
Re: Common Core snippet a little distressing
Posted:
Jul 18, 2013 6:31 PM



On Thu, Jul 18, 2013 at 10:36 AM, Joe Niederberger <niederberger@comcast.net > wrote:
> Kirby says: > >Here's an unconventional nonXYZ mapping: > http://en.wikipedia.org/wiki/Quadray_coordinates > > I found this to be more enlightening: > http://en.wikipedia.org/wiki/Synergetics_%28Fuller%29 > > Which led to places like this: > http://www.grunch.net/synergetics/ivm.html > > So have you found anything like a "branch" yet? You were saying you couldn't find one.
Alexander Graham Bell was another pioneer, with those octet truss kites.
> I don't see any applications of these coordinate systems explained. If to > were to go into business manufacturing geodesic dome kits and such, would > you use one of these to calculate your material and manufacturing > requirements, and resulting product specs, or something more traditional? > > One application I'm aware of was I used them to help compute what are called Waterman Polyhedrons (Steve Waterman found other collaborators after joining with our group). Watermans are maximally convex shells defined by IVM nodes of radius X or less from an IVM origin (i.e. some (0,0,0,0)).
Since all IVM vertexes have whole number coordinates (and all formed from them have whole number volumes visavis unity  even the irregular ones) there was streamlining involved in some of the computations.
But since all the algorithms out there (e.g. for finding a convex hull given a set of points) use XYZ, I of course needed twoway conversions.
The goal with the isomatrix may be "conceptual fluency" over "number crunching" i.e. in tuning in that scaffolding, we're doing what any chemist would do when imagining an FCC lattice. You can let computers do the busy work of plotting / rotation / scaling etc. (ala Pymol).
But the presentation along this branch is different (than dwelling on face centers of cubes) as Dr. Loeb helped point out. Rather than explain the FCC in terms of a cube, we start with a nuclear ball and pack 12 around it, to form a cuboctahedron (you have other options  though only one if all touching four neighbors plus nuclear with each ball), then another layer of 42, then another of 92... that's an important sequence, accessible to 8th graders.
http://oeis.org/A005901 (cuboctahedral numbers)
Follow links from there too: you'll get back to the same neighborhood (my web site, connected topics). See a branch yet?
Kirby
Cheers, > Joe N >

