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Re: Basic Pieces to MathWorld

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  • coyote_starship
    ... Orthoschemes are quad-rectangular. I was sloppy, thinking how a rhombic dodecahedron carves into Mites. I should not have used that term. I went to the
    Message 1 of 33 , Apr 3, 2010
      --- In synergeo@yahoogroups.com, "Alan M" <a.michelson@...> wrote:
      >
      >
      > --- In synergeo@yahoogroups.com, "coyote_starship" <Kirby.urner@>
      > wrote:
      >
      > > 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
      > you?
      >

      Orthoschemes are quad-rectangular. I was sloppy, thinking how
      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).

      Why?

      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
      Macmillan).

      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
      discussion?

      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
      schools?

      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
      basic information?

      http://mathforum.org/kb/thread.jspa?threadID=2057893&tstart=0

      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?).

      Kirby


      ** 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
      edge tetrahedron.
    • coyote_starship
      ... This 2nd line was removed. Here s the source, better indented: http://mail.python.org/pipermail/edu-sig/2010-April/009927.html ...
      Message 33 of 33 , Apr 12, 2010
        << snip >>

        > print decimal.getcontext()
        > print decimal.getcontext().prec
        >

        This 2nd line was removed.

        Here's the source, better indented:

        http://mail.python.org/pipermail/edu-sig/2010-April/009927.html

        > # long limo vip numbers

        <<snip>>

        Kirby
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