> -----Original Message----- > From: Kirby Urner [mailto:[EMAIL PROTECTED] > 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 _______________________________________________ Edu-sig mailing list Edu-sig@python.org http://mail.python.org/mailman/listinfo/edu-sig