UNIVERSAL REALITY
UNIVERSAL REALITY
Conceive a number line of all real numbers (i.e. all members of the set of Reals). Since we ought not discriminate between arbitrary equal numeric intervals, e.g. between any number and its successor, these are therefore given equal lengths on the number line. Our number line can now serve as a reference co-ordinate for vector v with modulus (magnitude) a, which may take the value of any real number and sign which upon reflection of the vector is multiplied by –1: a polar vector. The second attribute associated with a vector is its direction. Except for a reflected vector and the null vector, a vector’s direction introduces a degree of ambiguity as it can only be considered in relation to a reference such as a co-ordinate or second vector which would imply some as yet unidentified intrusion from two-space. Of course we could conceive a one-space in which vectors only lie on the co-ordinate from the origin to number a on the number line. It would seem that a vector will then be uniquely manifest at the end-point and its attributes will solely act on this end-point. However, such an approach only masks the enigma which, hence, we will presently address. We must acknowledge that thus far we had only paid lip-service to a necessary aspect of a vector as a concept: we cannot consider a vector in terms of any of its defining attributes unless we also define their relation with an observer who perceives them.
We will note that the observer’s line-of-sight can be perpendicular to the vector anywhere along its length, from its origin to its end-point for any a. The apparent length of the above-introduced numeric intervals (that provide a measure for the vector’s modulus) will foreshorten for any part of the vector that is observed at any angle from the normal (the line-of-sight perpendicular to the co-ordinate). Hence, it would appear that the only accurate way for observing the vector’s modulus would be to measure the length of a help-line to the vector’s origin and another to the end-point and their included angle. The angle will also provide a reference for the vector’s direction with respect to the observer. However, as a consequence we would imply that observation requires a "two-space", a triangle to which the ‘law of cosines’ can be applied in order to find its hypotenuse that adjoins the vector, yet that is not part of the latter’s one-space in which the observer had intended to perceive (the vector). One problem arises immediately with this approach: the only angle that can be determined is one of 180°, another that the help-lines cannot be calibrated unless they lie parallel to the co-ordinate. It would appear that both problems could be overcome by dividing the triangle into two smaller triangles with the line-of-sight as their abscissa, however, the latter’s length like their opposing angles will then be zero. Any other triangles would require measurement of one more side, since it would no longer be possible to determine the included angle. However, this third side is the abscissa that will always be perpendicular to the co-ordinate, thus its length can never be calibrated and, hence, neither can the length of the observer’s line-of-sight be determined. Thus it would appear that effectively the observer’s two-space conveniently reduces to the vector’s one-space. Not so, observation of the vector’s two attributes still requires the observer to rotate in two perpendicular directions (and their reflection- even though these will be superfluous at the end-point,). Accordingly, observation introduces an axis of rotation that is normal to the vector and that has two perpendicular components (and their reflection, both whose vector product is the rotation axis), one of the components lies in the direction of the vector. However, the second component it would seem cannot be real as it lies in the line-of-sight, an abscissa whose modulus will be either zero or indeterminate. We are drawn to the conclusion that observation of one-space requires two real dimensions (in the direction of the vector and the observer’s rotation axis) with the observer’s line-of-sight a third imaginary dimension which will be orthonormal at any point on the co-ordinate since modulus a can be any Real. More generally we can say that a one-space can only exist in a larger three-dimensional context, and as the above case demonstrates, that a real polar vector requires at least two of these dimensions to be real with the third imaginary dimension intersecting the two spatial dimensions at the Vector’s end-point. In effect, we have shown the observer to be an inseparable part of vector space, however, as a consequence it would appear that the introduction of the imaginary dimension exposes vector theory as incomplete.
In order that vector theory be reconciled with an imaginary line-of-sight, it must admit a component vector whose attribute ib is its imaginary magnitude. Accordingly, we let the modulus be any member of not just the set of Reals but in addition any member of the set of imaginary numbers. First we note that when for any av we associate any imaginary number, the aforementioned ib from the latter set, with a complex vector space of which av is the real component, there is no effect on v or its end-point (v lying on the real number line now represents the projection in one-space of complex vector va,ib) and thus we must regard all imaginary numbers ib as coincident in real one-space with a vector’s end-point for all a. Thus there exists a component vector ibvi that is orthonormal to av. This outcome gives a new meaning to a vector’s direction: we must also consider its orientation in relation to complex vector va,ib and consequently reject the tenet that direction vanishes for infinitesimals (only the latter’s real moduli are zero).
Next we will consider an angular change in the observer’s perception with regard to the original line-of-sight that will neither affect the apparent direction of the vector nor its apparent length while leaving the line of sight orthonormal to the vector. These three conditions can be met if the angle lies on a "virtual plane" orthonormal to the direction of the vector. In other words, the observer has complete freedom to rotate around the axis of a vector, or axially rotate the vector in the origin of the orthogonal co-ordinates of an imaginary plane, that is, without invoking an additional real dimension. To appreciate this mode of perception we must make a distinction between a right angle between two mutually normal real vectors (i.e. in two-space, e.g. that we can picture drawn on a sheet of paper), and that of the line-of-sight to any real vector. We will concede that the observer cannot ever visualise more than the infinitesimal cross-section of the line (of sight) orthonormal to the end-point of any vector. Furthermore, as the vector's axis or the observer’s line-of-sight rotates (the latter axially), we will acknowledge that the set of imaginary numbers (associated with the complex vector space), intersects real vector space (only) at the vector's end-point. Now, if the observer could adopt a perspective that would be orthogonal to its former line-of-sight (i.e. as if turned in the direction of the vector’s axis), we would find that the imaginary numbers lie concentric around the vector's axis on a plane that is as asserted not spatial but imaginary. Yet any point on this plane could in turn be defined by any two imaginary vectors in the plane and their included angle.
Accordingly, we have identified another part of our expanded vector space, one that has an axial vector as real component and two imaginary dimensions that can be represented by two orthogonal imaginary number lines on co-ordinates constructed in the manner described above for our number line of Reals. Note how from the orthogonal view toward the imaginary plane, mentioned in the previous paragraph, the origins of the two imaginary number lines coincide (intersect) anywhere along the length of our spatial number line confirming that the two imaginary dimensions form a complex three-space around the vector. We also note that, while the direction of the imaginary vector component ibvi on the imaginary plane is that of the observer’s line-of-sight (toward av), b will depend on the modulus and direction of composite vector va,ib, as does modulus a of vector v.
We have found that any real dimension is associated with two imaginary dimensions and, similarly that any imaginary dimension is associated with two real dimensions. It is clear that the normal to any real co-ordinate will similarly have an associated imaginary plane that shares one dimension with the first imaginary plane and that these imaginary planes will intersect at a right angle. Accordingly, owing to orthogonal symmetry we will find that, in addition to a vector space composed of three real dimensions that is congruent with conventional vector space, by analogy there exists another vector domain with three orthogonal imaginary dimensions.
We have demonstrated that component vectors can be real or imaginary. Thus va,ib is a complex composite vector whose modulus includes members from the set of real and imaginary numbers. However, this complex vector, while a one-space, does not exist in any real or imaginary plane of the expanded vector space, but lies in the complex two-space of its components in one real and one imaginary dimension that dissect spherical real vector space from its imaginary complement. It follows that complex space, such as av + ibvi = va,ib is hyperbolic. We also have shown that an imaginary plane may freely rotate about a real axial vector for any value of its modulus without affecting the latter or their mutual orthonormal relationship (i.e. the axial vector's end-point may be anywhere on a line in imaginary three -space which is orthogonal to the imaginary plane; the line and the axial vector thus being respectively the imaginary and real components of a complex axial vector). The same reasoning can be applied to real planes rotating about an imaginary axial vector component.