RE: [Edu-sig] RE: Concentric hierarchy / hypertoon (was pygame etc.)
-----Original Message----- From: Kirby Urner [mailto:urnerk@qwest.net] Sent: Saturday, March 19, 2005 11:07 AM To: 'Arthur'; edu-sig@python.org Subject: RE: [Edu-sig] RE: Concentric hierarchy / hypertoon (was pygame etc.) Basically, once you've got a tetrahedron inscribed in a parallelepiped as face diagonals, various affine transformations of said sculpture preserves the 1:3 volume relationship, i.e. this is not just about the regular tet and cube.
We can eventually get them to Klein's fusionist approach - fusing the algebraic and geometric, as well as the flat and spatial. Part One, Page 1 of the Klein "Elementary Geometry" book I have been referencing introduces us to the progression of formations of 2,3 + 4 points, which brings us from the line to the triangle to the tetrahedron. The length, area, and volume of the fundamental formations are a simple function of the determinate of the matrix of the rectangular point coordinates on the line, on the plane, and in space - respectively. He then points out that even further generalization can be achieved by giving significance to the sign of the determinate - so that given a consistent ordering of points, one can readily ascertain the volume of arbitrary polygons/ polyhedra by composing them into component triangles/tetrahedron from a given reference point either within or outside the form, and then by adding the volumes (which may be negative) of the fundamental forms. Klein's approach to geometry is to find approaches that move between dimensions and forms in such a way that best avoids the need to except any special case. Which is why projective geometry becomes the (nearly) fundamental geometry, and other geometries - affine, Euclidian - are specializations. In this view, a regular tetrahedron is a bit of a freak - perfectly placed and formed. And at least in some important senses is of much less interest than what can be said - and there is indeed a lot than can be said - of the geometry of 4 balls tossed arbitrarily into space. I choose to rarely think in terms regular forms. Besides seeming inherently less interesting to me, I truly get confused as to what traits I am observing (or calculating) which derive themselves from the regularity and which might be more general. Regularity is therefore dangerous, and potentially confusing - rather than comforting. No major mind damage is going to be done by a different presentation. But I would like to disassociate the notion of geometry and the regularity of forms as completely and as early as possible. And this is where I seem to be most non-Fullerian. Art
No major mind damage is going to be done by a different presentation.
But I would like to disassociate the notion of geometry and the regularity of forms as completely and as early as possible. And this is where I seem to be most non-Fullerian.
Art
Not claiming to follow, but yes, we appear to diverge here. The thing is, we don't disassociate "playing with blocks" from architecture, and by extension from geometry, at all, in current childhood education. We most intimately link a rectilinear format, with various cylinders, cones and balls, into the young (very young) imagination. Cubes are both prevalent and regular, without any doing from me. So Fuller's innovation is *not* with respect to linking shapes and geometry early (regular shapes included, in the form of blocks, toys using them), but in the particular assortment of shapes and their canonical relationships. It's much more 60 degree than we're used to (what with all the equiangular triangles everywhere), from a classical western perspective, which is more 90 degree, more into post and lintel perpendicularity. So where I think Fuller and our concentric hierarchy challenges the status quo is in the manifest non-rectilinearity of this approach (NOT that this conflicts with Euclid in any way -- it really doesn't, except when we get into abstruse territory, such as absolute and infinite continua versus discrete and definite manifolds and such (analog vs. discrete stuff)). My view is he went up against a huge bias, but since he based himself outside of academia, in a sort of business world place, it wasn't like he could be shut out of the game. He was independently capable of mobilizing a large network. This has relevance in that a lot of what we call the open source movement may be traced to his anticipatory design science revolution concepts of the 1970s and 80s. Engineering and a focus on artifacts trumps political efforts to block basic innovations in math teaching. There's really no stopping us, politically speaking (because we really don't care about politics that much (like, we're popular already)). Kirby
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Kirby Urner