomaly arises and has to be accommodated when this geometric relationship is considered alongside the equivalency principle.