Rhombic Triacontahedra etc.

This is a more recent version of a recruiting drive focused on "GNU
math teachers" (a pun on New Math), meaning we would be using a lot of
free and open source tools in the mix with our math teaching
(example: Python intermixed below).

Note that RadMath includes questioning the Cube and Square as the only
sensible models for 3rd and 2nd powering respectively. In order to
somewhat disrupt the reflex-conditioning that would have us always
think in terms of squares and cubes, the square root function is
rewritten as root2, while the 3rd root function is defined as root3
(and so on, should other roots be necessary).

Also by convention, I'm using phi for the golden mean 1.618... and tau
for its reciprocal. Some texts do precisely the opposite.

Here's some initial "stage setting" in Python 2.6 to get all this
working:

>>> from math import sqrt as root2
>>> def root3(x): return pow(x, 1./3.)

>>> phi = (1 + root2(5))/2.0
>>> tau = 1/phi

One approach (a basis for lesson plans). Start with the Platonic Five
and form combinations from pairs of duals:

dual(Tetrahedron) = Tetrahedron
dual(Cube) = Octahedron
dual(Icosahedron) = Pentagonal Dodecahedron

Tetrahedron + Tetrahedron = Cube
Cube + Octahedron = Rhombic Dodecahedron
Icosahedron + Pentagonal Dodecahedron = Rhombic Triacontahedron

<optional>

[ This information might be saved for a 2nd or 3rd spiral through this
same zone of interest. ]

Then use the MITE (minimum space-filler) to dissect the Cube and
Rhombic Dodecahedron.

Perform A&B module analysis of the Tetrahedron, Cube, Octahedron,
Cuboctahedron.

Perform T-shape analysis of five-fold symmetrics, per Koski and this
figure:

http://www.rwgrayprojects.com/synergetics/s09/figs/f86411a.html

T-shape analysis may involve phi-scaling said T-shape, perhaps
starting with the T-shape in the 7.5 volumed RT. 7.5/120 = 1/16 i.e.
half the volume of a MITE. Sometimes I refer to this T-shape as the K-
mod (a mnemonic to remind us of Koski's dissections, vZomes etc.).

</optional>

That brings students up to speed pretty quickly, plus starting with
Platonic Five is ultra-conservative, for those ticked off by any talk
of Radical (which means Root in math, as symbolized by the Surd
symbol).

Logo: backwards R under a Surd for RadMath.

The "starter kit" rhombic triacontahedron has long diagonals = those
of the icosahedron in the Platonic Five. We equate those edges to the
tetrahedron's, in order to get our standard "concentric hierarchy"
volumes of:

A,B,T: 1/24
MITE: 1/8
Tetrahedron: 1
Cube: 3
Octahedron: 4
Rhombic Dodecahedron: 6
Cuboctahedron: 20

The icosahedron may be developed from said cuboctahedron by means of
the "Jitterbug Transformation"

Youtube: http://www.youtube.com/watch?v=FfViCWntbDQ .

Its volume will be: 5 * phi**2 * root2 (2)

icosahedron + dual(icosahedron) = rhombic triacontahedron

Tetravolumes:

Icosahedron: 5 * phi**2 * root2 (2)
Pentagonal Dodecahedron: (phi**2 + 1) * 3 * root2 (2)
Rhombic Triacontahedron: 15 * root2 (2)

Using Python as a calculator:

>>> 5 * phi**2 * root2 (2)
18.512295868219162
>>> (phi**2 + 1) * 3 * root2 (2)
15.350018208050782
>>> 15 * root2 (2)
21.213203435596427

If we account the tetrahedron's edge as 2, then the icosahedron's is
likewise 2. The uni-radius IVM ball has a diameter of 2.

This could be done as a spreadsheet:

Entry level RT:
Radius: phi
Long diagonal: 2
Short diagonal: 2/phi
Tetravolumes: 15 * root2 (2)

BCC RT:
Radius: phi / root2 (2)
Long diagonal: root2 (2)
Short diagonal: tau * root2 (2)
Tetravolumes: 7.5

Using Python as a calculator:
>>> phi / root2 (2)
1.1441228056353685
>>> root2 (2)
1.4142135623730951
>>> tau * root2 (2)
0.87403204889764208

IVM RT:
Radius: root3(2/3) * phi / root2 (2)
Long diagonal: root3(2/3) * root2(2)
Short diagonal: root3(2/3) * tau * root2(2)
Tetravolumes: 5

Using Python as a calculator:
>>> root3(2./3) * phi / root2 (2)
0.99948333226234343
>>> root3(2./3) * root2(2)
1.2354293410542654
>>> root3(2./3) * tau * root2(2)
0.76353732347042169

The reason I'm calling the 7.5 volumed RT a BCC RT is because of
recent studies by Dave Koski which puts these shapes in a body
centered cubic packing with vertex connections (points of tangency).

Rhombic dodecahedra (RD), on the other hand, fill space in the CCP
arrangement.

http://www.4dsolutions.net/ocn/xtals101.html

The IVM RT is the one that shrink-wraps a uni-radius IVM sphere,
except the radius falls a tad short when the volume is set to
precisely 5.

Synergetics makes a big deal out of this "tad" suggesting it's a
bubble-thin difference between coherence and decoherence in some
primitive sense.

Kirby

<LORE>

GNU = GNU is not Unix and was the original inspiration for the
free software movement, centered on what become known as
the GNU Public License (GPL).

The GPL was a way for copyright holders to insist that their hard
work not be used only selfishly by others, that the fruits of their
labors achieve maximum positive benefit for omni-humanity, to
put a Global U spin on it.

Once the GPL was out there getting the work done (e.g. Linux),
a safe-enough ecosystem was developed for a bevy of alternative
licensing schemes, some of which allow shops to take something
inhouse, transform it, and release it under a more restrictive
license agreement, if they release it at all.

[ Python could be an example of this, in that developers of the
IronPython codebase have been enabled to start with CPython,
the original version, with the resulting codebase being free and
open source under a different set of rules from the GPL or even
from whatever the original Python was using. Actually, the story
is more complicated than that, in that Jim H. first worked on the
Java implementation before moving to C# in an attempt to prove
the CLR (common language runtime) engine was unsuitable for
dynamic languages. He came to the conclusion that the CLR
engine could work, hence the IronPython project. ]

</LORE>

Photostream:
http://www.flickr.com/photos/17157315@N00/4585728237/in/photostream/

Note that around the time I was recruiting for GNU math teachers,
I was suggesting we cast the rhombic triacontahedron as the
NCLB Polyhedron. This was to work in tandem with the NCLB
Polynomial, that which solves to the number Phi (1.618...) as
one of two roots. A:B :: A+B:A is the relationship to start with.

http://www.mathforum.org/kb/message.jspa?messageID=7080697&tstart=0

Some readers may have interpreted this inventiveness as mocking
the NCLB program, which focuses on "standards" without giving the
time of day to so many of our focal points, e.g. the A&B modules, the
Mites, Sytes and Kites, nor even ball-packing in general. &&

Like the NCTM's, the NCLB's administration is mired in some
distant past and self-marginalizes accordingly.

http://controlroom.blogspot.com/2010/05/more-on-public-education.html
http://worldgame.blogspot.com/2010/04/radical-math-and-python.html

One has to look to behind-the-scenes counselors and advisors to
figure out why such stuck-in-the-mud schoolishness was allowed
to prevail for so long, to the detriment of the student body. This
investigation is ongoing and I won't digress into a lengthy analysis.

My strategy, in disruptively butting in with new NCLB content,
contemporary and well-designed, was to provide contrast. Such
curriculum writing around the NCLB initiative seemed oxymoronic
to the point of ridiculous, like some kind of twisted joke. Yet why
should we take the standards bodies seriously if they can't stomach
any talk of a "minimum tetrahedron" (Mite), let alone a volume 1/8
one?

http://controlroom.blogspot.com/2010/04/suggesting-change.html

In other words, what's "no joke" is that we *do* take our polyhedra
seriously around here -- which doesn't mean always knowing the
Schlaffli symbol right off the bat. **

After the GNU Math chapter, I developed Radical Math and
continued with recruiting. I had better luck within my own zip
code area at least. Lindsey Walker was one of the first
RadMath teachers to come out of the woodwork, Glenn Stockton
another. Both are seriously talented individuals:

http://www.flickr.com/photos/17157315@N00/3725109167/ (LinZ)
http://www.flickr.com/photos/17157315@N00/4430540111/ (Glenn S)

The GNU Math campaign is still out there as well, working in
tandem. I think that FOSS culture is spearheading what Fuller
called a "design science revolution", and the ethics of "open source"
are permeating high technology culture.

If you're going to mess with the environment, you need to put your
cards on the table. This idea of externalizing consequences
while hiding the mechanisms is contrary to engineering
ethics in so many ways.

Dave Koski has been a consistent supplier of RadMath curriculum
materials, especially where the NCLB Polyhedron is concerned.

Kirby

&& More about tetravolumes:
http://controlroom.blogspot.com/2010/05/another-chart.html

** see Section 3-6 of The Geometrical Foundation of Natural
Structure: A Source Book of Design by Robert Williams
(Dover, 1976) pg. 61 ("Descriptive Properties of Polyhedra").