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Do Points
Have Area? Atomic Points Subject: Re: Reply to "Do Points Have Area?" Author: Kirby Urner <pdx4d@teleport.com> Date: Thu, 08 Jan 1998 10:56:15 -0800 > = Jesse Yoder >> = John Conway = Kirby Urner >Thinking along these lines, do the points on a Euclidean number line >ever touch? I assume the answer is "No", since you can always put >another point between two points on a Euclidean number line. Could >they ever touch, and how much space is between them? > Again, I'm thinking it possibly a misattribution to take relatively late-in-the-game real number lines and the associated Cartesian-style R^n n-tuple games and backloading these onto Euclid, in effect piggy- backing them on his good name. Did anyone get his permission for this? I find it highly relevant to remember another Greek thinker at this juncture: Democritus. His atomic picture of reality is a source of countervailing imagery vis-a-vis all these 1/infinity "no gaps continuity" dogmas. If updating our picture of Greek thought as a whole, perhaps we should recast it with a Euclidean "wrapper" encapsulating Democritus-style atomic "insides". We could still do stick- tracings in the sandy beach, ala Euclid's Elements, but the sand grains would continually remind us why we never bought into "real number lines" as later concocted. We have (and might as well again) mention Zeno in this connection -- his explorations of our concepts around motion presaged many later debates about the ultimate "smoothness" (not!) of all phenomena. Bishop Berkeley is another name to remember in this connection, as his brief was we could just as well do the calculus under the heading of discrete mathematics and not hinge anything central on any anti-Democritus teachings. I find many in computer science agreeing with this view, as their job has been to implement nuts and bolts calculus in an engineering sense, and it always turns out that dx is a definite increment, relatively miniscule vis-a-vis a larger scale phenomenon, but never "infinitely small" -- a concept with no operational definition on a computer (computer = "metaphor for energetic reality" as well). Lots of schools of thought have fought "continuity" (lets call them the "pro Democritus" schools). What I'm arguing is that it is intellectually dishonest to simply presume that Euclid would have been in the "anti Democritus" camp for all eternity, given that an atomic or discontinuous model *can* be reconciled with his original explorations, as operationally practiced (e.g. on sandy beaches). >A better way to view this is as 3-dimensional, in which case the >points or "spots" as you prefer to call them become 3-dimenensional >balls with physical space between them. > >You then continue: >>"Aha! So your geometry fails to be isotropic, and has a preferred >>system of coordinate-axes! Jesse, I don't much like this square-packed circles model as a home base "peg board" for doing geometry, and moving to spheres -- I'd encourage you to think about laminating pool balls, stacking layers upon layers in hcp or, more isotropically, in fcc format. You've likely already done this I realize -- so far I like your game BTW, though I don't claim to understand it completely. In synergetics, the fcc packing defines the vertices for a skelatal lattice of edges dubbed the "isotropic vector matrix", with tetrahedral and octahedral voids (or cells) in a population ratio of 2:1, with octahedra having volume 4 vis-a-vis the unit volume tets (so far the only aspect original with synergetics here is the assignment of unity to the tet's volume). Synergetics also investigates the shapes of the voids when we use baseballs (vs. just their centers), mapping the interspheric voids to the vertices of the space-filling rhombic dodecahedron of volume 6. You have more deactivated sphere packings latent in these voids: the long diagonals of the rh dodeca define one, and the short diagonals define two alternates, for a total of four (including the currently active one). The term "isotropic" in relation to fcc sphere packing can be found elsewhere in the literature than in Synergetics certainly e.g. Bonnie DeVarco copied the following to Syn-L last August: "...We can obtain an isotropic point-lattice in space in starting from four series of equidistant planes cutting each other under the same angle -- that is, parallel to the four faces of a tetrahedron. But oddly enough the partitions thus obtained (Plate 28) do not correspond to a division of space into close-packed tetrahedra (this in euclidian space cannot be realized) but to a division into tetrahedra and octahedra (twice as many tetrahedra as octahedra), or in cuboctahedra and octahedra in equal numbers. The point-lattice is identical to the one obtained in filling space by the most dense possible system of equal tangent spheres, and in taking either all their centers, or their centers and points of contact. "As in the plane we can place outside a circle six tangent circles identical to the first (figure 58) and repeat the process indefinitely, (the centers of the circles are then part of a triangular or hexagonal isotropic point-lattice), so in space we can ahve twelve spheres tangent to an identical inner sphere (figure 59). It is in this perfectly isotropic close- packing of thirteen spheres indefinitely repeated, that the centers (or else the centers and points of contact) produce the cuboctahedral point-lattice, because in relation to the center of each sphere the centers of the twelve surrounding tangent spheres (also the twelve points of contact) coincide with the vertices of a cuboctahedron (figure 59). [from Matila Ghyka's The Geometry of Art and Life; first published by Sheed and Ward, New York, in 1946 -- she was quizzing us as to who might have written this (clearly not Fuller), and Richard Hawkins took the prize]. Note that Cartesian axes are also "handed" or "chiral" in that you have to pick which of the 8 sectors you plan to make your (+,+,+) region. Usually people go for what's called "left handed" Cartesian coordinates, but this is of course an arbitrary convention. Four- dimensional quadrays are more balanced in this respect, as all four quadrants are expressed by {+,+,+,0} where { } means "all permutations of" -- none of the quadrants are "more positive" or "more negative" than the others. In fact, one of Coxeter's criticisms of synergetics is it wasn't chiral enough (right Bonnie?) -- but we should member the inside-outing transformation through the origin, into the space of the dual tetrahedron, has the effect of making a right handed glove into a left handed one, and vice versa. Kirby Curriculum writer 4D Solutions
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