[Edu-sig] RE: Concentric hierarchy / hypertoon (was pygame etc.)

[Arthur]

> -----Original Message-----
> From: Kirby Urner [mailto:urnerk at qwest.net]
> Sent: Saturday, March 19, 2005 11:07 AM
> To: 'Arthur'; edu-sig at 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