Congruence

Congruence

In the illustration I took a very old poem of mine, cradle song, to show how a poem might be written with a set of lines that may be intelligibly rearranged. In this sense some of these topological twists and turns are poetry. The question is if we are aware explicitly of these parallels as we explore some logical space or even a space of feelings in which the music or the rhyme tries to move in a coherent picture.

Again, I come to post saying things when I had little to say. I see Kea has a most interesting post in a language I could not grasp at all at first reading. She is obviously a very good teacher, we are getting it for free guys! take note!

But the more I try to read it I see that it must apply on those deeper levels of structures I have talked about- namely the relations within something like the codon bases or even benzene rings of carbon itself- something I must look up again in Coxeter as for a long time now I have not had access to my earlier writings and have to google my brain or repeat some logical reconstruction again. But I could be wrong for after all there are so many cases where we skip the obvious proofs and in the enumeration we cling to finite counting when he general theory can be so powerful
as concentrated methods that covers more than tangible examples. Yet we can lose sight of the down to earth applications and the path to wider and more elegant theory if we dwell only in the higher wisdom entranced by touching its beauty.

Congruence is an especially good idea, but it can be limited. For example Whitehead extended Euclidean geometry (going outside the lines in his colors) such that it fit better with the ideas of relativity. But his was of limited use- as for the logician Russel, he too felt more was needed than projective geometry to explain things.

Gauss, clearly he made the great advance in quadratic reciprocity... in the realm of numbers and such congruences he realized the, well sifting in the dark, this idea of the fractal nature in them (but the truth of this needs a more quasic conception of space wherein we can show the things within things in a more geometric picture and that we can feel secure in a certain ordering for the operations of addition and subtraction- and as Gauss shows and even designates as a project for Riemann, the same sorts of consideration for those who develop the complex aspects of this when in lesser algebra we designate the "residues" as plus or minus- eventually to find new complex ideas of what we mean by primes. If this is so basic, why has it taken this long to understand this apparent fractal and logical nature of numbers? What in this beautiful general picture have we not found the parallels of its poetry?

Two is a world of its own in these matter of modular numbers that is comprehensive over a field of them. I mention this because, again in the background of my intuitions, we find 18 attached to it- my question of the symmetries of the 18 faced deltahedron that certainly looks like it applies to our counting and geometry.

In Kea's post today (although putting things into sinh or any other exponential notation seems to me to defeat a more accurate description by traditional methods) the diagrams are clear, and I might ask can we have a horizontal up and down where there are three or more branches of trees in such a notation. We again have that problem in descriptive notation of how to distinguish the operations of multiplication and addition without the excluding of one style of applying them. Or we may think that above a certain complexity of our theories these things do not apply to the physics of the familiar world. What do we gain in the general logical picture anyway if we say order the square roots of numbers as far as the general fractal picture is concerned- is it not enough that two orthognal linear motions may describe a circle?

My thoughts of yesterday in the mathematical physics of vortexes. The horizontal of mirrors, the dummy planes and zero probabilities filled or containing real particles, are these not the pairing of tori beginning in four space? Is not the benzene ring, with its aromatic properties (as with certain square rings) not that stuff within the two higher dimensional but unseen tori? Of course we should not be surprised a the exotic properties of graphene and the analogies to physical theories of particles and the cosmos. In a sense, in the flatness at least at infinity implied can we not see the sides of the graphene sheet as if the two branes of the Ekpyrotic theory and the nothing between them as the carbon sheet itself?

I had a stray thought that somehow the next level of numbers, the irrational and in itself the transcendental have an analog of such properties as quadratic reciprocity even beyond the first blush of considerations of complex numbers. Something is needed to explain why things are not actually or perceived as whole numbers, primes.
This is conjecture but no more a speculation in compass than say the Ekpyrotic model.

After all, the logarithms in Gauss's treatment of numbers are as he says an exact analogy to his modular residue considerations. And for one who envisioned non-Euclidean geometry, perhaps not just the range of error in our experiments in so trying to show this, but the lack of reach of our imaginations limited by the logic at hand that sets the trend for a new generation of researchers but stifles or delays breakthroughs of those who took the effort but did not establish priority.

Before one can paint by the numbers or color outside the lines, one should have a vision, at least in their heads, of what is the outline of regions on the multidimensional plane. It is not enough to divide things into regions and copy to the accuracy of some region expanded to make a credible sketch of the confabulations and the facts of what is physical within a total intelligible picture where the creative parallels what is natural and common sense in the rime and reason of reality.

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