Complexor Arithmetic

The inventor of quaternions remarked, All is pure time; And space-time relativity has a quaternion-like definition; But that's neither the beginning nor the end of it

Considering first the complex numbers, having the solutions to general real coefficient polynomial factoring, eg. 2x + 1 = (x+i)(x-i), and as well general complex coefficient polynomial factoring, -whence the classification of complex numbers as an algebra,- mathematicians also discovered (post exordium) the hypercomplex numbers, the quaternions (multiplicatively non-commutative) and the non-associative octonions, also exhibiting this algebra-closure: requiring no new type numbers for fully monomial-factored solutions. But the younger mathematics student will sooner enjoy her own discovery of complex extension by examination of the basic complex number itself: Noting that each coordinate within is a real number, by implicit induction letting each become a complex number, the result, applying appropriate rules, is the quaternion complexor 4(u)ple, with a more obvious notation structure: a pair of pairs.

We build the 4-complexor, quaternion:

Let a = [ar|ai] : that is, a is complex, and ar,ai are the coordinates, initially tagged as the real and imaginary parts, Re(a), Im(a), of the complex number, a. Then extend this by letting ar = [arr|ari] and ai = [air|aii] where ari,aii are imaginary at the next level of complexity, so that ar,ai are themselves complex numbers: Whence a is the complexor a=[ar|ai] = [[arr|ari]|[air|aii]].

The rule for 4(u)ple complexor multiplication is a familiar extension of complex multiplication, now with a choice of left-handedness and right-handedness dependent on the order of the operands in process, augmented by further complex conjugations (herein denoted, a', etc.), the first-handed being (delineated with hind-order cross products) a b = [ar|ai] [br|bi] = [ar br-bi' ai|ar bi+br' ai] checked for necessary consistency in preserving the modulus calculation at this extension, whereby a,a' being conjugate symmetric about the real axis, and therefor multiplicatively mutually compensative, and commutative, a a' = [ar|ai] [ar'|-ai] = [ar ar'+ai' ai|-ar ai+ar ai] = ar ar'+ai' ai is the real-only modulus squared (the zero imaginary component inferred the hind-order though commutativity fixes what should never have been broken); And modulus of product, (a b) (a b)' = (ar br- bi' ai) (br' ar'-ai' bi) + (ar bi+br' ai) (bi' ar'+ai' br) = (a a')(b b'). [The other terms cancel obviously under commutations]

The other-handed a b = [ar|ai] [br|bi] = [ar br-bi ai'|ar' bi+br ai] preserves modulus equally because its complex factors are commutative.

But we notice two cogent notions in this construction: The additional calculatory conjugations apply to both coefficients in one operand against the other's imaginary coefficient, and, the order of operand coefficients appears notionally reversed in one term of the imaginary (zero) part of the result, but which is hidden in fact calculating equal, as these are thus far commutative -- which suggests the first operand intrinsically embraces the second, as though seen edge-on: the nearer factor surrounding the farther: a b = [ar| [br|bi] |ai] = [ar br-bi' ai|ar bi+br' ai] -- which we forenote, preserves the modulus at the next, 8-complexor level, though not directly for the modulus of a product: the octonions demand further entwining -- notions having basis in higher mathematics.

4-Complectors (delineated 4-complexors) are equivalent. [Complex numbers are not, complects]

Hypercomplex 4(u)ples [astructured 4-complexors]

Another simple extension of complex numbers to 4-complexor quaternions is by appending an i-like j, jj = -1, distinct from i, ij=k not real, and therefor k also being appended imaginary, kk=-1, completing 4 dimensions: the real with 3 imaginaries. Commutativity adjusts in preserving the modulus, eg. (i+j)(i+j) = ii+jj+ij+ji = -2+0 necessarily, which requires ij=-ji, anti-cross-commutativity; Also, jk=i=-kj, and ki=j=-ik; And, i,j,k are cross-associative, eg. (ij)k = i(jk), because (explanatorily) i,j,k are multiplicatively dependent, ij=k, whence (ij)k = (k)k (which co-commutes) = -1 = i(i) = i(jk) -- or (canonically) i,j,k are additively dependent in log-space, ln(i)+ln(j)-ln(k) = 0.

Thus we find higher orders of imaginary numbers increasingly complexified in maintaining the simplicity of reals.

Additional topical discussions:

1. [under construction] Embracing quaternions not needing anti-commutivity . . . .

2. [under construction] Extractable properties, including, Quaternion bi-multiplication, eg. qzq', exhibits limited rotation, which we can show by evaluating hypercomplex numbers as real+vector (time plus 3-space): Letting unit q=r+u, z=s+v, qzq'=(r+u)(s+v)(r-u) = qsq'+2r v+2r u×v-(u·v)u-(u×v)u = . . . .

Beyond the quaternion:

Before we continue an extension to the octonions, we exact the linguistics involved,- the terms, real, and, imaginary: Real, comes from the general utility of numbers in linear continuum,- added, multiplied, divided, root-factored or transcendental. The imaginary, extension of the reals is linguistically interesting: If an artist holds the image of a snake in thought, she draws a snake, not a horse: Thus imaginary, means a temporary similar process of mental imagery which soon results on the real side -- even as 2i = -1 and ijk = -1, multiply imaginaries and produce reals: Nevertheless, if a mother, imagines a snake, she does not birth a snake, unless by virtue of being a mother snake: Therefore, imaginary numbers, -were mathematics not considered art but reality,- might be called, virtual numbers: This becomes apparent and usable in the octonions developed next: We find time-space is real plus virtual; And we will sometimes refer to 8-complexor octonions as, dual-4-complexors.

We designate the 8-complexor as, the octonion, in the nomenclatural taxonomy of the quaternion [qua+ter = 1+3 = 4] --and as a round onion consisting of concentric spheres suggests multiple free-rotating radial vectors: apt icon for the 8 basis vectors of the octonion.

Also, the mathematicians' 1,i,j,k notation for quaternions is a real+vector order appending to the concept of real number, 1, imaginary vectors, i,j,ij=k, extended to reach the complex and complexor numbers -- while the space-time physicists prefer alphabetic-order, i,j,k,l, notation appending time to 3-D space; Octonions will append L,I,J,K, or, I,J,K,L (capital of 1,l=L).

We will use the term, virtual, in general and specific meanings: Generally, the virtual, is the non real portion, including the imaginary and beyond; Virtual, generally means the venue of 8-complexors; Specifically, the virtual, is that we've appended to the real and imaginary; and, specifically virtual, requires three or more vectors more-complex-than, or non, quaternion.

We build the 8-complexor or dual-4-complexor, octonion:

[This paragraph is being rewritten for full consistency with the hind-order] We develop our complexor deeper by combinatorial examination of the basic product: Letting (ab)Tn = ab or ba, be a transposition depending on the Tn in combinations being tried, and letting (ab)Cm = ab or a'b', be a part-wise conjugation depending on the Cm being tried: Then, (a b) = [ar br-(ai bi')C1T1|(arC2 biC3+br'C2 aiC3)T2] preserves a a' for Cm,Tn, and, (a b) (a b)'=(a b)r (a b)r'+((a b)i (a b)i')C1T1 = (ar br-(ai bi')C1T1) (br' ar'-(bi ai')C1T1) + ((arC2 biC3+br'C2 aiC3)T2 (bi'C3 ar'C2+ai'C3 brC2)T2)C1T1. We drop the latter C1T1 as redundant to C2C3T2, and find combinations of Cm,Tn that leave zero difference upon subtracting the product of moduli, (a a') (b b'), leaving: - ar br (bi ai')C1T1 - (ai bi')C1T1 br' ar' + (arC2 biC3)T2 (ai'C3 BrC2)T2 + (br'C2 aiC3)T2 (bi'C3 ar'C2)T2 = 0, which can be solved for three hands, T2, C2C3, C1T2C3, paired for six by the forward-backward order of the multiplication line itself. The technique for solution is to separately solve for the Re(bi) and Im(bi) parts (or of ar), noting that the 4 terms either by pairs sum =0, or as one pair of identical conjugate sums being thus real times the Im(bi), therefore total =0.

What is fundamentally important, is to recognize the potential utility of such a provided order: If space-time physics recognizes the importance of 4(u)ple quaternions, but then quibbles over left-handedness versus right-handedness of the cosmos, while rejecting any aether, we must conjecture the absolute aether having both handedness inextricably bound at every point in dual-4(u)ple complexor time-space -- this is as generally reasonable as having two hands of the same individual embodiment. And this leaves us pondering one step ahead of the ordinary physicist, just what is the aether: Every point in this dual-4-time-space has real and, virtual perception.

In mathematical context the question of handedness simplifies as there is no significant difference whether the labeling of dimensions consistent among themselves, is assigned positive or, negative nomenclature: the intrinsic rules and precedent operations are retained; (i)(j)=something, and that something is typically called, k,---but it could be, called, (-k), self-consistently. Left- and right-handedness are noticeable because of personal both-handedness (not altogether so practiced) and process orderedness (read, left-right).

Whence there is no handedness except by exemplary juxtaposition; There is one basic process of the octonion: that characterized in its prototype. Handedness variants are usable only in systems having corroborative isomorphs needing distinction---such as fast map processing enhanced by simplest change of bases. The octonion is the answer, to the question of cosmic handedness: It has both.

[under construction] We further consider its special vitality exhibited at the dual-4(u)plet interface . . . .

8-Complectors [delineated 8-complexors] are equivalent.

Hypercomplex 8(u)ples [astructured 8-complexors]

Another simple extension to finding the octonions is to itinerate the 6-step skew-associations chain (skewed off, independent of, any cross-associative quaternion slice) for fundamental, strictly virtual unit vectors, i,j,L: With 3 of the steps anti-cross-commutative, the remaining interstitial 3 steps must be anti-skew-associative: (ij)L = -L(ij) = (Li)j = -j(Li) = (jL)i = -i(jL) = da capo, (ij)L.

Although the imaginary and virtual components of this prototype astructured octonion each partake in 3 cross-associative 3-product realizers (being quaternion slices), eg. i, (ij)k=i(jk)=-1, (iL)I=i(LI)=-1, (iJ)K=i(JK)=+1, and all are anti-cross-commutative, and the real component is associative and commutative with all, the non-realizer 3-products, if not co-associative, neg. i(iL)=i(I)=-L=(ii)L, are anti-skew-associative, eg. i(jL)=-(ij)L, and the application altogether is thus not associative, because the imaginary has a 3-product back to real, ijk=-1, while the virtual takes 4-product, LIJK=+1, back to real.

However, the general, restricted set of, 8-ary complexor generated of a single virtual (vector) v, a[v]=ar+avv, is cross-associative among itself: as an octonion has zero cross-product with any other virtually like itself. This yields an iterative exponential property ("alternative"), having a usage form, ka b = a(a(...[k-times]...a(b))...), -and similarly on the right-side,- useful for proving conservation of angular momentum, while steerable under integrated infinitesmal application of perpendicular force.

Also, same-imaginary complex numbers are associative with octonions: having no skew-association terms, eg. (c1[j] a[v]) c2[j] = c1[j] (a[v] c2[j]). And of course, any quaternion slice (multiplicatively dependent subspan) of virtual space is associative, eg. 1,i,J,K, is a quaternion; neg. 1,i,j,K, not quaternion but strictly octonion because i,j,K are multiplicatively independent, specifically virtual.

A diagram of the multiplicative relations among the prototype real, imaginary, and virtual components, labels one corner of a cube as scalar unit, 1, and 3 adjacent corners as fundamental unit vectors, i,j,L, and the remaining 4 corners consequently as, k,I,J,K, by the generative rules conducted in their original order: first i, then j, then L. We then notice the mirror-fact that while, ijk=-1, is real, IJK=L, is virtually opposite-handed, plane-paralleling the real. [Humorwork assignment: Do we dream in the same chirality?]

In this prototype there are 4 right-handed 3-product realizers, ijk=-1, iLI=-1, jLJ=-1, kLK=-1, and consequently 3 left-handed 3-product realizers, iJK=+1, jKI=+1, kIJ=+1, and the 4-product realizer, LIJK=+1. Octonion basic 6-handedness is 6 set-exchanges of right-handed and left-handed realizers, eg:--

ijk=+1 iLI=-1 jLJ=-1 kLK=-1 iJK=-1 jKI=-1 kIJ=-1 LIJK=-1
ijk=-1 iLI=-1 jLJ=-1 kLK=+1 iJK=-1 jKI=-1 kIJ=+1 LIJK=-1
ijk=-1 iLI=-1 jLJ=-1 kLK=-1 iJK=+1 jKI=+1 kIJ=+1 LIJK=+1 "prototype"
ijk=+1 iLI=-1 jLJ=-1 kLK=+1 iJK=+1 jKI=+1 kIJ=-1 LIJK=+1
ijk=+1 iLI=-1 jLJ=+1 kLK=+1 iJK=-1 jKI=+1 kIJ=+1 LIJK=-1
ijk=-1 iLI=+1 jLJ=+1 kLK=+1 iJK=+1 jKI=+1 kIJ=+1 LIJK=-1
Octonion handedness in this positive-negative sense is 6-wise, but the generality of assigning dimensions is far more multiplicitous -- whereas for quaternions these were the same, as ij=-k would mean kj=i. The consequentiality of just 4 of these 3-products realizers is shown: LIJK = (L)iL(-iLI)jL(-jLJ)kL(-kLK) = (ijk)(iLI)(jLJ)(kLK) = 1 else -1 (real).

While there is proof that no deeper than octonion complexor exists as an algebra, as it must have an associative subalgebra, and equipollently there is intuition that association on 4 independent factors has a 5-step cycle, and therefore cannot be anti, ((ij)L)X = (ij)(LX) = i(j(LX)) = i((jL)X) = (i(jL))X = da capo -((ij)L)X, this may be because we have expired our standard multiplication of anti-able diadic commutations, ab=ba or -ba, and triadic associations, (ab)c=a(bc) or -a(bc). Had we an anti-able quadriadic multiplication rule, we might find a higher complexor -- suggesting, segment-localized hand-switching: octonion 6-handedness might structure 6-legged and multi-segmented pair-legged symmetries, or, co-virtual dimensions might have mutual zero products, or, by revising our original approach of complexors, we might solve for tri-partite tri-conjugate multiplication yielding real.

Additional topical discussions:

1. [under construction] The dual quaternion notion, real plus virtual, (qr+qv) . . . .

2. [under construction] Extensible, extractable properties, including, Octonion bi-multiplication, eg. ozo', also exhibits limited rotation, because the component (space)vector-products are cross-associative, each term involving elemental reduplications, eg. (iL)i = i(Li) = L, and the anti-skew-associative terms cancel, eg. (iL)j + i(Lj) = 0, . . . .

Dynamic spinning

If we multiply a given octonion by an octonion infinitesmal spinner, that is essentially a real unit tilted but infinitesmally away in any proportion of imaginary and virtual, and conjugate back again, rsr', this results in turning that octonion infinitesmally about the spinner axis -- finitely the real axis: And infinitely repeated applications of the infinitesmal spinner then cumulate to a significant, smooth, spin around the same real axis ... perpetually.

If the spinner is nutated (itself spun about the real axis) then the compound motion depends on the relative phases of spinner nutation and octonion spin: At quadrature the octonion progressively deflects, in-phase or out-phase the deflection halts, reversing at negative quadrature. If the two spin rates are not interlocked, this oscillates perpetually: But this deflection takes infinitely many turns to accomplish non-infinitesmal progression: We have therefor rediscovered a well-known facet of time, not as another axis, but as cycles of cycles -- as DANIEL perceived. And this progression is twice-infinitely removed from the single application. And the real axis is thus representative more of eternity, than time.

This deflection progression may optimize at certain stable cyclic phase-interlocking, as for example electrons appear everywhere the same except for cosmic red-shift, yet in nuclear generation have certain stable energy values, eg. muons decay into electrons: Whence there may be a regular cosmic nutation resulting in uniform electron mass-energy definition and speed-of-light constancy.

Octonions also have the vantage that the virtual L,I,J,K are like a parallel space, a virtual quaternion, IJK=-L, or +L, paralleling the real quaternion, ijk=-1, or +1 (+,-, depending on selection of handedness),-- which lends it description as a dual-4-complexor; and assigns L a virtual time-like interpretation, though it is just as perpendicular to the real axis, and as simply imaginary, to it:-- rotation by a spinner infinitesmally off the L-axis rotates about the L-axis, by 90 degrees plus the infinitesmal, thus taking four hops to resume seeming similar to continuous infinitesmal rotation.

The commonality of the real axis for every real quaternion slice, suggests that such parallel real quaternion spaces are coexistent.

[under further construction] We might early conjecture that each point in time-space spins freely, as infinitesmal points have infinitesmal pointing vectors: a kind of magnetics. Another early conjecture is that each point in 3-D space is link-chained by its 3-pairs of fore- and aft-hands, allowing it to apply or be applied directionally upon or by its infinitesmally near neighbors. Yet another early, co-conjecture with other aether current theory is that, dual-4-space may be perceived either on the whole, or as a hyper-space completely cohabited of strings, properly called stringage (curvi-lineage), derived from the construction that there maintains one real time-like coefficient -- this leads back to the concept of the coexistence of eternity and immortal time. Furthermore, standard astrophysics deduction indicates that all aether micro-motion is at constant speed, tending to result in a constant speed of light in constant-gravitated vacuum. . . .

Applications:

Practical application of 4-and 8-complexors includes the solution of 2-and 4-complexor gradient fields: finding path-threads of force by holding to zero the analytic-harmonic complement (virtual) 2-and 4-complexor, in 4-and 8-complexor-equivalent gradient fields.

A premise discovery under the title,

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