Negative Dimensionality (negatively dimensional space)
operations in negatively dimensional space – for the case of integer dimensionality
products of #s of integer values of dimensionality would vary in sign by whether that # be even or odd – even powers of negative numbers being positive products; odd powers of negative numbers being negative products. |
instances :- in the instance of energy-dispersal (expansion of light or of explosive event), because involving the 2^{nd} power of terms, this product is positive (regardless of original sign) [and therefore acting outgoingly]; in the instance of nuclear forces (weak and strong), because involving the 3^{rd} power of terms, this product may be negative (must necessarily be so, if the original sign is negative) [and therefore acting incomingly, i.e. attractively]. |
instances for magnetism :- repulsion for : diamagnetics (BISMuth [= BHISMa, who was upheld (levitated) by arrows (vectors of magnetic moment), according to the Maha-bharata] & antimony [its "star" = star of Bethleem hovering in the sky]); rotating objects (ceramics, without eddy-currents); objects in oscillating magnetic fields (produced by alternating current); superconductors |
speciously exceptional circumstances :- the attractive effect of magnetism is not a result of 2-dimensional, but of 3-dimensional functions, its interaction with electromotive force constituting its 3^{rd} dimension of activity [electromotive force linearly perpendicular to the plane of action of the magnetic force].
operations in negatively dimensional space – hypercomplex solutions
for recursive functions, "Exponentiation is non-commutative" (http://www.tetration.org/daniel/math_bio.html) : even "defining tetration for complex numbers ... indicates ... quaternions." (http://www.tetration.org/Ackermann/arithmetic.html) |
by allowing for non-commutative, non-associative, and non-distributive operations, 3 kinds of hypercomplex products may be available [derivable through fractional powers of functions]. |
the significances of complex and of hypercomplex quantities
the ordinary imaginary measure may most oeconomically be assigned (as done by Minkowski, Einstein, et al.) to time. |
the 3 hypercomplex dimensions of non-commutative operations may be conveniently assigned to the quasi-independent [from the physical one] universe of dreams. |
the 5 hypercomplex dimensions of non-associative operations may thereupon be assigned to the subtle realm of dreamless sleep and of the antara-bhava (bardo) state. |
the 7 hypercomplex dimensions of non-distributive operations may be assigned the otherwise unallocated range of connectivity among universes as manifested in omens and other inter-communicative actions. |
there may also be an overall principle of ethics unifying beings : if so, this could be 9-dimensional (the 9-pronged vajra). |
of the 9, 3 may control dreams (which are basically an expression of ethical standards, of the scruples adhaered to by the dreamer). |
of the 9, 3 more may have a somewhat dominant effect on after-death decisions made by the dead [as, the choices made by the dead in the Republic by Platon]. |
of the 9, the remaining 3 may have a relatively minor impact on communications with other worlds (which messages are often little involved with ethics whether public or private). |
the significations of "negative dimensionality"
The greater the dimensionality, the more numerous the independent qualities; whereas, the lesser the dimensionality, the fewer the independent qualities. |
Where qualities are many but mutually dependent, there may perhaps be a proper-fractional dimensionality; where some are and some are not mutually dependent, there may perhaps be a improper-fractional dimensionality. |
But because the [ordinary] root (fractional power) of a positive number is real; therefore the sort of time to be derived thence could not be the moving sort (which is, according the mathematics of relativity-theory, not to be measured in reals, but rather in imaginaries). |
Real-measured time would apparently be the unmoving sort (of the [e.g., Platonic] World of Ideas) = the Zrvan Akarana [of the Zaratustrian philosophy]). Negative time would tend to partake of a motionless / instantaneous transmission duality. |
Imaginary-measured time would naturally be associated with space of negative dimensionality. |
Objects in such space, like the space itself, would be definable as purely functions of time. |
To be definable as purely functions of time would be [as conventionally described] to be praedestined. |
Further, since praedestination is always described as entailing judgeable ethical conduct; therefore "free will" is dependent on praedestination and, hence, on negativity in dimensionality. |
Then, insofar as consciousness is a a matter of having a sense of "free will"; thus consciousness is necessarily a concommitant of [and only of, as is fairly well manifest] the negative variety of dimensionality.
physical space is negatively dimensional
4-dimensional space (required by relativity-theory, etc.) cannot sustain planetary orbits, for "a knot cannot exist in any dimension >=4 " (where "knot:" = "closed loop", e.g., planetary orbit) http://mathworld.wolfram.com/Knot.html |
But if negatively 4-dimensional, it might sustain such. |
curved spaces
Interrelatedness, of a type stronger than in recursion, can be found as metalanguages. |
But only such metalanguage as hath a 1-to-1 relationship with the language with that language which it describeth can be regarded as homologous with flat space, which is of integer-dimensionality. |
Elliptically curved spaces can be considered homologous with metalanguages which are restrictive. |
These would be of dimensionality as improper fraction somewhat more than integer. |
Hyperbolically curved spaces can be considered homologous with metalanguages which are expansive. |
These would be of dimensionality as improper fraction somewhat more than integer. |
There are also parabolically curved spaces. |
Their dimensionality would lie at that improper fraction which is the division between the other two varieties. |
The parabolically curved spaces would be aequally weighted for restriction and expansion. |
The correlated balanced metalanguage might suffice to describe cohaerently macroscopic time-retrogression. |
There would also be space of proper-fractional dimensionality. |
This would be "time-like" = the basis of virtual (superluminal) particles etc. |
A proper-fractional dimensionality would correspond to a nigh-absence of any metalanguage. |
Nigh-lack of metalanguage might make for an aequivalence to a wordless trance-state. |
knotted spaces
The difference between spaces of positive dimensionality and of negative dimensionality is that ones of positive dimensionality of not knotted, whereas ones of negative dimensionality are. |
The "strange loops" and entanglements (involving "strange loops") that may be developed out of recursive processes may thus be reckoned by negative dimensionality. |
The Fermionic "path integral" cannot be directly found in positively dimensional space. ("Path integrals don't really exist ..." http://www.math.toronto.edu/~drorbn/papers/Aarhus/I/AarhusI.pdf ) |
It must be sought instead in negatively dimensional space. |
analogy of "dimension" in Feynman diagrams with spatial dimensionality
Fermionic integration intuitively behaves like Bosonic integration in a "negative dimensional space" http://www.mat.univie.ac.at/~slc/wpapers/s49abdess.ps (p. 41) |
Here, the monicker "negative dimensional space" is used to describe such "Feynman diagrams" as involve knotting (cannot be adequately depicted by a diagram confined to a surface). |
This is aequivalent to saying that fermions may be regarded as standing forms of real-photon paths interknotted with virtual-photon paths. |
The idea is that the fermion is permanent simply because the bosonic "Feynman diagram" repraesenting it is not unknottable. |
To construct this, a minimum of two paths (1 real + 1 virtual) are needed. |
Hence the minimum of 2 partons composing an electron. |
Why should a maximum of 3 partons (as quarks) be allowed per particle? |
The answer (as may be conjectured) may reside in a topological decomposition of dimensionality to the maximum of 3 (dimensions). If so, the use of "dimension" as monicker in Feynman diagrams is indeed apposite. |
parts written May 9^{th} (Fri), 2008) :
operations in negatively dimensional space – for the case of integer dimensionality
operations in negatively dimensional space – hypercomplex solutions
the significances of complex and of hypercomplex quantities
"Negative time would tend ..."
[other parts above written months earlier]
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the below written May 17^{th} (Sat), 2008
[I can remember writing to the same effect (on mind, telepathy, & deities) some months earlier in some other article, however ("Fascism-Baha>i" & "negative exponentiation" sections, though, are new)] :
the mind (soul, consciousness) can be regarded as imaginary (= i) in quantity (timelike); therefore its functions in formulae involving dimensionality, which [dimensionality] is to be expressed as (in terms of) exponents (powers of quantity) which must be expressed by the rules of exponentiation for imaginaries, thus :-
function |
mind |
telepathy = communication between minds |
telekinesis = interaction with physical objects |
dimensionality |
1-dimensional (along time-line) |
2-dimensional |
3-dimensional |
quantity |
i |
i^2 = -1 |
i^3 = -i |
The 2-dimensional could be regarded as involving negative energies (in the sense of producible from such physical situations as, e.g., temperatures below absolute zero). So, the mind itself could be treated as aequivalent to an inhaerent negative temperature (= [in thermodynamic terms] negative entropy).
The 3-dimensional (= –i) must be understood as relations with deities, who are deemed involved with the praedestined (inevitable) [= the path travelling backwards in time].
instance of misuse (abuse) of this reversal of sign for 3-dimensional world-interactions
numeric |
occult application |
abuse |
for binary numerics : the cube of 2 is 8 |
octagonal towers in antient <arabia & in Hellenic Orthodox church |
8-pointed Fascist star-decorations |
the 8^{th} prime is 19 |
hence the 19 rods in Palo (‘rod’) Mayombe (of Zaire) ritual bundle |
fasca (‘rod-bundle’) of Fascism |
[Since the Baha>i (of Iran) also use 19 as a numeric, it may be that Iran is a limiting factor in the expansion of lutheran Fascism; because of the failure of the German government in both World Wars to achieve lasting effective domination of Iran (losing to the Czarist invasion of Iran during the 1^{st} World War, and to the Allied invasion of Iran at the outset of the 2^{nd} World War), it may well be that this became a factor in diverting the goal of the Thyssen-Krupp conspiracy from world-conquaest to more modest aims.]
where the dimensionality itself is negative, the rule for negative exponentiation would be used, thus :-
function |
|||
dimensionality |
-1 |
-2 |
-3 |
quantity |
i^(-1) |
i^(-2) |
i^(-3) |