[Edu-sig] An Urner update on doings/undoings
kirby urner
kirby.urner at gmail.com
Sun Aug 27 21:14:28 CEST 2006
I did a quick coin of HP4E but haven't used it for much since:
http://mathforum.org/kb/thread.jspa?threadID=1437142&tstart=0
Basically, I'm refining the "gnu math" concept, addressing concerns
about the New Math resonance, which one teacher worried had "Titanic"
connotations (as in "went down").[1] But I'm pointing to what we've
saved from that shipwreck: an emphasis on bases (as in subtraction in
base eight), and on Venn Diagrams (boolean algebra), and on Sets
(unions and intersections). All that stuff is still in the
mainstream, from Saxon to Singapore (those're textbook allusions, much
talked about on the math-teach list I frequent).
Gnu math uses a non-proprietary substrate on top of proprietary
hardware (e.g. Intel's, AMD's...) to keep computer programming a "for
everybody" sport, i.e. we won't wall them out on the basis of price
alone (never a good way to promote a meritocracy). Yes, gnu software
also runs on proprietary operating systems, and yes, we might someday
have an open source chip, i.e. the designs are free and clear to
anyone with a fab (still a huge barrier to entry at this point --
clean rooms ain't cheap). But until that blessed day, when it's all
open source as far as the eye can see (the Debian ideal), I'm not
averse to using whatever public venues they leave open to us laypeople
(even as we keep fighting to make inroads).
In the gnu math I envision, sets are just another data structure
(added somewhat late to Python, as dictionaries with null values, only
keys, did pretty much the same work). We're not as beholden to early
20th century philosophies of mathematics, which made the set front and
center in some theory of types, with Russell's paradox about what
might be "empty" or not. Per Wittgenstein, all that Cathedral talk
does *not* underwrite what we do in the Bazaar (set theoretic logic
provides a painted foundation under a painted castle, to paraphrase
the guy). The genesis of mathematics is ordinary ethno-arithematic,
and as such is "all in the open" i.e. doesn't depend on special Harry
Potter powers to see in "higher dimensions" (another position of mine:
contra the HyperCross Dogmatists (but I'm fine with Coxeter's
polytopes and recognize their utility in higher mathematics (but a
hypercube is *not* the same thing as a time machine remember, per
'Regular Polytopes' on page 119))).[2]
So yeah, we've got sets (lists, dictionaries...), and math.pi, and pi
generators, and let's not forget phi.
Phi is actually at least as important as pi as a curriculum asset,
once we're into the greek letters and spooky greek metaphysics
("infinitely extended infinitely thin planes" and all that ghostly
stuff).
People tend to dismiss phi, out of bias against anything to do with
the pentacle. Holdover superstitions. But we can't deny the pentagon
its due, phi being the reciprocal of its edge-to-diagonal ratio,
although some call that tau. Five-fold symmetry *is* important in
Nature. Who are we, the naked apes, to argue with her wisdom?
In fact, we've done a lot of homework on pentagon math right here on
edu-sig, including in the form of continued fractions (a perennial
favorite, as phi is "just ones" all the way to the bottom, like
turtles). Plus you've got this convergence to phi along the Fibonacci
sequence, already paradigm generator in the Pythonic literature, right
up there with Guido's cute little gcd function.
I'm still liking the fact of fib.next()/fib.next() getting closer and
closer to the most irrational irrational, every time we hit the Enter
key (again, the bigger number will be in the denominator, given
left-to-right evaluation, so the convergence is to 0.618..., not to
1.618...).
>>> from __future__ import division
>>> def fibs(a=0,b=1):
while True:
yield a
a,b = b,a+b
>>> genfib = fibs()
>>> genfib.next()/genfib.next()
0.0
>>> genfib.next()/genfib.next()
0.5
>>> genfib.next()/genfib.next()
0.59999999999999998
>>> genfib.next()/genfib.next()
0.61538461538461542
>>> genfib.next()/genfib.next()
0.61764705882352944
>>> genfib.next()/genfib.next()
0.6179775280898876
>>> genfib.next()/genfib.next()
0.61802575107296143
>>> genfib.next()/genfib.next()
0.61803278688524588
>>> genfib.next()/genfib.next()
0.6180338134001252
>>> genfib.next()/genfib.next()
0.61803396316670656
>>> genfib.next()/genfib.next()
0.61803398501735796
>>> genfib.next()/genfib.next()
0.61803398820532507
>>> genfib.next()/genfib.next()
0.61803398867044324
A general thrust of gnu math is to use OO concepts to build awareness
of "maths as extended type systems" (class hierarchies if you will).
That's a bridge to our Fuller School geometry, as we regard the
Tetrahedron as a kind of superclass object or parent, for the whole
idea of Polyhedron in general. You have edges, vertices, faces (V + F
= E + 2), and an inside and outside. Every subsequent Polyhedron
inherits those elements, and the machinery to go with 'em. And we
don't have anything simpler. The sphere is actually much more
complicated to think about, given its reliance on a continuum
hypothesis. The tetrahedron is friendlier to discrete math engines
like Python. We don't need some infinitely perfect pi to think about
our what's in our tree.[3]
If you've followed my rbf.py thread's evolution, you know that it
builds on a primitive vector class (of the kind any first year student
could write), and face-tuples, to define a vocabulary of primitive
prefrequency polyhedra. If I go:
>>> from rbf import Tetrahedron, Octahedron, Cube
>>> mycube = Cube()
>>> mytetra = Tetrahedron()
>>> myocta = Octahedron()
and then ask for volumes I get:
>>> mycube.volume
3.0
>>> mytetra.volume
1.0
>>> myocta.volume
4.0
Whereas if you *haven't* been following my rbf.py's thread's
evolution, you'll probably just be asking yourself "Why?" at this
point. Because that's how we've designed our little zen rock garden
in Synergetics.[4] The tetrahedron is unit volume, given it anchors
the Polys class hierarchy (introduces the whole idea of an inside
versus outside for example). Two tetrahedra, per Kepler's stella
octangula, define a Cube of thrice the volume. And that Cube's dual,
with edge intersections at 90 degrees, is the Octahedron of volume 4,
and anchors our 6-rayed XYZ coordinate system (which, notice, comes
somewhere on down the hierarchy tree i.e. the primitive tetra is
topological such that no coordinate system machinery need apply).
OK, that's a lot of verbiage. You're likely happy I don't flood
edu-sig with all these threads. When I'm blabbing about Bucky a lot,
I tend to firehose in the direction of some Bucky-friendly e-list,
like Synergeo on Yahoo. Or maybe I'm on wittgenstein-dialognet
(pretty quiet lately), or just discussing stuff privately among
Wanderers (wwwanderers.org).
Finally, in addition to Glenn's global matrix, I've been caught up in
Sam's and LaJean's scenario. They've been trying to cobble together a
website, using a content management system. I've brought this up on
edu-sig because I thought it relevant to our community, given Python
has one or more of the flagship CMSs out there.
We shouldn't over sell them (CMSs) to people who just need static web
content and could get by with XHTML/CSS. Client side WYSIWYG tools
make it easy to avoid server-side dependence. Unless you're hosting
the server yourself, beware of getting captured by your own ISP.
However, in the case of flextegrity.com, I think it's turning out OK.
They're likely to need those fancy CMS features down the road
(implemented in PHP in their case), i.e. it was never all that
"static" a design (even though it may have started out that way).
Kirby
4D Solutions
Portland, Oregon
Notes:
[1] re 'Titanic' http://mathforum.org/kb/message.jspa?messageID=5055407&tstart=0
[2] Coxeter & 4d:
http://controlroom.blogspot.com/2006/08/more-dimension-talk.html
[3] More About Pi: http://groups.yahoo.com/group/synergeo/message/28550
[4] rock garden: http://www.grunch.net/synergetics/volumes.html
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