Wednesday, July 26, 2006

Practicing Multiculturalism

Originally posted to math-teach @ Math Forum. Copied here, and slightly edited, for an illustrated edition.

Background to teachers:

The lesson plan to which I am appending, a guided meditation and/or storyboard for sketching and transcribing to television, derives from a system of mensuration premised on the priority of the tetrahedron as our unit of volume.

The four intertangent CCP balls, mentioned in connection with their rhombic dodecahedral ball casements, define our 2R-edged tetrahedron of unit volume.

Fig. 411.05: Four Spheres Lock as Tetrahedron
(click image for context)

It is vis-a-vis this standard that our above-described XYZ-Coupler is likewise unit volume.

Given geometry has room for more than one model, we're not presuming any lack of familiarity with the orthonormal standard, wherein cubic mensuration reigns supreme (a convention for centuries almost unquestioned).

One convention for going back and forth between cultures is to consider the 2nd-root-of-2 edged cube, what we know as the volume-3 cube in the IVM (and/or concentric hierarchy) and 3rd power that to get its traditional volume i.e. pow(2, 0.5)**3 or 2.8284271247461907 in Python.

Fig. 986.210: Diagonal of Cube as Unity in Synergetic Geometry
(click image for context)

(pow(2,0.5)**3)/3.0 and its reciprocal are now available as conversion constants for going between namespaces. Feel free to use them as globals in your programs.

With younger kids, with no prior indoctrination in the orthonormal system, just holding up the volume-1 tetrahedron and saying its volume is enough to get started.

It's like a mixing bowl or other kitchen measuring device. In this locale (or namespace) this is just how we use it, no proof or "conversion constants" required.

Just let students know: other cooks may use different kitchen implements than we do. This doesn't mean their food is bad, although we may find their methods of preparation somewhat awkward and energy-inefficient (no need to be rude about it, remember your manners).