Wednesday, February 13, 2013

Super-Projective Planes and Three-Space Condenser-Capaction

Super-Projective Planes and
Three-Space Condenser-Capaction

L. Edgar Otto    13 February, 2013

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?

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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.

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  1. 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.

    Anyhow, 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.

  2. 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.

    I 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

  3. 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", ( I also write poetry - if that can be considered art . . .

  4. I 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.