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Topic: STEM and Euclid's Geometry (creating a context)
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kirby urner

Posts: 2,762
Registered: 11/29/05
STEM and Euclid's Geometry (creating a context)
Posted: Dec 26, 2011 2:39 PM
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Branching from:

Brainstorming about STEM (was About Functions)
http://mathforum.org/kb/thread.jspa?threadID=2323260&tstart=0

On Sat, Dec 24, 2011 at 3:38 PM, kirby urner <kirby.urner@gmail.com> wrote:

<< snip >>

> I also don't mind (highly encourage) having alternative axiom /
> definitions handy, more accessible ones than usual.
>
> Drawing on the work of Vienna Circle member Karl Menger, I follow his
> suggestion to define a non-Euclidean "geometry of lumps".
>


This is more necessary in STEM than "pure geometry" because
as we're taking a helicopter or crane view of Euclid and his pals
scribing on the beach with sticks in the sand, drawing proofs,
we're also noticing the curvature of Earth and the fact that the
sand is granular, their lines anything but "one dimensional"
(same as the teacher's chalk lines).

The powers of 10 scale spectrum video segments are diving
us into cell interiors, crystalline structures, atomic lattices, so
once again there's a problem of where Euclid is getting to his
definitional beginnings, which the books imply are intuitive
in some cases, but are anything but in the minds of some
students.

Every line found in nature seems to have a story to it and
not reduce to "one dimension" in any direct / apparent
manner.

This cognitive dissonance is usually played down with the
resort to a "Platonic realm" argument, which in K-12 is
usually conflated with "imagination". Axioms and definitions
are bootstrapped as self-fulfilling in promise, as we're
getting a lot of interesting theorems attributed to them.

But aren't Euclid's results valid even in this non-Platonic
world of physical inconveniences in which we live? How
tightly do these proved truisms depend on points being
truly dimensionless? Some pioneering authors give us
other ways to think about these things.

Since the discovery of microscopy, the Platonic polyhedra
and their relatives now have a permanent place in nature
as actual physical forms as well.

The crystals and many biologicals are polyhedrons (e.g.
viruses) the old masters could visualize and sculpt / etch,
but not find in such exquisite and precise form as naturally
occurring objects until the very small had been investigated
- -- and to a much greater degree than Plato and his pals
could pull off given the state of the art at the time.

In the older curriculum, we're supposed to get to "infinitely
small dimensionless points" in the privacy of our own
experience, and since Johnny can't show Sally his example,
there's no real judge of success here other than maybe
self confidence.

Those who really believe they can imagine zero, one, two,
three and more dimensions have the advantage. The
true believers move on while the skeptics fall by the
wayside and are left to wallow in their confusion and
doubts about the integrity of the analog program.

In the old system, kids who confessed to not being able
to visualize "dimensionless points" or "infinitely thin
planes stretching to infinity" were given lower chances
of success, given the prevailing metaphysics and
predominant ways of *not* questioning authority. These
students were either too dull or too obstreperous or both.

Engineers needed to be good conformists, willing to look
and be the type, whatever is in vogue. That was before
geeks developed newer subcultures, technologically
adept yet not necessarily invested in cloning the look
and feel of yesteryear's company ethnicities / stereotypes.
All math is ethno-math.

A few questions in elementary school along these lines
(questioning the wisdom of the ancients) were tolerated,
but delaying an entire class to talk about how points
could exist yet have no substance, or not exist yet be
the cornerstone of results relevant to things that did
exist... the discourse would not bear up under close
scrutiny and what was used as infill was mostly hand
waving and bad philosophy (contributing to the latter's
bad name among those argued into sullen compliance).

Karl Menger comes to the rescue here, as a dimension
theorist from the liberal Vienna Circle, well known for
questioning tired dogmas. Why not introduce an axiomatic
substrate more able to accommodate real experience
that was backward compatible with Greek metaphysics
(with its infinitely thin planes to infinity) and future
scaffolding of a more binary / digital / discrete culture,
not into "infinite" as much as "definite"?

Could it be done?

He called it a "geometry of lumps" and allowed the primitive
players in the topology to not fit on a "dimension ladder" of
0, 1, 2, 3 rungs, as is customary in K-12.

We (the gnu math teachers) haul this out for student
inspection, and talk about "claymation" while rendering
geometric scenes in POV-Ray and other open source
free software (liberated software).

Ray tracers bounce conceptual light off everything of substance
in a scene. The result is quite persuasive. Most of the optics
we learned from the old masters is here, so it's not like we're
losing any math by allowing for lumpy geometry. The points
were like balls. They were allowed to have size. We might
call it Martian Math to distinguish it from the Earthians'.

> Then there's just more shop talk about geometry generally,
> independently of the scaffolding.
>
> We have reason to talk about the rhombic triacontahedron and rhombic
> dodecahedron having to do with chemistry and crystals.
>


Where some of my best teachers are starting today is
with the rhombic dodecahedron of Kepler fame, a space-filler.

Getting there may require a genesis story of sorts, with two
operations. Taking the dual of a polyhedron results in
another polyhedron, and the Platonics are all duals of
one another. The second operation is to combine two
duals by sizing them to have crossed edges, and taking
the result of their combination. The stella octangula is
a well known example: tetrahedron + self-dual = cube.

These presentations of information are not proofs so
much as "reminder trails" through a complex inter-weave
of relationships. Our emphasis is volume and organizing
the polyhedra in volumetric relationships, with overt links
to origami and flower arranging.

Our so-called "concentric hierarchy" is a sculpture imagined
and/or rendered in a "zen garden", meaning a lot of our
most fanciful renderings are replete with Chinese / Asian
and Indian themes.

The lions / dogs at the gate, with their paws on various
shapes, such as the rhombic triacontahedron, serve
an iconic function. Portland is packed with such memes,
being an Asiatic city on the Pacific Rim. This curriculum
is somewhat place-based from our perspective, but
also works as an import from someplace exotic -- or
localize to suit.

The rhombic dodecahedron and triacontahedron
interweave, though not as duals, and may be given
volumes 6 and 7.5 respectively. The latter is the
result of dual-combining the other Platonic duo, the
icosahedron and pentagonal dodecahedron.

The process (genesis story) kick-started (booted)
with the Platonic tetrahedron dual-combining with
itself to give the cube, the same cube as the one
forming the short diagonals of said rhombic dodeca-
hedron, the combination of cube and octahedron
(both Platonics), and half the RD's volume, so
weighing in at volume 3. Octahedron is volume 4.

We begin to develop this mental sculpture (of many
outward renderings) early on in Digital Math, as it
forms a basis for STEM geometry in many branching
directions (see Martian Math on Wikieducator).
Our elementary level spatial geometry is so greatly
superior to conventional / standard K-12 that it has
to stay underground a lot of the time -- too
threatening to the status quo.

We've formed an alliance with geek cultures and
rely on hackers to propagate it to some degree
(some with criminal records, as I was explaining
on mathfuture). Spatial geometry doesn't work
on calculators very well.

> This is STEM after all, so not just the needs of the geometry teachers
> matter -- we have geography to think about too.
>
> Kirby
>


Which again is partly why "infinite planes" had to
have that "grain of salt" added (they're taken less
seriously after our comparing and contrasting
lessons). We're very Earth focused, looking both
in (satellites pointing in ala Google) and out
(ala Hubble). The Earth is only locally flat and
overcoming the miss-perception of "infinite
flatness" or "flatness to an edge" was a big
accomplishment, a major conversion among the
landlubbers, a victory for astronomy and cosmology.

Dwelling on "infinite planes" and pretending our
drawn triangles are on them, seems like a slipping
back to the dark ages. We'll do it for backward
compatibility and do some pure Euclid, but we
won't sacrifice STEM itself to obsolete notions
(we bake a counter-culture right in, to keep up
the yeasty ferment of a true dialectic). Humans
deserve to take advantage of their own heritage
and subsequent thinking since "the old days".

Kirby



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