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