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

Kirby Urner urnerk at qwest.net
Sat Mar 19 17:06:47 CET 2005

```
> Might they also be interested that in Egypt, long-ago, in the shadow of
> the Pyramids, folks not only understood this to be true, but were not
> content with this knowledge. A way of thinking was developed that allowed
> them to become satisfied that the truth of this observation "makes sense"
> - that we should in fact *expect* it to be true.
>
> http://babbage.clarku.edu/~djoyce/java/elements/bookXIII/propXIII15.html
>
> What seems surprising - given a logical progression of thought - can be
> found to be not surprising, really.
>
> What is the lesson to be learned from that?
>
> To me, it *is* the lesson.
>
> Art

Yeah, once the relationships are established ("burned in") as I put it,
there's every reason to go over the reasoning in various ways.  There's not
just one way to cover this ground, but certainly a goal is to supply any
logical apparatus (including Euclid's) that works.

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.

Once you've accepted the octahedron of volume 4 (same edge length as
tetrahedron), it's easy to do the tet:cube in special case:  fragment the
octa into eight equal chunks around the central angle of 90-90-90 (1/2
volume each) and apply four of them to the faces of a regular tet to build a
cube (1 + 4*1/2 = 3).

http://www.rwgrayprojects.com/synergetics/plates/figs/plate03z.html

(there's a theorem you need for the octahedron in this picture:  that
tetrahedra with equal bases of the same height have equal volume -- you can
pull that out of Euclid if you like).

But with the pre-K groups etc., I find the measuring cups work well.  My
attitude is *not* that they should be surprised.  I'm very matter of fact.
My surprise is over the adults not teaching this material, not over the fact
that spatial geometry has a simple, rational, whole number core.

Kirby

PS:  ordered a new laptop (Toshiba Satellite A60), which should arrive
either this morning or Monday morning (leaving for Pycon on Tuesday).

```