Here closer to the original Starbabies: In the process of looking for centered hexagons I came across this link which of course is an interesting abstract idea in the realm of some logic, concrete or not as the link wonders for its utility. But such logic also intelligibly connects with basic ideas of geometric modeling:
http://www.seop.leeds.ac.uk/archives/spr2010/entries/logic-deontic/
After all this is a creative philosophy blog too.
This Side of the Mirror (Steradian Stars)
In trying an idea for stacked star candles (there was not much else to do) I realized the principle could be useful and it lead to some general thoughts on space of higher dimension. This in a sense the Mystique of Multi-Dimensions part II. But the paper drawn star is only similar to the crude paint sketch above as it was difficult to draw and I do not have the time.
It is based on the idea that to a chord in a circle the angles on that circle drawn to the chord would be all the same. In fact this seems true infinitely close to one end of the chord. Naturally it occurred to me to ask if this principle might hold in three or four dimensions and in what sense.
In this simplest of diagrams, the triangle, it turns out that the angles mesh together as if in a lattice of say triangles and hexagons linear, discrete, and straight.
We observe this difference again in the symmetry of a circle or a polygon as a matter of space filling that compresses or expands around a polygon chord. In a sense, and in the colorful labels of the stars obeying four fold symmetry, What is up or down is more significant than the "starfish" of right or left symmetry. (This however may be an artifact of our evolving to so see as Pinker suggests). But by up and down I mean in and out directions- for higher space perception this can be thought of as contraction and expansion.
Imagine playing golf in three space. The holes would vaguely be spherical rims as if we might imagine the surface of a black hole. (I hesitate to use terms like N-sphere or N-ball as I feel them misleading so as to hide the generality.) Obviously the golf ball can fall into it in a sense into the forth dimension.
Another way to see things, by this common aid to visualization, is to imagine it like a potential well and volcano. If the ball is charged it may be repelled before it gets to the circular rim to fall in. If it is neutral but going fast it will overshoot the rim. At just the right speed it falls in. This analogy is used to describe what happens when we shoot neutrons into a nucleus. The spherical rim of it can have the neutron pass thru it and touch nothing- so we slow them down that they fall in.
1. From the viewpoint outside such a sphere or a polygon, such a centered object can be thought of as a chord on this side of some mirror of things opaque or hidden underneath. Thus, at some shell or wider, even expanding sphere around it we have all the steradian angles (cones) the same.
2. This depends on the diameter of the "hole" or chord. Thus no matter how far away from the center in expanding there is a constant or minimum value akin to the ideas behind Euler's constant where the constant angle really never vanishes in theory.
3. Nine dimensions are not the only dimensions where the orthogonal and transcendental spaces have the same volume. This is true of a chord of one dimension and possibly at zero dimension. (How is it then that we observe integral values in the shells of things this side of the mirror?) The Cyclic group of polygons stand out here as if structures of infinite number and complexity. On this the relation between the chords forming them define questions of star density.
4. I imagine that when we ask the general shape of the universe as if it a Sphere we are not asking for an abstract conception- we may say that if we travel in a straight line we come back again because of "intrinsic curvature", but some really mean it is literally curved concretely only the scale is too vast for ordinary measuring. In this model the density of things assumes a default to something of total compass and in a sense finite. It is not accident that among the current models that the dodecahedron one is the only finite universe in the running.
5. Yet we can have an abstraction many ways as to how something straight may intrinsically curve even in a flat space. In a sense the current debate as to unification of the physics is one that approaches from the idea of an invariant intrinsic curvature as if from such cyclic chords (or even strings) in dealing with higher dimensions and those who strive to see it more from a totality where we have to begin to unify the properties with at least nine natural dimensions. (How then do we decide if such dimensional ideas are physically abstract or physically concrete?)
6. It is of some interest that we explore things like quasiperiodic five fold tiling as the golden ratio is a key one in four space. We observe this ratio in nature say in the stacking of certain crystals. Many see the evidence of the Penrose tiles at work here in the idea of darts and kites. So let us not be surprised (despite the general fact that deviations from a recurrence formula for hyper-volumes is here a matter of default and this in a sense linearly intrinsic and invariant to the supposed differences and flows of things as or in the universe- that is beyond the questions of genus structure where these can be hidden on one side of a mirror intrinsically or even expanded in principle- the 24 cell the best structural example) that in three and four space of these steradian angles that the internal to such polyhedral chords (let us not forget how long it took to see that if we divide the angles of any triangle an equilateral triangle is defined inside it) We have analogs to such kites which in general, at least as Euclidean, show the natural doubling of such angles.
6. But these polyhedral, polytopal, spheres as chords describe but one internal angle as the steradian angle- we are not necessarily subdividing the internal ball as if it were composed of many normally defined steradians cones. In a sense we may find the stuctures on such compression principles stable or compactified with a multiple density of such cones superimposed intrinsically.
7. In the original steradian star we imagine also the hexagon with its 15 connections of its points and this is a good representation of triangle or simplex groups of things where everything is connected to everything else. Thus this five dimensional representation may contain four dimensional simplexes. If we exclude one point and those to which it is connected (this may determine if we are dealing with a flat or round space but here internally everything is already round) that we have one such simplex of five points. But in this structuring we have the representation as if a rectangle of the square root of three and on that a quarter of it such that the hatted pentagon appears as if in three space a pyramid of five faces. This particular shape can lead to some interesting lattices in two space. But let us not forget as in the rhombidodecahedron (of two varieties and from and internal dissection of a ten sided polygon Coxeter found both the flat and three space quasi-tilings.) is also as a four space shadow and "chord" hiding one of its points ideally aligned in the center, thus the two of the count of things in odd dimensions, and the 14 abstractly on the invariant chordal ground symmetrical to the outside of this rim or flanged compressed sphere.
8. With such considerations we clearly see why in the imagining of multi-dimensions we intuit what is mystery or what is trivially intelligible structures. So when we apply such intelligible notions to ideas of general physics and space we can have a wide variety of possibilities in determine what is the concrete and what the abstract.
I find he so called near miss polyhedron truncations most interesting in that in a sense these can be impossible structures- in particular the internal volumes in dividing them up may in some way be abstractly but not concretely regular from some view- and that where it is obvious one cannot say have a square, a hexagon, and two triangles meet at a point and the combination not be flat- but perhaps in a sense from some view this is abstractly intrinsically curved. These also come close in actual construction say of paper models where certain ones almost close like those concerning 11 points for an impossible 18 faced deltahedron. In building on such polyhedra as in my Triaconway game, it is possible to arrange holes down from some of the faces in which to embed the thirty cube puzzle.
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I think it would be worthwhile to look at particle decay from the old picture of they being composed of so many levels of bricks with small variations between them. I keep coming close over the years for haunting integer numbers much too soon to say they actually apply or how. Clearly the Pion can be counted a few more electron masses than the muons and so on. In a lose sense some more fundamental idea of a particle as integral yet for general reasons intelligibly diffuse could be what determines some of this discreteness. A lot of interesting things have come from experiments, angles of incidence of refraction for example, and these in a sense with a negative value.
I do not think a general concrete sounding description of say some change in our normal group invariants without a more general theory can prove anything. From the very start, way back to Descartes, we know a hyperbolic lens (say around the z axis) can focus light just as well as the more elliptical varieties. We know also that flat mirrors can concentrate the heat if slabs of them are arranged in a parabola that focuses on a smaller set of them also made of flat slabs mirrors in the opposite x y direction. Beyond that a mirror as the inside of a cone can focus light diffusely so as to in theory concentrate the light hotter than the surface of the sunlight going through it.
Part of the general geometry models then concerns this depth and span of distinguishable ideas of contraction and expansion of what is within or without some general idea of a membrane, be it a plane or a sphere or some difference such that we can take some models of geometry as concrete when we consider if the primary ground of a geometry is not a general mixing of what is hyperbolic verses elliptical. In general the multi-parallel but no-perpendicular geometry is that which is in the inside of such a boundary and although but a sketch to which some say we cannot have the total vision, from some perspective we can take these as a concrete representation of the space as if we view it outside some mirror. All the geometries like this stand and fall (logically) together and so apply. In a sense that we are on the outside of such objects guarantees that properties we observe of them seem to reduce to concentric spheres- so to the description of how nature realizes the phenomena of action in regards to real and abstract symmetries.
An iota particle may just do the accounting abstractly as things tend to fall to some center, or such may be as real as anything. But what are such abstract structures really save a vibration or duality between or analogous to such in say the polyhedra and their duals? This same general idea on some level of the steep climb to a more general physics may be said of any levels of particles- and some of the resonances or even jumps to higher shells as abstract become concrete or physical. But from a more limited view it is not clear that we understand how light can determine the space and space the path of light. I suppose this means that our logic of CPT, even if these are conserved only as the three-ness together, is not as fundamental as we hoped. But hen how many logical systems that treat notions as if in some sort of polygonal relation, if the assignment of the notions to the structures are exact enough, are logically possible even with all this abstract geometrical reduction. Surely, with the polyhedra as the ancient model there are whole new area of apply our (partial) differential equations in an intelligible manner in a more general and fundamental theory. Like many other such theories part of the problems are filling the steps on steppingstones to reach a clearer path to the more intuitive and general picture. Nothing actually forbids that we cannot have interconnections between the successive shell structures of objects- but nothing assures us that in the enumeration such it can give us a total picture of the topological terrain.
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Coffee shop closed tomorrow and day after for maintenance, a rare event like for some holidays. It is possible I hang out in the library- but for now, I again think I have nothing more to say. Still, the shut down for some train of thought is not always a radical break as the ghosts of ideas like old lovers linger in the streets of cobblestones and your dreams. Happy Spring!
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