> 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). _______________________________________________ Edu-sig mailing list Edu-sig@python.org http://mail.python.org/mailman/listinfo/edu-sig