Re: Basic Pieces to MathWorld
- --- In email@example.com, "Alan M" <a.michelson@...> wrote:
>Orthoschemes are quad-rectangular. I was sloppy, thinking how
> --- In firstname.lastname@example.org, "coyote_starship" <Kirby.urner@>
> > This bit about "orthoscheme
> <http://www.clowder.net/zubek/dblcrnr.html> of the rhombic
> dodecahedron" is not
> > correct. I was using the term too loosely. Orthoschemes
> <http://www.clowder.net/zubek/dblcrnr.html> are
> > quad-rectangular, not tri-rectangular
> <http://www.clowder.net/zubek/octane.html> . I need to go back and
> > refile I guess. Damn.
> > Kirby
> Maybe, Frank Zubek <http://www.clowder.net/zubek/zubek.html> can help
a rhombic dodecahedron carves into Mites. I should not have used
that term. I went to the Wikipedia page on orthoschemes recently
(thanks to you I think it was) but didn't immediately catch the
error of my ways.
In any case, that throw-away clause in my sentence was not some
kind of deal breaker or show stopper. It was like a 3rd arm, not
important to the argument (is there an argument?), where if you
cut it off, I still have two arms (Koski's analogy).
I was trying to characterize the Mite shape.
When you criss-cross the face of a rhombic dodecahedron, the
resulting 4 tetrahedra to the center are Mites (not orthoschemes).
When you criss-cross the faces of a cube, the resulting 4 tetrahedra
to the center are also Mites (not orthoschemes), pretty cool.
The same process applied to a rhombic triacontahedron gets you
the T module (or T-shape, depending on relative size within
the concentric hierarchy).
I see basically two issues being addressed by my contribution, aside
from that "a a" typo, which I notice I didn't mention in submission
two (I bet they fix that at least):
(1) the MathWorld page on Space-filling Polyhedra attempts to be
somewhat complete, yet avoids the foundational "bottom layer" of
simplexes. It currently only mentions tetrahedra to say:
(a) Aristotle was wrong about regular tetrahedra filling space and
(b) regular tetrahedra do fill space in complement with octahedra.
But then when it comes to hexahedra etc., there's no requirement
for "regularity" -- that's not a criterion space-fillers need to
obey -- at the tetrahedral level either.
(2) the MathWorld page, were it to mention irregular tetrahedra in
any way should give special attention to this "Mite" and even
mention Fuller by name (a truly radical suggestion).
Does Fuller deserve mention at all?
I think so because:
(a) he has this original dissection in terms of A and B modules **
(b) he's very clear this tetrahedron is a first space-filler without
handedness and so is a limit case in that sense. By "first" I mean
"bottom layer" or tetrahedral (simplest polyhedron).
(c) he cites 'Regular Polytopes' and is aware of the surrounding
literature. He knows he's making a real contribution but is not
wanting to hog credit. He's playing the game by the rules while
putting some high cards on the table.
(d) the Mite, Syte, Kite progression is well designed nomenclature,
good mnemonics. He didn't just come up with the rhyming names, but
developed all the plane-nets, commissioned the graphics (including
as color plates), got everything published (with a lot of help from
his friends, Ed in particular when it comes to shepherding through
If we wanted to get serious about including more spatial geometry
in K-12, I don't see why we'd want to completely avoid this "lookup
table" especially as it relates to tetravolumes, which is one of
Fuller's chief specialties.
Do we want to do anything with tetravolumes though? So far, the
implicit answer seems to be "no", as no textbooks have picked up that
ball and authoritative sources on mathematics such as MathWorld
have nothing much about it.
Published histories and recaps since Fuller have tended to avoid
all mention of the 1:3:4:6:20 volume progression, with the
phi-related interspersed five-fold symmetric shapes. We don't
see much about 2 * P * f * f + 2 either, even though Coxeter
thought that was a legitimate and accessible discovery. He wrote
a paper about it, generous to Fuller.
What did he say about the A & B modules though, if anything?
I still don't really know.
Tetrahedral accounting is an issue I'd like to see raised and
discussed, which is why I gave it such front and center treatment
in the Wikipedia article.
If it's just me raising it though, in these public archives, fora
etc., then it's pretty easy to ignore. Some single dad in Portland,
a former high school math teacher, your average joe, thinks
so-and-so is important. Big deal and so what right?
Of course I'm not the only one bringing it up, it's just that I'm
not seeing that much activity. It's discomforting to just have a
few threads on Synergeo going, a web site here and there, some older
titles. This has nothing to do with the mathematics itself being
wrong or incorrect. So why throw it all away, after such little
Where are the secretaries of education, the deans, the other
curriculum writers? Who wants to risk putting a professional voice
behind this question? Isn't the concentric hierarchy, relating
sphere packing to basic polyhedra, worth at least a few hours in
the first eighteen years of life? What's the source of the big
delay in phasing it in? Why are we this retarded when it comes to
passing on our heritage? If one *is* a professional in this area
and has nothing to say, is that helping one's career? Silence is
not always the safest course or path of least resistance. From
my angle, the deafening silence seems damaging, given what appear
to be the extraordinarily high stakes.
NCTM (National Council of Teachers of Mathematics) has that buried
lesson plan on non-traditional volumes, entitled Tetrahedral Kites,
but that's just some needle in a haystack. It's enough to show
acceptance of the basic concepts (which is why I hyped it in the
BFI Forum), but how does that translate into what goes on in the
Dr. Arthur Loeb signed on, after expressing all that skepticism
about Fuller per 'The King of Infinite Space'. That should count for
something. The reaction to Amy's book was quite hostile from several
corners though (she was his protege in a lot of ways). Clearly
there's sincere resentment against any perceived "partisans" aka
"disciples" ("buckaneers"). We're all aware of this situation.
But what's it about? What's the operative psychology? Inquiring
minds want to know.
Because the cost of *not* teaching more about Fuller's contribution
is really quite high -- so at least we should have some more open
discussion (coming up on 30 years after his passing).
Here's me on the Math Forum arguing (implicitly) that all this
stuff the Obama administration wants to accomplish regarding
nuclear disarmament would be so much more doable if people started
seeing more Fuller on the K-12 syllabus. Nuclear disarmament
requires mutual trust, as well as a verification regimen. How does
one trust a curriculum that actively suppresses vital and relevant
Imagine: a subversive writer, decorated and embraced by world
leaders, one who makes basic contributions to math, philosophy
and architecture, actually gets studied in schools, and not just
in terms of being a "genius kook" or some other easy dismissal.
That'd be a different world in a lot of ways. But are we so tied
to this one, infested with killingry, dying on the vine, that we
don't dare make the leap? I'm not suggesting over-hyping the guy,
but at this point I'd say Fuller's stock is still way under-valued.
It's a matter of connecting the dots (dots of light?).
** Robert Williams cites Synergetics, still unpublished and in
manuscript form, in the context of developing his own meaning for
A and B modules. Actually, the A module is the same (the
orthoscheme of the regular tetrahedron), but then he develops
the B as the orthoscheme of the regular octahedron without
subtracting the A already inside it. So the Williams B (Williams.B)
has twice the volume of the Fuller B (Fuller.B). In the concentric
hierarchy per Fuller, A, B, T = 1/24 vis-a-vis unit-volume unity-2
- << snip >>
> print decimal.getcontext()This 2nd line was removed.
> print decimal.getcontext().prec
Here's the source, better indented:
> # long limo vip numbers<<snip>>