On Wed, Jul 17, 2013 at 10:06 AM, Robert Hansen <email@example.com> wrote:
> > Primarily, because it isn't intuitive (at all) as an illustrative aid for > students learning volume and cubing. That is the reason it didn't nor will > ever catch on. > > It did catch on, in the 1900s, and there's a branch of mathematics exploring this new territory.
The tetrahedron (simplex) is a topologically minimum volume in having the fewest edges of any edge-built structure. The cube has 12 edges and 6 faces compared to the tetrahedron's inventory of 6 and 4.
For an alien civilization to think a topologically minimum volume could also serve as a unit of volume does not seem that far fetched -- great science fiction could be made from this.
The tetrahedron arises more naturally when you consider sphere packing, a topic that got a lot of attention in the 1900s as well, especially in n-dimensions.
Three spheres (equal radius) make a triangle (connect their centers) and a fourth sphere, nesting in the valley, above or below, makes a tetrahedron (connect centers again).
That's the first time we get the edges to enclose a volume (the edges define an inside separated from an outside).
The octahedron we get, of four balls in a square, one above, one below, has a volume of 4, relatively speaking.
Young students have no problem with these concepts.
The tetrahedrons and octahedrons fill space together, to make even bigger tetrahedrons and octahedrons.
Playing with this skeleton is worthwhile because it's what we call the iso-matrix, also CCP and FCC (same thing).
It's of core significance in chemistry and crystallography, a kind of "holodeck". Molecules may be (are) defined with reference to it.
> But when its oblique coordinate system applies to a problem, then by all > means use it. Once you learn all the math to apply it. > > Bob Hansen > >