
Re: Article on Quadrays in March 1999
Posted:
Feb 8, 1999 8:40 PM


>The permutations of quadray (2,1,1,0) are similar to >my system. Subtract one from each coordinate and you >get (1,0,0,1) and the twelve unique permutations >are the vertices of a cuboctahedron in my system.
Yes, same number of permutations, given one of the terms appears twice, other 2 uniquely.
>But the permutations of (1,0,0,0) in quadrays are >not the centers of closest packed spheres.
Actually, in quadrays {1,0,0,0} (permutations) *are* at the centers of closest packed spheres. But (0,0,0,0), the origin, is not  it's in the hole between the 4 intertangent spheres, at the center of the regular tetrahedron.
Do you include volume calculations in your system? Makes sense that if your cubocta has {1,0,0,1} for coordinates, that your related tet would have fractional terms, as their volume ratio is 20:1 (or is it, in your gizmo?).
>They are something that requires fractions in my system. >The idea is to use integers mod a prime), and get only >the centers of closest packed spheres.
If you move my (0,0,0,0) to the center of a closest packed sphere, then you're correct, {1,0,0,0} are in the holes around that sphere.
{1,0,0,0} + {1,1,1,0} = the vertices of the volume 3 "duotet" cube, with vertices in the tetrahedral holes of the fcc (at the termini of the short diagonals of the spacefilling, sphere containing, rhombic dodecahedron of volume 6).
So if (0,0,0,0) is aligned with an IVM (or fcc) sphere center, then all the other sphere centers will be linear combinations of {2,1,1,0} i.e. add from those 12 at will for a "random turtle walk" that "connects dots" in the fcc (cuboctahedral) sphere packing arrangement.
> By the way, is arithmetic remainder (as well as modulo) > a prime a field?
Not sure I understand the question. Remainders needn't be primes, merely less than the divisor.
> Bucky wrote that the tetrahedron's four dimensions refer to the > distances from the midpoint of the tetrahedron to the centers > of the four faces. The advantage of that is to represent any > size (positive or negative) tetrahedron anywhere with four > coordinates.
I essentially agree with you here. Note that if you depict 4 vectors to the vertices of a tetrahedron, you're likewise showing the same vectors as penetrating the _face centers_ of the dual tetrahedron (the invert). So quadrays _do_ go through tetra hedron face centers. Where they 'stop' doesn't really matter that much (you can always grow or shrink any tetrahedron).
But yes, Fuller's idea was to begin with volume as the starting point for conceptualization. Abbott's 'Flatland' neglects the observer, i.e. if you imagine a line, you also imagine the space in which that "observerobserved" relationship occurs. Synergetics brings the camera (which originally meant "room") or the "mind's eye" back into our thinking.
The separation of relationship, between a point and an observer of that point, creates volumetric twoness (the twonesses of poles i.e. axial rotation, and of concave/convex). Experientially, our awareness of geometric objects is in this context of volume. This is all philosophical content, at the definitional level. You have to start somewhere and synergetics starts here, with the concept of volume, of context as containment  minimally a container vs. contained relationship.
The next step is to give a minimal shape to that context, to signify containment. Just as 3 points define a plane, making the triangle the minimum signifier of a flat 'two dimensional' surface (which one observes in a spatial context), so four non coplanar points define space, which gives us the 6 lines of relationship, and the separation of volume into 'internal' and 'external' sets (plus the systemdivider itself). Of all polyhedra, the tetrahedron has the minimum inventory of edges, vertices and faces. And this becomes the new unit of volumetric mensuration.
In stressing that conceptual volume is '4D' in Synergetics, Fuller is deliberately going against the grain of standard academic usage, according to which we're all drilled. We all know to think of volume as comprised of three "linearly independent" dimensions: height, width and depth. In 'Synergetics' you find passages designed to counter this programming, suggesting that we have no conceptual experience of volume minus one of its "independent" component dimensions  any more than we have an experience of "four mutual perpend iculars" (or more), I would add. This is not an empirical limitation so much as a logical one, or what the later Wittgenstein would call a "grammatical" aspect of our experience (language and experience ultimately having no "internal boundary" to keep them apart: Physus = Logos).
Fuller thereby sets up a kind of "noise" in both directions: neither subtracting perpendiculars, nor adding them, gets us away from primitive, tetrahedrally defined volume  which he defines as 4D, just to help break the spell of the older paradigm. He does add further dimensions however. But these are in the direction of "more reality" i.e. dimensions having to do with time, energy (more of the "physics meaning" of that key term). The added dimensions, which differentiate an abstract, purely imaginary cube, from a physically realized one, come under the heading of "frequency" in Synergetics.
This attempt to embrace the philosophical distinction between the Platonic cube and the Empirical experience of special case cubes, within a consistent language, is partly what makes 'Synergetics' a philosophical work and not a mathematical one. Mathematicians invoke the "Platonic versus Real World" distinction in their writings, but don't consider such chatter properly a part of their formalisms, which are strictly on the "Platonic" side.
To talk about the "difference" between conceptuality in pure principle and energized spacetime, is generally not undertaken as an exercise in mathematics per se  this "difference" corresponding to an interdepartmental interface within the academic context, a boundary between discourse A and discourse B.
But Fuller's commitment was to subsume the multiple academic languages and anchor them within a consistent gridding system or conceptual framework  a kind of old school commitment to comprehensiveism which we rarely find on the university scene these days, in any department.
Which isn't to say we can't think along the old lines in whatever special case shop talks, e.g. we needn't deny ourselves the freedom to write computer programs wherein nodes and edges have as many tuples as we like. And we will continue to apply spatial metaphors to socalled hyperdimensional polytopes. But we might also choose new metaphors just as well e.g. look at ntuple encodements of visualizables (the 3D shapes extractablebyalgorithm from ntuple data storage formats) as a kind of "data compression" (hyperspace as a kind of encrypted "zip" format, wherein methods such as "rotation" are operationally defined (metaphors consistent with how ntuple mathematics is actually used to support cell phone channel multiplexing in the engineering department)).
To accept Fuller's language as internally consistent in its own way is not to break from usage patterns which characterize the bulk of contemporary mathematics  but it _does_ serve to help counter some of the dogmatic hardening of the mental arteries associated with the latter, symptomatic of the unquestioning acceptance of various authorities over the centuries i.e. it's not "God given", nor even "a priori", but "cultural" that we say "space is three dimensional". An intelligent, wellschooled layperson might reasonably say otherwise, and not be less sane therefore.
I am hopeful that a more enlightened, Renaissance era will ensue, as we learn from the case history surrounding 'Synergetics' and realize the price of overspecialization, which almost succeeded in preventing future generations from appreciating the many life supportive advantages contributed by this important century thinker and engineerphilosopher.
> There is a way to transform from my system to > quadrays just as there is a way to transform to XYZ. (or if the four > coordinates do not sum to zero, to XYZT).
I don't doubt it.
Kirby

