[Edu-sig] Rapunsel, Rapunsel

kirby urner kirby.urner at gmail.com
Mon Apr 3 17:18:26 CEST 2006

```> So if you want, just consider what I am talking about as Fuller
> prerequisites. ;)
>
> Art
>

Yes and no.

One of Fuller's big breakthroughs is at the really basic level, where
you don't really need any background.  People just assume that, this
late in the game, there couldn't be anything new that doesn't require
prerequisites to understand.  They just assume all the action as at
some distant frontier and it'll take years of study to reach the front
line.

The breakthrough consists of nesting polyhedra (an ancient game) such
that whole number volumes emerge, thanks to new emphasis on the
tetrahedron as the most topologically primitive volume (spheres, in
the other hand, aren't defined as such, as it's a discrete geometry
with no continuaa, no solids that we're talking here).

In my various math versions of rbf.py (RBF = Fuller's initials), my
Pythonic stash of simple polyhedra, I exploit this breakthrough in the
constructor by setting self.volume = 1 for the regular tetrahedron.
The stella octangula defines a cube of volume 3, octahedron of volume
4, rhombic dodecahedron of volume 6, cuboctahedron of volume 20.
There's also a space-filling Coupler of volume 1 and space-filling
irregular tetrahedron of volume 1/8.[1]

In a 2nd or 3rd grade classroom I'll simply have kids pour beans from
one shape to another (or I'll do it, asking them to guess the
outcomes).

Also, when we scale a poly, i.e. multiply all edges by a common scale
factor, I simply multiply volume by a 3rd power e.g. newvolume =
self.volume * scalefactor**3.[2]

You don't need to read any Klein to accept this basic innovation in
early pedagogy.  Maybe Piaget would be more relevant?

None of which is to say higher level geometry is irrelevant.  When
Fuller goes on to dissect his shapes into A and B modules, we get more
into Coxeter country.

HSM Coxeter was one of the great 20th century geometers and Fuller
dedicates his Synergetics to him [3].

But it doesn't follow that Synergetics is geometry. Just cracking the
cover and reading for a few minutes should persuade anyone of *that*
simple fact -- unless you buy that "explorations in the geometry of
thinking" (the work's subtitle [4]) is an academic geometry of some
kind (an uphill battle I wouldn't care to fight).

But a coherent-enough philosophy?  Sure, definitely.

Kirby

[1]  http://www.rwgrayprojects.com/synergetics/plates/figs/plate03z.html
[2]  http://www.4dsolutions.net/ocn/python/hypertoons/rbf.py
[3]  http://www.math.toronto.edu/coxeter/
[4]  http://www.rwgrayprojects.com/synergetics/synergetics.html
```