Chaotic manifolds






E-mail: paper0@byandbygones.com                                                               IE6x: Text optimized at: up to ~ 700px




I: Introduction:
    Tensor analysis assures us of the existence of its rank two tensor of dimensionality n, n an integer, n³2. Therefore, by
its process/operation of tensor inner product we are guaranteed of the existence of rank one tensors of dimensionality n.
Rank one tensors are identical to ordinary vectors (in euclidean space/cartesian frame). Therefore there exists ordinary
vectors of dimensionality n (and in fact the n-dimensional vector is posited as a priori fact in tensor analysis).
   Tensor analysis further assures us that the inner product of these posited or derived n-dimensional rank one tensors is
defined, meaningful, and valid: that is, the inner/dot product of vectors of dimensionality n is an indispensable fact of n-
vector/tensor analysis. In fact to smother any implication whatsoever that the existence of the rank one tensor/vector of
dimensionality n is in any way dependent upon the existence of any higher or lower rank tensor whatsoever of whatever
dimensionality (and vice versa) tensor analysis a priori postulates the independent existence of rank one tensors/vectors
of dimensionality n³2.
   The linear orthogonal transformation in Enof an n-dimensional (virtually always cartesian) reference frame of n linearly
independent direction axes thus defines/posits the arbitrary n-dimensional tensor of arbitrary rank. The linear orthogonal
transformation is a linear rotation of the reference frame/tensor in n dimensions - n-space. Thus when the n-dimensional
unit basis set vector, say, i=e1, is rotated in n-space, En, the new vector e'1, as a consequence of the linear orthogonal
transformation, is some linear combination of the orthonormal basis set ei, ei.ej=1, j=i, and ei.ej=0, j¹i.
   Therefore the linear orthogonal transformation/rotation of a vector/tensor, and the resulting vector/tensor of the linear
transformation is each totally independent of dimensionality.
   Like the linear rotation operation, the differential operation is a rigorously defined, valid, linear difference operation
totally independent of dimensionality: what is generated/derived by the linear operation in m-space, m>1, is preserved
in n¹m - nothing qualitatively new or novel created, nothing substantive wiped out. Thus there is no substantive/analytic
structure difference at all in the instance,



dR = (dR1)e1 + (dR2)e2 + (dR3)e3                                               

dR = (xR)dx + (yR)dy + (zR)dz,       R=R(x,y,z)                   

dR = Sni=1(xiR)dxi,       R=R(x1,...,xi,...,xn)                   (1)     


or even

dR = (tR)dt + (xR)dx + (yR)dy + (zR)dz,       R=R(x,y,z,t)

                              dR = (tR)dt + (xR)dx + (yR)dy + (zR)dz,       R=R( x(t),y(t),z(t) )               (2).





II: Differentials of R:
    The premise that an observer stationed at a point fixed relative to an n-space n-dimensional cartesian (x1,...,xi,...,xn)
coordinate system with origin O rotating wrt an (x'1,...,x'i,...,x'n)-coordinate system taken as fixed in n-space and having
origin also at O, is logically and analytically valid. Indeed, it is the raison d'etre of the basis of the equations of the linear
orthogonal transformation of orthogonal reference frames and the Klein definition of the vector.
   The linear combination of two successive linear operations is a linear operation. Thus to the fixed observer, due to its
linear rotation, the orthonormal basis/direction vectors e1,...,ei,...,en in the moving (x1,...,xi,...,xn)-frame change during
the duration of the rotation (infinitesimal or discrete). Therefore, to take the 4-dimensional case for algebraic simplicity,
the observer computes the differential of R=Sni=1Riei as


dR = Sni=1[(dRi)ei + Ridei]                 (3)

To determine the dei in terms of the linearly independent ei we note that ei.ei=1 implies ei.dei=0, which in turn implies
that ei and dei are perpendicular (real euclidean) n-vectors. Thus dei, simplified, must be an n-vector restricted to the
i¹j (e1,...,ei,...,en)-surface of En. Thus Ridei is (in array format):

or


[Sni=1Riei] = [1n]T[Rii][dei] = [1n]T[Rii][sij][ei]              (4b)

in which [1n], utilized for the convenience of summing the array entries to ordinary vector format, is an nx1 row array in
which every entry is 1, [Rii] of R=[1n]T[Rii][ei], is the nxn version of the nx1 [Ri] array for R, Rij=0, i¹j, and sij=0, i=j,
i,j=1,2,...,n.
   Evidently the sii=0 are the ei.dei=0 from ei.ei=1, and evidently too the sij tab the ei.dej. That is, from ei.ej=0, j¹i,
we have [ei.dej+ej.dei]=0. Since we define ej.dei=sij and ei.dej=sji we see that sij=-sji. Thus, the [sij] array is an
antisymmetric array for all n³2, but we add, the array is a computing and calculating picture, not quite a formal matrix.
   For the instance n=4 the first order differential dR in (3), fully written out is:

dR = [{(dR1)e1 + (dR2)e2 + (dR3)e3 + (dR4)e4 + ... + (dRn)en}

+ {R1de1 + R2de2 + R3de3 + R4de4 + ... + Rnden}]

= [{(dR1)e1 + (dR2)e2 + (dR3)e3 + (dR4)e4 + ... + (dRn)en}

- {(0 + s12R2 + s13R3 + s14R4 + ... + s1nRn)e1

+ (s21R1 + 0 + s23R3 + s24R4 + ... + s2nRn)e2

+ (s31R1 + s32R2 + 0 + s34R4 + ... + s3nRn)e3

+ (s41R1 + s42R2 + s43R3 + 0 + ... + s4nRn)e4

                                                  + ... ... ... ... + (sn1R1 + sn2R2 + sn3R3 + ... + sn(n-1)Rn-1 + 0)en}]              (5)

and the second order differential d2R of (3) for n=4 is:

d2R = [{(d2R1)e1 + (d2R2)e2 + (d2R3)e3 + (d2R4)e4 + ... + (d2Rn)en}

- {(0 + s12dR2 + s13dR3 + s14dR4 + ... + s1ndRn)e1

+ (s21dR1 + 0 + s23dR3 + s24dR4 + ... + s2ndRn)e2

+ (s31dR1 + s32dR2 + 0 + s34dR4 + ... + s3ndRn)e3

+ (s41dR1 + s42dR2 + s43dR3 + 0 +... + s4ndRn)e4

                                      + ... ... ... ... + (sn1dR1 + sn2dR2 + sn3dR3 + ... + sn(n-1)dRn-1 + 0)en}        

- {[d(0 + s12R2 + s13R3 + s14R4 + ... + s1nRn)]e1

+ [d(s21R1 + 0 + s23R3 + s24R4 + ... + s2nRn)]e2

+ [d(s31R1 + s32R2 + 0 + s34R4 + ... + s3nRn)]e3

+ [d(s41R1 + s42R2 + s43R3 + 0 + ... + s4nRn)]e4

                                    + ... ... ... ... + [d(sn1R1 + sn2R2 + sn3R3 + ... + sn(n-1)Rn-1 + 0)]en}        

- {[(..)s11 + (s21R1 + 0 + s23R3 + s24R4 + ... + s2nRn)s21 + (s31R1 + s32R2 + 0 + s34R4 + ... + s3nRn)s31 +

(s41R1 + s42R2 + s43R3 + 0 + ... + s4nRn)s41 + ... ... ... + (sn1R1 + sn2R2 + sn3R3 + ... + sn(n-1)Rn-1 + 0)sn1]e1

+ [(0 + s12R2 + s13R3 + s14R4 + ... + s1nRn)s12 + (..)s22 + (s31R1 + s32R2 + 0 + s34R4 + ... + s3nRn)s32 +

(s41R1 + s42R2 + s43R3 + 0 + ... + s4nRn)s42 + ... ... ... + (sn1R1 + sn2R2 + sn3R3 + ... + sn(n-1)Rn-1 + 0)sn2]e2

+ [(0 + s12R2 + s13R3 + s14R4 + ... + s1nRn)s13 + (s21R1 + 0 + s23R3 + s24R4 + ... + s2nRn)s23 + (..)s33 +

(s41R1 + s42R2 + s43R3 + 0 + ... + s4nRn)s43 + ... ... ... + (sn1R1 + sn2R2 + sn3R3 + ... + sn(n-1)Rn-1 + 0)sn3]e3

+ [(0 + s12R2 + s13R3 + s14R4 + ... + s1nRn)s14 + (s21R1 + 0 + s23R3 + s24R4 + ... + s2nRn)s24 +

(s31R1 + s32R2 + 0 + s34R4 +...+ s3nRn)s34 + (..)s44 + ... ... . + (sn1R1+sn2R2+sn3R3+...+sn(n-1)Rn-1+0)sn4]e4

+ .... ... .... + [(0 + s12R2 + s13R3 + s14R4 + ... + s1nRn)s1n + (s21R1 + 0 + s23R3 + s24R4 + ... + s2nRn)s2n +

(s31R1+s32R2+0+s34R4+...+ s3nRn)s3n+ ... ... +(sn1R1+sn2R2+sn3R3+...+sn(n-1)Rn-1+0)s(n-1)n+(..)snn)]en}]                                                                                                                                                                                 (6)

the generalization of which to all n is quite obvious - making it clear that rather than being some derivation of some sort,
(4) is literally a one to one description in array format of what's already in (3). Clearly, exactly as for n=3 (and even for
n=2) which is obvious from (3,4,5,6) the sijRj terms of (5) result from a non-scalar/non-inner product of vectors since
a rotation and differential of R and ei cannot create non-vector objects and structures. From actually working through
the algebra for n=2,3,4 we calculate from (4,5) d*R.R=0, where d, ( akin w, n=3), is meant to represent the angular
motion n-vector constituted of the sij of (5), and the * is the generalized product for all n³2. From blue {..} of (5) and
red {..} of (6) we further calculate that (d*R).(d*R)=-d*(d*R).R - and which is true for in particular n=3. It is clear
from extending the pattern in (5,6) to all n that these results hold too for all n³4. [The blue subscripts of (6) may facilitate
computing a typical sum term in d*(d*R).R]




III: Arrays in dR and d2R:
    Recasting/literally copying the calculated product in blue {..} of (5) into array format we have for n=4 - extended to
arbitrary n:

or


d*R = [1n]T[Rii][sij][ei] = ([sij][Ri])T[ei]                              (7b);

in which Rij=Ri, j=i, and Rij=0, j¹i; and similarly for the actual calculated product in red {..} of (6) for n=4 - again
extended to arbitrary n:

or


                            d*(d*R) = [1n]T[Rii][sij][sij][ei] = ([sij][Ri])T[sij][ei]                              (8b);

in all of which we tucked in the generalization to all n since the generalization of (5,6) from n=4 to n>4, and hence (7,8),
is so instantaneously and transparently evident; for (7,8) the sii=0, sij=-sji, i,j=1,2,...,n, and T is the 'transpose'.
   The s=[sij] arrays are antisymmetric for all n, including for n=3, the basis for which can be seen in the four arrays for
A*B - which is AxB for n=3 - in the fully written out first principles form converted to array format for arbitrary three-
dimensional vectors A, B - and extended to arbitrary n:

(9a,b) showing (A*B)T=-B*A if [ei*ej] is antisymmetric.
   Evidently there are other array permutations that equate the expanded expression for A*B; or AxB for n=3, and n=2.
And evidently too we can also use the very same arrays to represent the dot product instead of the *-product. Thus the
presence of the antisymmetric array in an analysis of vectors appears to be somehow a result of the perspective in the
analysis rather than the logical preclusion of the non-antisymmetric array, and vice versa. In fact as written, since (7) is
not the standard array for d*R in standard matrix analysis - i.e., since the array [sij] is not the array form of the vector
d, as is [Rii] of R, (7) is evidently not reflected in (9), nor can (7), without specifying whether [sij] results from d, or
from * or from both, handle d*R becoming redundancies like d*d or R*R, though it evidently will logically handle, as
a matrix under matrix properties, self-repeat symmetry like R.d*R and (d*R).(d*R), and analogously (8) will handle
the likes of R.d*(d*R), d*R.d*(d*R), and d*(d*R).d*(d*R). But since we cannot extricate d from R in the basic
d*R root/nucleus as it were to get [d]T[Qij][R] in conformity, as at (9), with [y]T[Qij][x] or [x]T[Qij][x] of standard
matrix analysis, neither (7) nor (8) can accommodate the likes of d.(d*R), d.d*(d*R) or even known array equivalent
of structures of (5) and (6) and further mth-order differentials like, unsymmetric with (7) and (8), (dmd)*R, in which the
(dmd)={[1]T(dm[sij])[ei]} and which is obvious in (6) for m=1. Actually from actual calculation all R.(dmd)*R=0, and
so too all d*(dmR).(dmR)=0, with (dmR)={(dm[Ri])T[ei]} - that is, the ei are not included in the differential operation
in these terms.
   Returning to (5,6) again it is clear by actual calculation that from {..} of (5) that dotting d*R with itself is exactly equal
to dotting R with d*(d*R) from red {..} of (6) for all n. Thus,


                                              (d*R).(d*R) = - d*(d*R).R                                                (10),

and the layout of relevant terms in (5,6) and (7,8) makes it evident that (10) is true for all n.
   In terms of the representational arrays (7) then becomes,

                                                (d*R).(d*R) = [ei]T[Rii][sij][sij][Rii][ei]                                 (11),

again with diagonal [Rii] such that Rii=Ri, Rij=0, j=i; and dotting R=[ei]T[Rii] into (8) means (8) now becomes,

                                                R.d*(d*R) = [ei]T[Rii][sij][sij][Rii][ei]                                   (12),

and here because of the evident symmetry in the arrays making use of the transpose properties of [Rii] and [sij] in (11)
the arrays of (12) and (11) make (10) evident for all, En, orthogonal frames, n>1 - though without necessarily making
transparent what the *-product incontestably necessarily is.
   Since the linear rotation (orthogonal transformation) of an orthogonal frame is per se evidently independent, as for
n=3, of dimensionality (and since even if we linearly rotate an (n-m)-dimensional surface of the n-dimensional frame/
manifold the linear rotation is still by definition a linear infinitesimal rotation dq), analytically, the inherent properties of
the *-product are, for n=3, evidently identical of the intrinsic properties of the x-product.
   Thus, if we linearly rotate an n-dimensional frame in one sense (counterclockwise?), then the vector ei changes from
an initial time zero vector ei to a time t final frame position vector ei+dei: that is, the change is [(ei+dei)-ei]=dei. Now,
since there is no preferred/absolute (reference) plane, and, in general, no preferred/absolute (n-m)-surface, in/of an n-
dimensional space or frame and in an arbitrary plane/surface no preferred/absolute direction and since every plane in
any/every En space can a priori likewise be orthogonalized as the (e1-e2)-basis of E2, then the nature of 'angle' in any
arbitrary plane cross-section is the same in all spaces. Thus, linear (infinitesimal) rotation is linearly reversible. Thus, if
now the ei+dei vector/frame is now linearly rotated back to its original time zero orientation then ei+dei must return to
a final t' equal time zero state. Thus {[(ei+dei)-dei] + [ei-(ei+dei)]}=0 is the vector sum of the changes ei to ei+dei to
ei in every space n>1.
   Thus when an arbitrary linear rotation compounds arbitrary n-dimesional vectors A and B, n>1, to yield the resultant
n-dimensional vector A*B then an arbitrary counter linear rotation that compounds B and A to yield the n-vector B*A
means [A*B+B*A]=0. Thus A*B=-B*A for all n³2 since dq¹dq(n).
   For A*B=-B*A setting B=A implies A*A=-A*A. Thus A*A=0. Thus since A and A are parallel, A*B=-B*A=0
when/implies B is parallel to A. Thus for arbitrary A and B if there is a component of B (that can be resolved) parallel
to A then the vector A*B cannot have any component in any way parallel to A. Similarly for any component of A (that
can be resolved) parallel to B. Thus the n-vector A*B is perpendicular to (the plane of) both A and B.
   A more analytical derivation of the (same) properties of the *-product is perhaps more concrete. To begin with we
are assured of the existence as analytical mathematical fact of the n-vector A*B by (5) and (6) and all dmR beyond (6).
The n-vectors A, B exist a priori, and the vector operation we characterize A*B, or more precisely, d*R does yield -
in fact, are - the vectors of (5), (6). They are hard facts of valid analysis of a logically and mathematically non-optional
fundamental a priori of vector analysis and the very definition of the vector. The n³2 n-dimensional vector d*R is not
a priori decree, not notional fancy, not axiom, not dictum, not surmise not premise, not hypothesis, nor even airy-fairy
abracadabra wand-waving; it is an irremovable fact of the linear rotation and the linear differential operation.
   The n-vector R is arbitrary and since rotation is susceptible of arbitrary rate, of arbitrary increment/infinitesimal dq, of
arbitrary inclination, of arbitrary direction, of arbitrary angle displacement then evidently d is arbitrary. In fact not only is
d an arbitrary n-vector, since it comes from the ordinary sum of ordinary scalar, Ri, multiples of ordinary n-dimensional
dei, it must necessarily be an ordinary n-dimensional vector as evidenced at (4) and (3). Granted, a d that would seem
transparently independent of and totally separable from the Ri of R, as for n=3, would be ideal, but its independence of
and separability from the Ri transparent or not evidently the scalar Ri cannot affect the vector nature of the dei by simple
scalar multiplication Ridei. Thus, even a d=d(Ri,sij) even with scalars, Ri, of R, must be an arbitrary vector. Thus given
that the n-vector d*R is a non-optional, analytic, fact, and d arbitrary and ordinary, then the vector properties of all/any
arbitrary A*B, regardless of origin context of A, B are the properties of d*R.
   Now, since under linear rotation (regardless of sense) the length/magnitude of an n-vector (by definition of the vector
under linear orthogonal transformation/infinitesimal rotation) is preserved (a scalar) then A/|A| and B/|B| are (even with
components variable) unit n-vectors, and since the absolute angular distance relation of A to B is the same as that of B
to A then these unit n-vectors must satisfy the scalar relation |(A/|A|)*(B/|B|)|=|(B/|B|)*(A/|A|)|£1 unless * is a left-right
untisymmetric boost function. But evidently since by definition sense of rotation can change only their direction it cannot
affect |dei|. Thus by (5,6) * has no such properties. Thus:

                              |A*B| = |B*A|                               (13)

since |A|, |B| are scalars.
   Now, since d, R arbirtrary and by (5), (6) d*R is an ordinary n-vector whose unit n-vector evidently exists and can
be calculated from its actual components in (5) then A*B/|A*B| exists and is a unit n-vector uo parallel to A*B. Thus,
from uo.uo=1 and (13),

                       A*B2 = |A*B|2 = |B*A|2 = B*A2                           (14)

   From (A/|A|)*(B/|B|)=u'*u with unit n-vectors u'=(A/|A|) and u=(B/|B|) and (A*B/|A*B|)=uo it is evident that u'*u
and uo are parallel since |A||B| is a scalar multiplier of u'*u and |A*B| is a scalar multiplier of uo. The dot product of the
two A*B yields

A*B2 = u'*u.uo|A*B||A||B| = |A*B|2       (15a),

and thus

|A||B||u'*u| = |A*B| = |B*A|= |A||B||u*u'|       (15b)

in which |u'*u| is the angle or some function of the angle between u', u or A , B.
   Clearly by (14), (d*R)2=-d*(d*R).R, which is (10), by symmetry, must imply

(R*d)2 = - d.R*(R*d)                (16a),

and thus

d*(d*R).R = R*(R*d).d                     (16b)


for all n-vectors. That is, evidently * and . are interchangeable for all n-vectors in the manner of n=3. In fact that, the
interchangeability of * and ., and additionally the relation A*B=-B*A, is what (10) always stated - and it taken straight
out of (5) and (6) evidently requiring virtually nothing more than mere observation.
   Thus, the results, by actual calculation, (dmd)*R.R=0, etc., and, in particular, (d*R).R=0, also by actual calculation
from (5) or (7) now, and noting that by actual calculation too, (d*R).Q¹0, does imply (d*R).R=d.(R*R)=0, which in
turn, since d is arbitrary, implies R*R=0. Evidently too, by interchange of . and *, d*(d*R).(d*R)=0 for all n-vectors,
a result by actual calculation at (5) and (6). However, although the analysis at (5,6) seems to know, or to define, what
d is, (d*R).d=(d*d).R=0 is not computed from (5) as is its parallel (d*R).R=0, since, for n>3, how to construct the
n-dimensional d n-vector from it's organic (1/2)n(n-1) sij 'components' is unclear. Nevertheless, d explicit or implicit,
as a consequence of the interchange of * and ., (d*d).R must be trivial since non-trivial A.(B*C)=B.(C*A)=C.(A*B)
becomes, when B=A, say, A.(A*C)=(A*A).C and A.(A*C)=B.(C*A)=A.(C*A)=-A.(A*C)=-(A*A).C, thus forcing
a regulation 0 since +(A*A).C should logically be retrieved if that is not necessarily analytically the case.
   Now since R*R=0 now means the *-product of parallel (or anti-parallel) n-vectors is 0 it implies that for arbitrary A,
B the n-vector A*B is perpendicular to A, B since A.A*B=0 and A*B.B=0.
   From R.R=R2 we have R.dR¹0 in general. Therefore [R*dR+dR*R]=0 implies R*dR=-dR*R. For the case R
parallel dR with R*dR=0 there is the observation that |R*dR|=RdR|u'*u| can yield 0 for |u'*u|=sinq but not cosq; and
for R perpendicular dR sin(±p/2) satisfies R*dR=-dR*R. In fact since it's a determination of |u'*u| that is required, in
effect u'*u is the situation ei*ef which implies |ei*ef|=|ef*ei|=|f(±p/2)|. Now, since ei*ef=-ef*ei=(-ef)*ei=ef*(-ei) we
then have |ei*ef|=|-ef*ei|=|(-ef)*ei| which implies f2(p/2)=f2(-p/2)=f2(-(p/2)-p). Thus, along with |ei*ei|2=f2(0)=0 and
|(-ei)*ei|2=f2(p)=0 these imply f is evidently a periodic function in the (ei-ej)-plane such that f(±p)=0 - evidently the sine
function.
   For arbitrary q, we take A, B separated by angle q. Since we know only the perpendicular component Bn of B to A
contributes to A*B we find |A*B|=|A*Bn|=|A*un|Bsinq=|u'*un|ABsinq and similarly |A*B|=|An*B|=|u'n*u|ABsinq for
which u', u'n are the unit n-vectors of A, An and likewise u, un are those of B, Bn. Since |A*B|=|u'*u|AB we find for
|u'*u|2=f2(q) the relations |u'*u|=|u'*un|sinq=|u'n*u|sinq yield

f2(q) = f2(±p/2)sin2q       (17).

   Now, (u'*un)2=u'*(u'*un).un=un*(un*u').u' and similarly (u'n*u)2=u'n*(u'n*u).u=u*(u*u'n).u'n. The n-vector u'*un
is perpendicular to u', un, thus the triple product n-vector u'*(u'*un) is parallel, or antiparallel, to un. Similarly for each
of the other three such analogous n-vectors, and they are all equal; the dot product angle in all must be the same 0 or p
since cosq¹cosp. Thus, for each of the n-vectors, equating their magnitudes to the magnitude (u'*u)2, that is, from the
fact that (u'*u)2=(u'*un)2sin2q=(u'n*u)2sin2q we now find f2(q)=f2(p/2)sin2q=f2(-p/2)sin2q, or |f(p/2)|=|f(-p/2)| as well
as |f(p/2)|=|f((p/2)+p), |f(-p/2)|=|f(-(p/2)-p)|.
   Now, for n-vectors A, B we can always find A=(B-C) with coplanar A, B, C comprising the law of vector addition
triangle and parallelogram. For the parallelogram so formed the angle between A and B is f, that between B and C a,
that between A and C then will be a+f such that the angle between the image n-vector of A at the opposite side of the
parallelogram and C is (p-a-f)=b. Thus in their n-vector triangle A is opposite a, the angle opposite B is b, and that
opposite C, f. With |A|=A and |B|=B we have then |A*B|2=A2B2f2(f); when A=(B-C), |A*B|2=|-C*B|2=B2C2f2(a),
while for B=(A+C), |A*B|2=|A*C|2=A2C2f2(b)=A2C2f2(p-a-f)= A2C2f2(a+f). Equating terms therefore now yields
[A2/f2(a)]=[B2/f2(b)]=[C2/f2(f)].
   In the n-space triangle the n-vector normal to C from the vertex at f is such that (Bsina)=(Asinb). Similarly for C.
Thus [A/sina]=[B/sinb]=[C/sinf] with, since the n-vector triangle is a plane triangle, the clear inference is f(a)=sina,
f(b)=sinb, f(f)=sinf.
   However, noting from R*R=0 that A*B is reduced to A*Bn or An*B and that in the ABC-parallelogram/triangle
these *-products are actually the area of the parallelogram or its two triangles then we see |A*B|=|B*C|=|C*A|, which
implies A2B2f2(f)=B2C2f2(a)=A2C2f2(b), which we unnecessarily derived earlier trespassing just a bit on a distributive
law for the *-product. But, a distributive law [A*(B+C)]=[A*B+A*C] can be verified taking, say, A=R, B=d*R,
C=d*(d*R) given at (5,6), without evaluating ei*ej - which means all such verifications (including n=3) taint, even for
A*A, just that bit of circular logic at the level of the ei. However, taking the inner product distributive law as given for
A.A it can be verified following the familiar method of (dot product) components An,p, Bn,p, Cn,p as necessary, where
n, p mean the perpendicular/normal and parallel components respectively.
   Now, in the relation A2B2f2(f)=B2C2f2(a)=A2C2f2(b) from the ABC-parallelogram/triangle going by the dot product
cosine law for plane triangles C2=(A2+B2-2A.Bcosl), evidently valid for all n-vectors A, B, C, n>1, clearly, increasing
l to p/2 results in the pythogorean triangle. Increasing the angle f between A and B in our ABC-triangle results in the
right n-vector triangle with n-vector C its hypotheneuse. Thus, f2(p/2)=(C/A)2f2(a)=(C/B)2f2(b), showing that f2(p/2)
is the maximum value of the f2(q) since in area |A*B| we have Bn=B, or An=A, that is A or B is already the maximum
'height' of the triangle/rectangular parallelogram with respect to the one or the other as its 'base'.
   Since the magnitude of the *-product of two n-vectors, n>1, is shown from R*R=0 to be the area of a plane rectangle
(actually in fact calculated from only the scalar length of n-vectors, thus rendering the vector concept a mathematically
superfluous concept in such cases) it is evident that the area of the plane rectangle formed by mutually orthogonal unit n-
vectors ei, ej must be of unit area since merely |ei| and |ej| suffice since ein=ei with respect to ej, or vice versa. Thus
|ei*ej|2=|ej*ei|2={f2(±p/2)}=1. Thus (17) becomes f2(q)=sin2q. Thus |A*B|=ABsinq.
   We may also come to (16a,b) geometrically. For d*R.R=0 evidently nothing of d parallel to R can survive * - or
d*R.R¹0. Thus dp*R=0. That is, R*R=0=d*d=A*B - here due to the .-product - and therefore R*dR+dR*R=0.
Thus d*R.R=dn*R.R=0. In that relation we must have dn*R and d*R perpendicular R, but clearly we also have dn
perpendicular to R, with d and R coplanar since two vectors always necessarily define a plane in any space. Thus d*R
(is not necessarily, but) can be coplanar with d, R - that is d*R=ldn. Thus,

dn*R.R = 0 = (d*R)*R.R       (18).

However, we know from (6) and (5), or (10), d*(d*R).R¹0. Now since d*R=ldn and thus dn*(d*R)=0, then
d*(d*R).R=dp*(d*R).R¹0. Since dp=l'R, then

dp*(d*R).R = R*(d*R).R ¹0        (19),

thus contradicting (18). Thus d*R¹ldn - that is, d*R, d, R are not coplanar. Alternatively in d*(d*R).R¹0 we can
substitute d*R=ldnR to get d*dn.R=[dn*dn.R+dp*dn.R]¹0, again showing d*R, d, R are not coplanar. Neither
d nor R is special, and since d is independent of R, it being constituted of no Ri or dmRi but the dei only, then d*R is
perpendicular d.
   Of course, like d*R.R=0 there are the infinitely many (dmd)*R.R=0, d*(dmR).dmR=0 already mentioned. They are
all necessarily coplanar triplets since two vectors are always identical. Like d, R in d*R each pair of n-vectors in the *-
product necessarily constitute a plane, each plane in general orientated differently in n-space. Thus from above, and in
particular, d*R.R=0 from (5), we can infer that since all the vectors and their planes are arbitrary but each triplet, two
vectors of which are identical, is necessarily coplanar then necessarily A*B.B=0. Thus d*R.d=0 since it's a triplet with
two identical members and thus necessarily coplanar.
   Thus the three non-coplanar n-vectors d*R, d, R clearly constitute an n-space parallelepiped of sides d*R, d, R,
and d*(d*R).R its volume. Now, since the volume of the parallelepiped must be independent of which face is labeled
the 'base' in the 'base area x height' definition then its volume is d*(d*R).R, or R*d.(d*R), or d.(d*R)*R - which is
evidently the interchange of * and . in (d*R)2=-d*(d*R).R, which is (10), and which by symmetry, must imply

(R*d)2 = - d.R*(R*d)                (16a),

and thus

d*(d*R).R = R*(R*d).d                     (16b)

for all n-vectors.




IV: Basis Vector Products:
    Noting that the linear rotation-differential calculation at (6) somehow knows how to turn the triple n-vector product
d*(d*R) at (6) into the single vector at (6) it means that wrt to the formal vector analysis there is a 'bac-cab' rule or its
equivalent to the *-product; thus


(A*B).(C*D) = [(A.C)(B.D) - (A.D)(B.C)]        (20)

must somehow be consistent with *.
   The inner product d*R.d*R we know by actual calculation at (5), (6) is (10), d*R.d*R=-d*(d*R).R for all n
-vectors n³2, by symmetry, is (16a), d*R.d*R=-d.R*(d*R) for d, like R, an arbitrary n-vector. Now, from the
result |A*B|=ABsinq we see,

(d*R).(d*R) = d2R2sin2q = d2R2(1 - cos2q) = [(d.d)(R.R) - (d.R)(d.R)]        (21),

which is evidently a two-vector case of (20).
   Thus from (10) and (21)

- d*(d*R).R = [(d.d)R - (R.d)d].R         (22)

and from (16a) and (21),

- R*(d*R).d = [(R.R)d - (d.R)R].d         (23),

the rhs of each being evidently the A*(B*C) triple product 'bac-cab' rule for, by the lhs, the special case of a two-
vector triple A=B, and C, with two perpendicular but otherwise arbitrary n-vectors d and d*R, and R and d*R
respectively, with arbitrary angle between d and R, and their third n-vector d or R as per (22) and (23). Therefore,
-R*(d*R)=[(R.R)d-(d.R)R] - up to one or two arbitrary n-vectors C, C' perpendicular to R such that R®(R+C),
and d®(d+C'). However, at the outset, by hypothesis, R in (3) is the only n-vector and by consequence of the dei of
(3) - not even of R per se, not of C, not of C' - the d n-vector in (5), (6) and implicit in (3).Thus C=C'=0.
   Given this triple product special 'bac-cab' rule for all n-vectors n³2 we now utilize it to analyze the basis vectors triple
products ei*(ei*ej) and the products ei*(ei*dei) in which ei is perpendicular to ei*ej and ei*dei and the angle between
ei and ej, and ei and dei is an 'arbitrary' p/2 - but evidently ej and dei can be any productive n-vector since even q=0 is
valid in the ei*ej or ei*dei vector. Now, ei*(ei*ej)=ej and ei*(ei*dei)=dei=sijej. Next we write ei*ej=arsek, k¹i, k¹j
and ei*(ei*ej)=arsei*ek and then write ei*ek=bmnep, p¹i, p¹k. We then sort and collect the arsbmnep to the (n-1) ep
basis vectors in ei*(ei*ej) and compare its terms with ei*(ei*ej)=ej to obtain (n-2) Sarsbmn=0 and one Sarsbmn=1
equations. We will now have (n-2) ars and (n-2)(n-2) bmn. Next, for i=1, say, we can replace j=2 with j=3 to have by
(22) or (23) ei*(ei*ek)=ek, k=j=3, i=1. There will be a new set of ars thus we label the a's aijrs, aikrs, etc, the ij, ik,
etc identifying the specific basis n-vector pair that generated them but evidently the bmn will remain the same but for the
ei*ej that comes from ei*ek - and evidently this ei*ej when it appears is labeled the same ars used for it in ei*(ei*ej).
Similarly, every 'new' set of ars in ei*ek of ei*(ei*ek) is in fact already the bmn in ei*(ei*ek)=arsei*eq. Thus eventually
every equation is a bmn equation but for one ars - or if we prefer, an ars equation but for one bmn.
   From ei*(ei*ej), j=2,3,...,n we can go to ei*(ei*ej), now from i=1 to i=2,3,...,n, all permutations still in the same bmn,
only with some sign changes in the bmn equations as ei*ej becomes ej*ei in the permutations. Similarly, for the analogue
schemata for the [nearly-as-convenient] triple products ei*(ei*dei) and permutations, this time the coefficient equations
being Sarsbmn=sij, sii=0, which is evident since dei=sijej and thus ei*(ei*dei)=Ssijei*(ei*ej).
   Perhaps the results of all the permutations of ei*(ei*ek) can be more vividly arranged in an nxn or (n-1)x(n-1) grid/
array. At the top of the grid the orthonormal basis set e1,e2,...,en is arranged in order horizontally; on the left of the grid
the same set of unit n-vectors is arranged in order vertically. Next we make every element of the first row and the first
column 0. This first row is e1*(e1*ei), i=1. The remainder of the grid is the (n-1)x(n-1) identity matrix nestled/nested into
the trivial first row and first column. Every element of the grid is 0 except for the diagonal elements but the (1-1)-element
which is 0. The row elements of the grid are the components of the n-vector e1*(e1*ek)=(0e1+0e2+...+1ek+...+0en) of
the kth row; and the column marker of the column element, 1, for this vector is also k. The elements of this e1-grid are
Sa1krsbmn=0 or Sa1krsbmn=1, where the 1 is for the basis vector e1 and the k for ek in e1*(e1*ek). Again, the a's
are from ei*ek=Sa1krser, r¹1, r¹k, while the bmn come from e1*er=Sbmnem, m¹1, m¹r.
   The nxn ei-grid for ei*(ei*ej), i¹1, looks the same as that just described for the e1-grid e1*(e1*ek) except that now
ei*(ei*ei), that is, j=i, puts the row of all 0 elements at the ith row of the grid and the trivial column at the ith column of
the grid. The remaining n-1 diagonal elements are 1's, and the off-diagonal elements 0's; the (1-1)-element is now from
ei*(ei*e1). All elements of the grid are again Saikrsbmn=0 or Saikrsbmn=1. The bmn components in a column do not
change within a grid between pairs of consecutive elements but for precisely one b since a set will be brought in for one
omitted on the basis that ei and ej are not in ei*ej=Sbmnem but ej is in e1*er=Sbmnem while ek and ei are not. After
all the a, b components of ei*(ei*ej) are summed according to whether they belong to ep or eq in the ei-grid the bmn
of a column are the same in all the elements but one but in general change from element to element in a row; evidently the
row a's of ei*ej vary in general from element to element along a row and in general so too down a column as we move
from ei*ej to ei*ek of ei*(ei*ej). From grid to grid a set of a's or a set of b's may change sign as we go from an ej-grid
ej*(ej*ek) to an ek-grid ek*(ek*ej).
   However, whether we grid-format the permutations of es*(es*et) or es*[es*(es*et)] or es*(es*dei), etc they are all
the same n grids. What we notice in a grid is that the 1 element 'moves' along the row and down the column. However,
the a, b components that constituted that element do not 'move' with it. Now, at most one and absolutely only one bmn
of the sum of the arsbmn products will change/be replaced when the 1 'moves', and the original sum that was 1, now is
precisely 0. The a's also might change since the 'move' of the 1 actually means the n-vectors going, for specificity, say,
from e1*(e1*e2) to e1*(e1*e3) - that is, from row-2 to row-3 of the e1-grid. Thus the a's of e1*e3 need not be those of
e1*e2. However, the a's of each are arbitrary. If the replacement a's are all 0 to account for the now 'moved' 0 element
in the grid, then evidently e1*e3=0. Further, both sets of a's can be positive, and the one b replaced also positive, since
they are all arbitrary. Thus, the mandatory 0 upon the change e1*(e1*e2) to e1*(e1*e3) is impossible, and likewise the
change from the 0 e2-element to 1 e3-element. In fact being arbitrary both sets of a's can be the same. Moreover, with
e1*e2=[a3e3+ ...+anen] and e1*e3=[a'2+a'4e4+...+a'nen] - with (a32+...+an2)=1 and (a'22+a'42+...+a'n2)=1 - we
can even have all the a's equal each to each since e1*e2 with a3e3 cannot be the same n-vector as is e1*e3 with a'2e2,
even when a3=a'2 and the rest of each vector is common.
   Thus, all the coefficient elements in all the grids should be 0 or 1. The elements in a grid and thus the contradictions in
the grid are of course the consequence of the 'bac-cab' rule in (22), (23). Thus just one element in a row of a grid being
1 and the other n-1 being 0 means e1*e2=liei and ei only, evidently i¹1,2. Thus, e1*e2 is not an n-vector, n>3. Even
e1*e2=[liei+lkek] won't do, let alone e1*e2 with, by hypothesis, the other n-2 basis n-vectors it can be expressed in,
given that e1, e2 perpendicular e1*e2. For e1*e2=[liei+lkek] the triple product e1*(e1*e2)=[lie1*ei+lke1*ek]. Now,
e1*ei=[e2e2+ekek] and e1*ek=[q2e2+qiei]. Thus e1*(e1*e2)=[(lie2+lkq2)e2+liekek+lkqiei]. However, by the 'bac-
cab' rule in (22), (23) e1*(e1*e2)=e2 only. Thus (lie2+lkq2)=1 and qi=ek=0. Evidently then e1*ei and e1*ek must be
the same vector to within ±1. Thus, since this analysis holds for n>4, there can only be three linearly independent basis
vectors.
   Thus, whether by straightforward algebraic analysis, or the method of the ei-grid, there is no n-dimensional vector or
vector space, n>3. The ei-grid of course shows vividly the implications of n³4 - and in fact for n=3. The ei-grid handles
any arbitrary n-vector A of ei*(ei*A) in the same way it handles ei*(ei*dei) if what we do is the entire sum, in each case
expressing ei*ej as n basis vectors, then add them before we test each to see if it can really be expressed as an n-vector.
And, actually adding the grids gets us a row of 1's or column of 1's. For n=3 there are three 3x3 grids and the one bmn
that swaps with bnm is precisely the one that 'moves' with the 1 - that is, there is just one element per column (or row)
that is not trivial and that nontrivial element is the one b, one a in one ab product. Thus the importance of one and only
one b, a that 'moves' with the 1 is clear.
   The foregoing analysis a la triples of the sort ei*(ei*ej), ei*(ei*dei) is evidently highly labor intensive. However, a less
labor intensive triple product of the ei is implied in another special case 'bac-cab' rule for a triple product of the vectors
d*R of (5) and d*(d*R) of (6). By actual calculation of their actual n-vectors in (5), (6) we find d*(d*R).(d*R)=0
for all n-vectors as is evident from the layout in and their array template constructs of (7), (8). Utilizing (20) we find for
A=d, B=R, C=d, D=d*R,

0 = (d*R).[d*(d*R)] = [(d.d)(R.d*R) - (d.d*R)(R.d)] = 0        (24)

since R.d*R is not only, like in fact the genre (dmd)*R., etc, already cited, a calculated 0, it is a regulation 0; and since
d must be, as already shown, an ordinary n-vector like R, then d.d*R is also a regulation 0: that is, for A=B=d, and
C=R in A.B*C, d.d*R=d*d.R, and d.d*R=B.C*A=d.R*d=-d.d*R=-d*d.R means it is impossible for d.d*R¹0
whether or not one bothers to find an/the expression for d.
   Analogously to (22), (23) we find from (24),

d*{(d*R)*d}.R = [(d.d)d*R - d(d.d*R)].R         (25)

and

R*{d*(d*R)}.d = [d(R.d*R) - d*R(R.d)].d        (26).

As for (22), (23) any possible C, C' perpendicular to d, R is C=C'=0. The n-vector d*{(d*R)*d} of (25) is evidently
another ei*(ej*ei) triple product. However, in the n-vector R*{d*(d*R)} of (26) R, d, and d*R are now all different
n-vectors for all n>2. Evidently, R, d is each perpendicular to d*R, and the angle between R, d arbitrary. However, in
R*{d*(d*R)} we know the dot product R.d*(d*R)¹0, and thus R is not perpendicular to d*(d*R). Therefore the
'bac-cab' rule in (26) is not now as severely restricted as that in (25), and definitely does not now exclude three mutually
perpendicular n-vectors - though of course R and d optionally so, a fact that means the 'bac-cab' rule in (26) is too not
necessarily an absolute regulation 0. Further, we know R, d are not compulsorily trivial since (d*R)*{d*(d*R)}¹0 -
as can be verified by (A*B)*(C*B)=[C(A.B*D)-D(A.B*C)]=[B(A.C*D)-A(B.C*D)], since it is derived from the
'bac-cab' rule and (d*R)*{d*(d*R)} is covered by the 'bac-cab' rules in (22), (23) and (25), (26).
   The 'bac-cab' in (26) turns out to be quite convenient to the analysis of triple products of the n mutually perpendicular
orthonormal basis set n-vectors of any n-dimensional orthogonal frame and thus of n-dimensional space, n>2.
   Clearly, by the rhs of (26), (ei*ej)*ek=0, i¹j¹k. Thus, ek must be parallel to ei*ej. No matter which of the other n-3
ef we put for ek each must be parallel to ei*ej. Thus, each must be parallel too to ek, and thus not all the ef are mutually
linearly independent as postulated. Every possible permutation of the n basis orthonormal n-vectors/set in (ei*ej)*ek is
evidently trivial. However, n-2 ek are linearly independent by hypothesis. But since ei*ej must be parallel to each if it is
expressed in terms of the ek, and ei and ej, then it must be the n-2 ei*ej=Snk=1(lkek+0ei+0ej), k=1,2,...,n, k¹i, k¹j,
which implies ei*ej.ek=[lk/(n-2)]ek.ek or ei*ej=Snk=1[lk/(n-2)]ek. Thus ei*ej is a smaller n-vector than itself unless
the n-2 such ek components are equal to it: that is, n=3. Thus there can be no orthonormal frame of greater than three
dimensions. Thus there can be no n-dimensional vector space, n³4.
   For the space/frame n=2 we still have d*R.R=0=d*R.d by calculation. Thus the 2-vector R, d is each perpendicular
to the 2-vector d*R=R', say, with all three 2-vectors d, R, R' all evidently in the same plane/surface of the same two-
dimensional space. Since d, R, R' are arbitrary we now take d*R'=R'', say. Thus R''.d=0, and R'.R''=0. But R'.d=0.
Thus R''.d¹0, which contradicts R''.d=0 from d*R'=R'', d¹0, R'¹0, R' not parallel to d, thus R''¹0. Thus, although a
two-dimensional surface in/of a three-dimensional space is evidently consistent, there is no two-dimensional space.
   Evidently a one-dimensional frame/vector cannot be linearly rotated in a one-dimensional space and a dot product is
defined for only q=0,p on a just so basis, thus the one-dimensional space is beyond the scope of our analysis.




V: Conclusion:
   It is clearly demonstrated that n-dimensional frames/spaces are mathematically impossible. Nothing mathematically
impossible can be physically or conceptually real; n-spaces/frames exist by untested and unanalyzed premise.
   The n-vector structures at (5), (6) are not what-if premise or what-if supposition; they are not mathematical inventions
- they are as valid as the vector itself and as real as rotation. The n-vector structures at (5), (6) are simply not dead-end
formalities to first principles n-vector analysis; mathematically they are as natural as numbers. At (5), (6) (and in vector
analysis generally) the dot product necessarily spontaneously defines a *-product as soon as it (the dot product) - in
fact as soon as orthogonality - is defined/postulated.
   Given n-vectors A, B, C continuous and A*(B+C)=(A*B+A*C) for n-vectors it is quite an elementary exercise to
show from first principles d(A*B)=[(dA)*B+A*dB]. The wonder at (5), (6), n³4 is how does the differential operator
know the difference between the n-vectors ei*ej and ej*ei when, not to mention even whether it exists, even given ei*ej
and ej*ei perpendicular to ej, ei they are still in an (n-2)-surface of n-2 perpendicular ek's - that is, there is no shortage
of possibilities to find two different perpendicular n-vectors even in the same surface. The wonder is not that dR even
knows that there is d*R - it's that d*R is calculated with such precision from the myriad possibilities for ei*ej that even
if it were the only such instance, d*R.R=0 for every n³2, exactly as it's supposed to be, according to n=3 analysis.
   The differential dR method is not the only analysis that points to the mathematical impossibility of n>3 n-spaces/frames.
It is commonly known that the antisymmetric matrices/arrays can be taken as the array of linear infinitesimal rotations,
and so too the orthogonal symmetric matrices. However it is also shown that the group of nxn antisymmetric matrices
form not an n-dimensional vector space like the symmetric matrices but rather a vector space of dimensionality ½n(n-1).
This then means that the n³4 n-vectors, or their equivalent antisymmetric matrices, necessarily forming a vector space of
½n(n-1) dimensions, necessitates infinite ascent to an associate ¥-dimensional vector space if its associate n-vector does
in fact exist, and thus Sni=1cos2qi=1 if so, which implies Sni=1sin2qi=(n-1). Therefore the orthogonal antisymmetric and
symmetric matrices/groups are mutually contradictory unless n=½n(n-1). Since this ¥-dimensional space is not optional
but mandatory given the associate n-vector this necessarily means every sinqi=[(n-1)/n]=1, n=¥ and cosqi=0, that is,
every ¥-dimensional vector is parallel to evey basis vector of the ¥-dimensional space - and thus in fact every arbitrary
¥-dimensional vector one-dimensional. Thus every n-vector n³4 is ultimately an ¥-dimensional one-dimensional chaotic
vector - apparently including the Hilbert space/vector.
   Thus in accord with our analysis no postulated manifold of postulated dimensionality n>3 can be spanned by ordinary
orthogonal frames. If they do exist then they must be spanned by chaotic frames (without a concept of dimensionality/
definite direction) and thus are chaotic manifolds. Such non-ordinary manifolds postulated to be spanned/spannable by
orthogonal frames must necessarily be flawed manifolds/spaces.
   There is no other space but three-dimensional space, unless, since linear rotation is after all motion, relativity, despite
its complication, if not its invalidity, of the definition of the vector, can, taken into account, change the analysis. So can
relativity change the analysis? No, definitely not - not as currently formulated; and furthermore there is no such thing as
spacetime. do see paper1.
   Now, since the complex n-vector has been postulated and routinely accepted as mathematical fact and theorems
concerning complex/imaginary n-vectors and complex n-spaces automatically apply to real n-vectors and real n-spaces,
are there complex n-vectors and n-spaces despite the impossibility of real n-vectors and n-spaces, n>3? Definitely not -
see paper4 section VII, paper2 section II. Thus there is/was no mathematically valid Riemann Hypothesis, to name one
celebrated problem. And, in fact, too, there is no mathematically valid Poincaré Conjecture, since the topological three-
surface of n=4 is mathematically impossible.
   Even as make-believe - or math-believe - n-space, n>3, is mathematically utterly impossible. It cannot be utilized with
mathematical confidence and certainty in math derivations, proofs, analysis, ... , anything else; nothing mathematical can
be proved on a false or faulty premise. If n-space, En, of orthogonal dimensionality, n>3, is still presumed to somehow
exist in some guise it must now be rigorously and definitively proved to exist.

[&]   paper0      paper1       paper2       paper3       paper4       ePR       PIE        home      indexx