**11 February, 2013**

*L. Edgar Otto*
Micro-tones of integer
ratios with respect to just notation allow a merger with the chromatic twelve
tone tempered system of music for consideration of the change of key theme hierarchies and this too is a method of understanding the intimate relation between
abstract space and numbers.

But as regions of the
quasic plane the discrete integer ratio values are quasifinitely expresses as
square or quadratic forms in the number theory.
The general idea of proof of Fermat's Last Theorem should not be an end
of interest in the sub depths and spans of
familiar low level dimensions and numbers. In particular and as a pattern parallel that
is the basis for that proof the answer sought for by integer counting methods
involves the abstract dimensions as represented, creating an abstract symmetry
hidden to the dimension of plane flatland,
between two and three space.

These are quasic as
they are issues of quasi-contiguity.
What may touch in some sub-space dimension may not necessarily do so in
another sub-space dimension. The idea
that what can be represent as triple integration adding or subtracting a
dimension is the same as partial integration does not necessarily unify the
descriptions between such contiguity as equivalent in some directed or
evolving degrees of freedom to which physical functions may set the bounds of
behavior.

As a very simple
exploration, the actual counting of areas in such abstract quasifinite ratios,
I looked at the comparison of actual pixel area count and methods related such
as our ideas of dilitation or congruence. From a foundational view on that
frontier it is not at all clear our intuitions unto this exploration will yield
mroe than trivial results- these moreover to be abandoned as we strive to
explore wider laws of mathematical physics beyond such trivialities of the
Euclidean plane to which we regard then as too obsolete a view as to consider
it progress in science to explore- in other words the research direction itself
is an enterprise of quasifinite understanding and imagination as self limiting.

This deeper view of
the plane region thus not only allows for the distinctions of symmetry
(including chirality and parity) but is the foundation at singularities for
asymmetry as well to which we may follow in some way the structure of our
theories.

We know from more
general considerations of space and number, expressed as questions or
paradoxes, usually, as with any perplexity of a truth in our decisions of
thought, that the whole (equal to its parts) does admit certain nodes in a closed
orbit or in a recursive sequence (irrational even as a span) to which our ideas
of complex numbers humanity has long computed and applied, even as sacred
numbers in a vauge mysticism or worship.

We know that beyond a
certain point so expressed as this mapping unto or at some infinity as complex
planes or more general abstract brane multi-dimensional representations, the
operational laws of mathematics run out before the continuity considered and
endless dimensions.

In these imagined,
(perhaps thus possible) conceptions of space and number operations, including possible new operations or symmetries imagined, we do not necessarily heed the
speculations of the inventors of such methods as Quadratic Reciprocity that
beyond a certain level of complexity the theory is not left comprehensive and
complete. Modular numbers with residues
go way beyond our simple programming algorithms of residues and so on.

We know that the real
number plane R, can be further extended *R. So why not the depth of its
differential geometry? Why not the consideration of a wider realm of p-adic
numbers to which in regard to limits and the infinitesimal the Great Dedekind
gave us a much deeper yet different view? Or in such abstract realms we may
light upon the ideas of a higher vision, treating the irrationals as surreal in
a calculus?

Other than our ideas
of what is the standing out as if from a different dimension, the whole
greater, equal, or less than its parts, this nature of insight that is the
substance of insight, the business of physics is intimate to the binary yet
vague or wild card considerations, thus the variety of numbers as in the power
continuum as if subsets and the unique span of a continuum. In particular much of physics, as well the
errors in measure or theory so resolved involves the half real in an imaginary
number ( just as there are hyper-reals, there are hyper-numbers complex, up to
nine dimensions in their calculation as imaginary...) To these various power continua the power
raised say as a prime with hierarchies of powers or with certain integral
differences on the same level we try to put physics into some side of its
quasifinite number forms.

This is lately the
postulated inclusion of so many numbers of holes that describe embedded or enumerated manifolds of geometry on the way to a trivial or unified theory
free from our quaint ideas of a foundation really shown just views of one
structure as in the first imagined enormity of string landscapes and
compactified dimensions.

Notation itself may
be extended formally yet have a use in the future application in the
sciences. Coxeter's reflections on
symmetry exhausting space up to 8 dimensions may be extended beyond this- so
too Conway's notions of 24 dimensions, apparently both levels of insight still
not allowing higher levels as possible in a one universe complete whole to
which the application to physicality may develop partially in its methods we
imagine valid but not reachable or provable unto some infinity as the universe
diversely evolves.

In the counting of
regions of the quasic plane we thus have to consider the sub-space grids and
holes. In theory we may discover some of
these as properties of particles, or we may acknowledge that some known
particles do seem to have their mirrors in space and time while asymmetrically
so and there exists at least half spin particles. Certainly, we see this general halving of
things in paradox a part of the properties of topology even when the parts are
jumbled. We see also the paradox of
mapping any such abstract intrinsically flat line as if a continuum into a
circle or a sphere. But one thing seems
sure, local time has many subjective and objective variations. After all, in the impossible clock measure of
time in the world it is clear that a stopped clock is right so many binary
times a day.

If this research
proves ultimately trivial, well, forgive me my mathematical recreations, it
makes for the beginnings of some interesting digital drawings in perhaps an
infinite variety of them...some of these ratios are pleasing in a sense of
beauty I suppose for instance 89 is Sophie Germain prime as well as 23 and so
on I consider as pixels. But in my art
like drawings one should not assume all of them have a hidden sense of color or
ratio, discrete or toward the square or hexagonal cube analog to decode- yet
for the actual drawing if not the sketches toward an idea I have not other than
some initial learning of skill on some level of method drawn thing related to
science as if free brush arbitrary or coincidentally discovered order in the
chaos. Where there is something to
decode is it set in the patterns carefully.

Perhaps we are
limited in many respects as to how far certain numbers may go such as the Mersenne primes so favored by
Pitkanen and his number 89 and so on (I am still trying to sort out the
parallels with his and my quasic theory) so we need to show why such limits- or
not necessarily for when we know the patterns we do not need to compute all the
possibilities to infinity in the inductions (or as mathematical inductions)
beyond a few first levels to know on some level something like pairs of primes
extends forever. But, let us hope some
viewers enjoy the illustrations for the art's sake if nothing else.

Part of defining
consciousness, by the way, is this creative process to which we express or
desire out engagement with the world and discoveries for those who may care,
that part of it made more than and less than in the empty whole, the substance
of it as made of insight.

* * * * * * *

Clearly, our stance
to holes and the bifurcation in them amounts to a global logic- as to what
happens in the brain.

This vaguely a higher
structure of which we imagine emergent wholes different as defining
intelligence that its parts, and with the idea of thresholds, that multiplicity
of a single photon cascading others in retina stimulus or in photosynthesis as
if a model of artificial intelligence, or as logic circuits the simple states
of signal or no signal for a zero ground- logic as algorithm or these holes as
non-algorithmic. But the higher level or
lower level over indefinite space may have this distinction top down and
bottom up that do not necessarily correspond.
Nor does there need to be a material structure for such dynamics of
space.

If the eye as part of
the brain (Penrose) then the brain too has this process of merging or halving
what we imagine as visual illusion in which to distinguish what is outside our
quasifinite apparently closed surface as we contemplate what in the dynamics is
separated space like in the speculations on relativity and black holes unto
singularity.

We see patterns
develop where these abstract vector like empty entities can merge or
separate. Two lovers may break up and
still live together for all the outside world as if defined in a Platonic
memory past the state of drastic change of relationships if in the world the
imbalance only to be compensated by the interpretation of patterns of their
dreams.

Such a state my work
together both ways...given some formal way on a four way world, quasifinite in
that with time itself we may not strictly define what is inside or outside,
just as the binary helices that merge or separate in biology, that is time
needs not be absolutely one directional outside or inside some past or future
or middle zero stance of a state of experiencing and learning mind, in complex
space many issues of the grounding of theory is what are the space like or time
like dimensions. In complex space the
abstract count of so many seemingly isolated singularities (holes so to speak)
these may make knots possible in matters of rotations and so on just as they
are only possible in real three space as stings or lines. We double such structures and double them
many times as these are abstract doublings but have to be justified in the
formalism not by the simple quantum directionality with its own uncertainty and
coherence but the ensemble of global logic that contains yet evolves the
unaware subconscious machine brain as well an adult intellectual assessment, so
too the teleology as far at it goes as our intuition.

While there appears
something deeper than quantum theory the parallel or many-world view as quantum computers adds little to the understanding or manufacturing of the whole, say
not as something relevant to military use, but quasic ideas do seem, sometimes
close to the middle ground of triviality, to justify the classical and quantum
mechanisms where they work as physicality.
We do not find, as a matter of probability perhaps, what we are not
looking for... and we do not conjure from the nothingness what we do not have
the threshold of awareness to imagine.
But the ideas of Godel and Turing in the ensemble of machines suggest
something more than a last Omega involves, thus choices beyond that at the
frontier of natural law yet universal freedom as structural systems as
universal law to contain, abstractly, the one and the many, the biochemical
like pathways of delay that forms a utilization of processes linearly or in
mazes of time.

We should not make
such machines save for the experiments sake in times of sufficient funding
knowing such principles nor expect results from complexity of grouped machines
alone constrained by Omega universal machines or ever receding last absolute
ideal infinity as with Cantor's Omega.
We may only reach some sort of teleological Omega in the sense of time as
such a direction of progress, of intuitive time with this question of centers,
beginnings or ends. But as things are in
our time, if we are concerned with vulnerability between states to which
disputes sometimes in the balance at the fulcrum of the seesaw reach the states
where surprises come outside a system, such basic research if neglected or suppressed in a society amounts to the new dark ages in the main, dark in the
reflections of our minds and mindless behaviors. To such an unknown frontier even as we have
the ability to judge scenarios, to abandon this sort of project and fundamental
research makes us more vulnerable than we have ever been.

We are vulnerable
also to close minded self assured ideas where they cannot distinguish right or
wrong so decides if an artificial machine is immoral as to its maintenance or
saving as it not being human but being subconsciously so in its architecture
that does not need clockwork time to evolve, for can the mentally ill be so on
this wise if in a sense they have no mind?
Yet let us not hold sane minds away from free inquiry and weapons, to
disarm them is to make the general state of our society less than sane, to
declaw the cat as acceptable so they do not scratch the furniture.

On the other hand,
that we, even an individual exists so as to awaken to an new concept it may set
off parallels in the social whole for these paradoxes and debates continue the
interpretations as to how consciousness engages with the universe and its
deeper realities. Communication and
signals are but parts of the divinity of intuitive patterns.

* * * * * * *

Footnote: considering the mapping of functions, and that certain things we cannot follow due to difficulty between odd and even numbers (I mean sometime or other we might ask if zero is even for example) but like the sine we have functions that lead (in a simple oscillating way) to its negative. I must think more on this as that distinction is not clear in my algebra...the numbers have gender idea. But do not we have a core issue with particle and wave as conceptions? Something in this idea suggests to me a key issue at deeper foundations- in any case it may relate to this post as to how we treat the simple division of these abstract and countable creative hole objects- still the concept of a hole may in a sense be a non-existent idea unless we mean in a positive (reduced to zero) sense it counts genus as holes all the way thru. Stashef once agreed with my comment that there were no such things as holes back when our local stringers consulted him on topology (interestingly he also was drawn to work on quantum bits.)

Some sort of structure (stereonometry) goes beyond the metaphysics of physics foundations. Far beyond such a use of these for quantum computation speculation. Even there forces and particles may come in nonnecessary innate pairs. Either such negation is a matter of convenience that undermines necessary redundancy or it is a sort of physical information itself that does compare intervals irrational and transcendental- perhaps the first frontier to find new mathematical physics beyond.

* * * * *

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