So as I've been discussing with some friends (and Friends) on other lists, a lesson plan for our children involves discovering the dodecacam, a real piece of equipment warehoused around SE 10th in our coordinate system, with a website worth visiting.
The geometry angle is of course the camera itself, with 11 lenses occupying facets of a pentagonal dodecahedron, in terms of how they're oriented, with the 12th facet reserved for the pedestal, which attaches the camera to the camera person, who also lugs the recording media (a hard drive).
We use SQL to pull up tables of duals, starting with the Oregon state standard V + F = E + 2 (search for details), columns in our Polyhedra table. By searching on (V,F) == (F,V) pairs, E constant, you find which polyhedra interpenetrate in a dualistic manner -- check any geometry book for this idea (tetrahedron self dual).
The dual of the pentagonal dodecahedron is of course the icosahedron, of volume 18.51 relative to the tetrahedron of same edge lengths (icosa: E = 30, V = 12, F = 20; dodeca: E = 30, V = 20, F = 12 -- just switch F and V).
Of course actually field testing the equipment is of interest, as one of our mantras is you can't do math if you don't have access to a night sky free of light pollution, i.e. no certification by an "indoors only" route anymore, unless you're disabled.
So our math lessons have shifted to calories and joules, their expenditure, in the context of physical work. This may sound like a detour, but a mathematical algorithm, such as sorting alphabetically (or by any dimension -- no limit on how many dimensions SQL might sort by) takes energy, and less efficient equipment or programming tends to squander energy (and/or time). Important lessons for anyone yes? You body is a mathematical instrument (Keith Devlin good on this).
Getting our hands on actual dodecacams is maybe more feasible here in Portland. I've met the CTO and know this company is eager to partner with urban studies type people, has a long track record in that respect already.
So to keep this properly academic, I will be going through the professoriate. In fact, the University of Rochester is responsible for getting me into this loop in the first place (actually a Reed connection, but academia is like that -- lots of criss-crossing affiliations).
Speaking of which, congratulations to the VPython team for the new release announcement (Bruce Sherwood et al), I'm looking forward to testing it soon.
For those of you just tuning in, VPython is our tool of choice for actually showing these duals I've been talking about, i.e. once you're ready to see the progression, of tetrahedron, cube etc., we have a graphical engine at the ready, our big departure from TI culture and "technology in the classroom" ala NCTM, which is rarely anything but flat, "two dee". We don't go for that. No can do.
Gotta have "shapes" (polyhedra) as 3rd graders called 'em in the high Himalayas (I used to call this "Bhutanese Math" when writing for Father Mackey in the 1980s). Of course you've probably guessed my fantasies, of where I'd like to take these dodecacams (Paro anyone?) That's hard country to beat, when it comes to "immersive".
So, to put it another way: I've been working with my Meeting committees to put together opportunities for our teens that would involve a field trip to one of our respected Portland based companies, in the context of a gnu math unit on duals and Euler's Law for polyhedra (in it's easiest form -- proofs by cartoon very doable). The Columbia Gorge is a likely outdoor school for practicing the camera skills, even with dummy cameras some of the time (expensive gear can't be wasted). The goal of the outdoor segment is to fulfill our commitment to keeping math from being physically wasting, a kind of disease in that case. If you don't have a physical education component, you won't get it about calories and joules. We have lots of gym equipment. Gnu math teachers tend to be pretty fit.
PS: my daughter is having trouble with Spore suddenly not working, a very fun biology game, a computer fantasy of course (long in the making) but good for putting down hooks for later science, recommended (despite this sudden glitch...). School is out at this time. Doesn't mean there's no learning curve.