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Topic: Conway on Trilinear vs Barycentric coordinates
Replies: 13   Last Post: Feb 16, 1999 8:05 AM

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Kirby Urner

Posts: 803
Registered: 12/4/04
Re: Article on Quadrays in March 1999
Posted: Feb 8, 1999 8:40 PM
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>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 "duo-tet"
cube, with vertices in the tetrahedral holes of the fcc (at the
termini of the short diagonals of the space-filling, 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 "observer-observed" 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 system-divider 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 so-called hyperdimensional polytopes. But we might also choose
new metaphors just as well e.g. look at n-tuple encodements of
visualizables (the 3D shapes extractable-by-algorithm from
n-tuple 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 n-tuple mathematics is actually used to support cell
phone channel multi-plexing 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, well-schooled
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 over-specialization, which almost succeeded
in preventing future generations from appreciating the many life-
supportive advantages contributed by this important century
thinker and engineer-philosopher.

> 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





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