**Super-Projective Planes and**

**Three-Space Condenser-Capaction**

**13 February, 2013**

*L. Edgar Otto*
Where we imagine a
filled and continuous space or region of some natural dimension even as a
universal empty span, the idea of condensing of dimensions not like their
discrete breaking in perception we arrange as in compactification theory, the
'condenser-compaction" I have long called

*flangelation.*
This has analogs to
what we cannot see or construct, or paint on a three dimensional canvass
representation (n-D printing) by our intuition of continuity and discrete (thus
quasifinite) ideal entities remote and in close proximity to our experienced
scale and quasi-oriented centered
coordinate position

*.*
Given the
impossibility of construction along the lines of such ideal super-projective
spaces even in the two dimensional case, in the third dimension we take the
mapping of a curve as of two such impossible line elements, a sort of inverse
of the idea of holes and genus. By
flange I mean just beyond the ideal of isolated quasic observed flatland
(branes or limits reached exponentially, log curves like ends of a pseudosphere)
nature's unbiased singularities exhibit this same condensing description even in
negative angle geometries or saddle shapes where our ideas of such manifolds
may evolve to be distinguished.

In interpreted as
vacuum as force in the two dimensional case of gravity defined by the emphasis
of space over matter we may conclude that such flanged or condensed abstract
structures into three space is to be interpreted as charge on the general
exponential grounding of approach to the self differential convergent and
divergent branes.

Painters generally to
not represent on a canvass a spherical object as to what is actually seen or
drawn by the specific methods of projection that would be ovals. The mind processes these, in the artist and
the viewer, as solid objects with the widest general but detached ideal
symmetry.

Evidently, the mind
already with a given sense of counting and space relations that lead to the
formation of an orderly language in diverse development so grounded is thus possible.

Our literal
comprehension of particles and space reflect this sense of geometry in the
modeling of decay modes and accompanying charges in a wider four space ideal
but interrelation with ideal three space familiar geometry. In what sense in expressing relations to such remote touchstones of ideals do we assert that all or any empty but extended
string like intervals may be looped, open at one or both ends, or closed? The condensing of ideals escape the general
concept of scale yet are constrained by such general laws at any level of ideal
dimensions.

Can we then say that
such geometries represent discreteness as general laws of particle decay,
energy vibrations, and also escaped gravity? Moreover, can these not be seen
and yet mathematically represented intelligibly in terms of information or
perhaps is changes conserved, arising, or vanishing, a structural uncertainty
that also defaults to some idea of the impossible or improbable remote?

A model of a cloud of
particles exhibiting symmetry, for example smoke rings set to spin (and thus
sort the cloud as if Maxwell's demon if constrained in a wave guide soliton
thru a cylinder) may also from the center of the space it surrounds exhibit
discrete topology structures with the simple laws of connectivity- these may
also oscillate between the duality of geometric shapes.

The ideas of such
super-projective planes reduced to our perspective as we centered, outside, or
going with the viewing as if in motion, suggest the idea of genus and its count
of holes is not a bifurcation strict in the description of surfaces and volumes
as to the nature of the number of sides or orientations ideally contemplated or
realized. In the discernment of indiscernible chirality can lie in space and time as symmetry and asymmetry
right and left as in between.

Consider as a constructable approximation of the idea geometric object or objects (no actual difference here philosophically of one or many worlds) a simple thin steel ring
which may have a link where if disconnected the ring folds down into a
segment. If we apply outside forces, the
experimenter may reconnect it with a twist. If left on its own as a ring or
such a segment it seems to persist in that state. Obviously such rings may connect together or
even be subject to the idea of a minimum distance in a certain space that can
be quasifinite rather than the discreteness of time or space at the
foundational levels of existence.

As we proceed to
increase the numbers of twists at that single ideal node the resulting rings
are alternately one or two sided surfaces. The higher the number of twists (not
to mention if we were to cut them in to so many strips that are really opposite
each other locally the result is not a surprise if these so form links that
describe particle decay and the nature of the physics of those decayed, as in
quarks, closer to the intuition of our first ideas of these as nuclear strings)
we find if the spring like tension is allowed to spread out and be balanced at
least along its own edges against itself and each other of the pointing, we
find ever increasing geometry- first trefoil like, then tetrahedral in the
general direction of its loops... then more complex structures presumably
reflecting higher dimensions of symmetry.

The same idea given
the possibility as realization by open or closed knots, internal tension or
external compression of force or this in a sense mirrored - for the presumed
physical tangible freedom from scale where the intervening space may be thought
of as full or empty, the end of a closed string a naturally occurring thought it
bound or contained within some surface brane generalization of a string- I have imagined a circular string guitar to
play on it the sliding shapes of chords with the chromatic tempered keys, but
how would you tune it and in what sort of space if the physics of a string
within our reach of scales for playing, would make such an instrument possible
as if weightless or bounded in a disc in free space?

* * * * * * *

In the accompanying
illustrations, really made before the essay to which I considered posting only
as if art, I was inspired by Darling's book and an illustration by John Sharp
called the pursuit curve. In walking
thru Randal park to the coffee shop, the diagonal crosswalk are roughly this
shape to which the important thing is that the distance along the curve to the
center equals a square side along the park's edge. So why take the shortcut? Anyway, at the center this is supposedly
never reached so a construction in the illustration of a 5-cell or alpha 4D
simplex has such ideal holes at some segment ends, or it is described as an
octahedron opened in two directions.

If you make a candle
of the great pyramid and allow it to freeze rapidly outside on a cold Wisconsin
winter night it seems to split into two parts, or four parts, but not totally
as we superimpose another square of the log twisting ones.

I found it surprising
in an experiment long ago, and I did all these experiments in the flesh to some
degree of relaxed results of which I now realize these quasi-thought
experiments were significant for my ideas later development.

All right angles are
equal, and the idea of another dimension at right angles to the next lesser
ones... that and the direction of a gyroscope.
I wondered what would happen if we put a system of gyroscopes together
to express this outside right angle?
With magnets (and Lenz's law as well as conservation of momentum) not as
complex as one from an old shortwave radio of toroidal construction that allows
for the redirection of forces say from induction of house wiring that things in
it stay in perfect distances between each other like steel ball bearings if put
in the ring it seems to wind up certain metal spirals where I imagined tapping
the spin of electrons so directing them- and many such ideas that from some
view may be wrong as well surprisingly true.

I have passing
thoughts lately as if I am in a second childhood where I played with my fathers
microscope and telescopes and resistors- one of the first transistors in my
hand although crystal radios more magic needing no batteries. Guess it was because I found such a
microscope in an antique store... longed for it although it could not do much
in that technology other than the sentiment.
It was beside a globe from that time too which still seems familiar to
me having learned the map at an early age- and yes tried to copy North America
tracing over it only to find the trace would never close.

* * * * * * *

Yeah, you know, it's funny, Cardinality was created in order to conceptualize infinity and infinity, by definition, exceeds Cardinality - countably so in one sense and uncountably so in the other! In the end, mathematics is just an Ontology agreed upon by the community of inquirers . . . and even then not all are in agreement! But without mathematics it is exceedingly hard to maintain deductive consistency - it's a function of deductive complexity.

ReplyDeleteAnyhow, I would definitely characterize your art as math . . . and being an artist myself, it seems my art only became formally mathematical after I came to understand a certain degree of math. Artists who work without a functional knowledge of elementary calculus really fail to fully appreciate geometry - just my opinion.

I had an artist friend long ago who tried to make structures from the straws and struts of an organic molecular model set I had purchased that day in town. Now I remarked that what he was trying to do, to make a ball of hexagons, was not possible- he worked a long time trying to convince himself this was the case. But realizing these limits he then used the knowledge to make a rather interesting abstract structure that had beauty irregularly on its own. Of course with the straws the pentagon angle is only close to the octahedral angle, 108 and 109 54 54. Still, Pauling imagined structures akin to say today's buckyballs so as to think about the chemistry of it all withing a range of what is possible with a fuzzy range of interplay of definition. But if we look more deeply it is clear to me other things at the foundation may distinguish art as such from mathematics as such and this is hard to see- only possible in physics if we extend and not diminish our ideas of induction and deduction and stuff outside or in between. Sometimes if one looks hard enough there are ways to balance things.

ReplyDeleteI am continuing with this theme in the next post as to our relation of such illusions or mirrors- where art of course meets somehow to the math. Then, making a metaphysical leap (as it seems at this point) I explore principles even beyond our ideas of some ideal infinity or geometry. As usual we face new paradoxes that affect us, perhaps how we see ourselves or others as beauty...

What sort of art do you do? PeSla

I paint (mostly with oils) and I sculpt. The masthead on my blog as well as the Self-Portrait just below are oils of mine. You can see a couple more works on my post "Artworks and Such", (http://atomicdecompositions.blogspot.com/2012/10/artworks-and-such.html). I also write poetry - if that can be considered art . . .

ReplyDeleteI think topology, a subject I am just now beginning to explore, is at the intersection of math and art . . . Although while a mathematician can't comb the hair on a hairy ball without forming a cowlick, I believe any decent artist probably could.

ReplyDelete:-)

ReplyDelete