URL: http://www.inetarena.com/~pdx4d/ocn/chaos0.html
For those of you not familiar with our approach to geometry, this
essay will appear dense, owing to some unfamiliar terminology. Thanks
to the web, we facilitate the student's deciphering process with
plenty of explicatory links, so those of you still stopped by
"IVM" or "concentric hierarchy" will have a quick on-ramp for
getting up to speed.
I encourage teachers to cannibalize my curriculum for ideas relevant
to their own. You don't have to buy this stuff "whole hog" to get
a lot of useful material. At the very least, feel free to point
some of your kids to the above URL for an independent research
project for some extra credit.
Kirby
Curriculum writer
Oregon Curriculum Network
http://www.inetarena.com/~pdx4d/ocn/
===============
OF CHAOS AND INCOMMENSURABILITY
IN SYNERGETIC GEOMETRY
by Kirby Urner
August 7, 1998
In synergetic geometry, the nested systems of the concentric
hierarchy (tetrahedron, duo-tet cube, octahedron...) divide
into two regimes: the 4-fold symmetric and 5-fold (noting that
the tetrahedron is also 3-fold symmetric).
The 4-fold family is coordinated by the A and B modules,
equivolumed irregular tetrahedra of volume 1/24. The 4-fold
systems all have simple, whole-numbered volumes relative to the
nuclear tet of volume 1.
The 5-fold family sets up chaotic tension within the hierarchy,
introducing dynamism and incommensurability. The Pythagorean
demonstration of root(2) as chaotic is transferred to the 4D
jitterbug transformation: a root(2) diagonal of the
cuboctahedron's square face shortens to 1 as the cuboctahedron
jitterbugs to icosaphase: the jitterbug is a smoothly
transformational bridge over incommensurability (chaotic
waters).
The 5-fold family has incommensurable volumes relative to the
4-fold family of A/B modular assemblies. Koski's recursive,
phi-scaled modular set, slicing irregular tets from the golden
cuboid (phi x 1 x 1/phi) is what works well here. But where
do 4 and 5 fold families come into closest proximity, how
shall we "tune chaos" within our concentric hierarchy model?
Fuller, like Koski, is using the so-called T-module cut from
the rhombic triacontahedron (RT, 5-fold) and finds an
interesting feature. If the RT's diameter from face-center to
opposite face-center is set to precisely 1 (circumscribes an
IVM sphere), then the 120 E-mods have an incommensurable volume
relative to the A & B mods. But if you shave the tiniest hair
off the RT's diameter, letting it shrink (relax) ever so slightly,
then you've got an RT with a whole numbered volume of precisely 5,
and T-mods of volume 1/24 (same as A and B modules).
Incommensurability has not been removed from this picture, merely
focussed to the relationship between the PVR RT and the PVR- RT
(PVR = prime vector radius, the radius of an IVM sphere; PVR-
means PVR with a bit shaved off; RT = rhombic triacontahedron).
Chaos is attuned in the relationship between the T-mod (PVR-)
and the E-mod (PVR). The difference is tiny (PVR-/PVR is small),
but from an energy containment viewpoint, critical. T-mods
conserve energy while E-mods deliver it back to Universe.
The chaotic tension between 4- and 5-fold symmetry permeates
Universe and "making sense" has to do with knowing how to be
energy-conservative in a chaotic regime. Synergetics itself
makes sense from an E<->T point of view (ET) -- which doesn't
keep it from being chaotic (a feature, not a bug).
References:
Synergetics Modules (A,B,T)
http://www.teleport.com/~pdx4d/modules.html
The T Module in Synergetics
http://www.teleport.com/~pdx4d/tmod.html
Synergetics 987.413 (incommensurability & jitterbug)
http://www.servtech.com/public/rwgray/synergetics/s09/p87400.html#987.413
[note: use sqrt() for 2nd root is a departure from the
original text and is somewhat inconsistent with the synergetics
philosophy -- should perhaps be updated to a radical sign in a
future upgrade. Note reference to S module as well.]
Synergetics 986.541 (volume ratios of A,B,T,E modules)
http://www.servtech.com/public/rwgray/synergetics/s09/p86520.html#986.540
Synergetics 986.548 (picture of PVR vs PVR-)
http://www.servtech.com/public/rwgray/synergetics/s09/p86520.html#986.548
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