Thursday, January 5, 2012
How Far and High does the Falcon Fly
How Far and High does the Falcon Fly L. Edgar Otto January 5, 2012
Horus, the god of writing, man walking upright,
counts with his raptor head
Of ropes & tally marks & cycled rods for wheels
chance golden measure built higher the pyramids
Picture words & numbers one at history's dawn
As in the gyres of Yeats, phases of the moon,the
shadow geese migrate at their nineteenth autumn
The prairie dogs warn each other, surface peek
between the holes & mounds, underground their tunnels
The pure colors of my fleshy soul calls out blind to
the trace of falcons
A part of our dreams, our sun & grain give back to the gods
They weave & knead again, rejoin shadows & dust to Olympus.
* * *
It is not clear at all in the absoluteness of p-adic or in the surreal (where say the square root of 2 can be treated as if rational) that e, which can be divided by any number (prime?) - so we have a general color or quality of "primacy"- that in these rarefied & distant trans-Euclidean dimensions there cannot be a final, large, prime number presumably in a sea of the further idea of composites.
So can we have a view of an ultimate Omega, the last infinity- or all counting complete in algebraic time? How far can we dream & reach with our words and numbers?
The idea of proof has its place & utility but we should realize its context, scope. Certainly we do not want to abandon (even if logic can prove inconsistencies) its spirit of reasoning. Even our fantasy's can be true in its poetic parallels to its inner laws. Common sense & practice as far as we can see in low numbers & complexity suggests for a two dimensional flat plane four colors suffice for distinguishing adjacent regions. So to distinguish say, states on maps. If Illinois is blue and Nebraska blue, well, that is still philosophy. Yet, should we develop higher & superimposed (where the point is the plane as higher spaces cycle and begin at four space) dimensions with time-like concepts an Euclidean plane needs five colors.
These can be squared to to equal 25, or rather 24 + 1 of permutations. How dimensions, groups, & symmetries not seen develop from here is much how we develop measure & counting; how more than that given finds unity again & the grounding operations at coordinate singularities that combines cardinality and the idea of ordinals. A three dimensional space as such requires seven colors.
What are we to make & how was it done, someone decided that four of the Einstein variety of dimension is equivalent to the ten dimensions of the Euclidean variety?
Surely the question from either view will be difficult for most people. Higher dimensional space & its directions, inverses (compliments) may only appear simpler in symmetry beyond the lower three, four or five dimensions. This is also a question of applying distinct operations like division across things like fields. So we consider again the integers & the idea of primacy, also the duality & six fold congruences over models (a sort of primitive gender of perfect number idea in these unique numbers whose sum is 8 and not necessarily the view to put all things into six space.)
In particular, sets of sub- or super- color may move together or be exchanged resulting in 4D symmetry or anti-color symmetry globally, the compliments of colors locally interchanged depending on our choices, of view, of permutations combined.
These concerns seem to me intimately connected to close packing of spheres in all the dimensions (for now still in a numerology of intuitions). 64 + 16 = 70 the Catalan number here concerning points of 5 hypercubes. 24 x 4 by 4 is 384 the number of rotations and inversions of a hypercube. 40 (the number of 5D close packing) + 24 (the number of 4D close packing of spheres) is 4^3...by this I imagine 40 colors so composed as higher analogs of the triality of 8x3 hypercube faces and 8x2 further sets hidden for the total of 5! permuted cubes over time-like motion. Clearly what is unity and what are thirds derive integrally from such numbers. 240/40 = 6.
At some place we can imagine a mixing or something between how we divide alternate and simplex factorial even and odd permutation groups. The concrete description of color, moreover, as scales over unity is transitive over our defined dimensions (perhaps in a fractal like non-degenerate analog hierarchical manner).
A hypercube of 24 faces, moving and surfaces contiguous, can be considered as two cycles of ten objects so to get things like the Icosahedral groups of inversions and rigid rotations (which is a four way deal or actually 2^n super- or sub-duality analogs.
Recall that it is said that analogy is only true to the extent it reflects the underlying geometry of things- even if proofs of theorems is an art form, let us recall also that questions of minimum entropy, distance, action & so forth are also questions of art as related to statements of minimum lines and then the enhancement of color space as comprehensive style and imagination. We can be expert tracers keep in the main to some degree of error between the lines, or we can with some variation as to what we see in our own perceptions or those shared of color paint by the numbers too.
Triality of 5 such objects are deficient by one for the 4 by 4 matrix. Assembling 3 pentacubes x 2 in a 3 by 3 by 3 matrix gives partial sets of many such collections.
Each dimension, uniquely, has three or four abstract associahedra by such 2^n analogs. So too the 6 = 3+3 color cubes surround one of 15 tri-axial labels & the orientations or twists. Yet in a sense these structures may together be prime and unique in a given quasic state space. We only see 12 of the sets of 30 cubes.
Now, we know something happens at 9 dimensions- that is at 8 the volume of the square boundary analog space equals the volume of the sphere in the same dimension. I imagine also (as there are many ways to branch off from these informal intuitions so there is work that can be done) that in the deficit of the good old e to the i-pi equals one for the division of say a disk into so many parts where a curve meets some line at infinity so to speak, (I mean does the center of a spinning wheel move abstractly- or more to the point the poles of the earth to which the fixed star of their days the Aegyptian thought the direction and goal to some heaven- but holy Loxidromes and God said let there be Riemann! And God saw it was good and made him at a distance so to speak- said it was Gauss.) That at the zeros and the powers of two where this deficit occurs, say 256 (257 surreal nimber game like) that at the tenth such number, at 256 the values are equal again... I wonder if this continues for higher powers of two?
Continuous space and counting numbers may meet again where it is not clear that anything can ultimately cycle or be repeated when it comes to reduction to some zero singularity. I would caution also, despite the brilliant concept of the difference of cubes, 19 in particular- or even that coincidence of Euler to use the pentagonal numbers for some proofs of series- that in these matters of 2^R as Mersenne or Fermat numbers the addition or subtraction of ones and the divisions by 2 or 1/2 may have to be adjusted for more general number property values, perhaps for any such use of number to define better particle mass in the still hidden shadows of the beginning of symmetry in our new particle zoo.
Such a Higgs-like entity has at once a continuous body and a shadow for what is hidden beneath the discontinuous for its conception of a god with a falcon's head that can swoop down perhaps or be forever out of sight & only its wake seem faster than time- meanwhile, we watch the shadows & the direction of flight of the silhouettes that can be the goose or the hawk, take cover in our familiar planes and give warnings, dig and connect the tunnels of our languages and lifetimes.
* * * * *
Turtles again (note also Orwin they have 5 receptors, 4 for birds, three for us for the primary colors.) newscientist
Is this a matter of evolution or a fixed concept to take advantage of somewhat beyond our primitive ideas of a magnetic field- perhaps the turtles see in this five colors primaries so deeper into the surface of the Atlantic gyre...
* * *
I begin to get a glimpse of the complexity and various issues of groups and strings and symmetry in the usual manner- especially the work of Conway and related authors of some theories including those not yet settled. Terminology also: the "kissing number". Interestingly, I followed a slightly different path- even today I just typed how important the kissing numbers were for close packing- we all have the same dream we get a hint of I guess...
for reference I list this (but I would like to recall the recursive formula again:
Riemann surface resulting from taking the quotient of the hyperbolic plane by Γ0(p)+ has genus zero if and only if p is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71.
5 40 44
6 72 78
7 126 134
9 306 364
10 500 554
11 582 870
12 840 1,357
13 1,154 2,069
14 1,606 3,183
15 2,564 4,866
16 4,320 7,355
17 5,346 11,072
18 7,398 16,572
19 10,688 24,812
20 17,400 36,764
21 27,720 54,584
22 49,896 82,340
23 93,150 124,416
196882 or 196,884