From: kirby urner <kirby.urner@...>
Date: Thu, Sep 30, 2010 at 5:13 PM
Subject: Re: unearthed an old letter
To: David Koski <dbkoski@...>
Cc: synergeo@..., CJ Fearnley <cjf@...>


Apropos of your FAQ Chris, I'll give a little of my perspective.  I'm
sitting in the Quaker meetinghouse office waiting for Sara.  Per
today's journal entry, I'm somewhat in a vortex today.

http://worldgame.blogspot.com/2010/09/switchboard-activity.html

What's clearly an addition to the canonical concentric hierarchy is
this 7.5 volumed rhombic triacontahedron David talks about, has been
talking about for over 20 years by the looks of it.  Fuller doesn't
mention it.  Why it turned out to seem so relevant has to do with a
closing theme in Synergetics 2, which was the end of his search for a
volume 5 member for his hierarchy.  He puts a lot of eggs in the
T-module basket, even though the radius is 0.9994 and change, which
might seem capricious in an already mecurial work.  The 120 T-modules
make the RT of volume 5, as you know.

The volume 7.5 RT interlaces with the rhombic dodecahedron of volume
6.  It's not a dual obviously, is from a different symmetry family,
but they're clearly related.  Each "T-module" now has a volume of 1.5
that of the original T, just as overall volume is greater by a factor
of 1.5.

1.5 x the T module is the volume of half a Mite, or a "smite" a Koski
dubbed it.  In terms of shape, this would be a "double corner"
(Zubek's term for Schlafli's orthoscheme of the cube -- comes in left
and right handed).  However, the same 1.5 volume is characteristic of
the 1/4 Rite as well, i.e. two-Mites / 4, for four half-Mite volumes
(Rites and Bites are both Sytes, of two Mites).

The Rite is a space-filling isosceles tetrahedron (named by Fuller but
known to Sommerville and others who'd explored this same territory)
and its 1.5 volume "quadrants" are same volume the K-modules, i.e. the
analog of the T-mods in a 7.5-volume RT),  Renenber E-mods have a
radius of exactly 1, versus the T-mod RT of radius 0.9995.

What excites David and I about the quarter Rite (our "fourth-Rite") is
it gives us a space-filler that doesn't need handedness, *and* has the
half-Mite volume.  Why is that important?  It just feels that way I
guess.

The other revelation stemming from the 7.5 RT, which seems just
beneath the surface in the 1991 post below, is that the exact
algebraic expression for the T-module altitude (RT radius) is 3rd root
of 2/3 times phi/sqrt(2).  You may have read Kappraff's 'Connections'
in which he makes phi and sqrt(2) sort of these "pillar ratios" for so
much of art and geometry.  Having both in the same ratio...

OK, gotta go, Food Not Bombs is happening at the meetinghouse...

Kirby


On Thu, Sep 30, 2010 at 1:26 PM, David Koski <dbkoski@...> wrote:
> I could scan this at another time.  This is from a letter I sent to Steve Baer
on Sept 2, 1991.  Pre-internet days.
>
> "If the rhombic triacontahedron = 5 PVR tet units, the long diagonal is
equivalent to third root of 2/3 * second root of 2 = 1.235429341.
> The shorter diagonal, which corresponds to the enneacontahedron (long diagonal
of thin rhombus), that can inscribe within the triacontahedron, is equal to
third root of 2/3 * second root2 * phi to the negative first power = .763537323.
 The volume of the enneacontahedron will have a volume of 4.4427191.  This
volume is enticingly close to  pi * second root of 2, the volume of a PVR
sphere(diameter 2).  The difference being .000163838."
>
> I did not include the volume of the sphere at 4.442882938, so the
enneaconatahedron is smaller in volume.  This was over 19 years ago.  Twas a wee
bit ahead of myself methinks.
>
> On the same page I have the rhombic triacontahedron as having a volume of 7.5
PVR - long diagonal is second root of 2.  All of these numbers were worked to
retain the volume 6 rhombic dodecahedron which resides within the
enneacontahedron and is composed of 4 of Baer's B cells.
>
> Also, on the page I have the volume of the enneacontahedron at "119.5820393
Prime vector radius tetrahedral units (tet edge = 2)"
>
> And more:  "The volume of the rhombic triacontahdron, long diagonal = second
root 2 * phi to the second power, is 134.5820393 or 15 PVR tet units greater
than the inscribable enneacontahedron.  15 PVR tet = 2 rhombic triacontahedron -
 long diagonal second root of 2."
>
> 15 = 2 * 7.5 was the point on the last part.
>
> Dave Koski