The captain just called about what's on Science Friday, something about mathematicians high in the "help wanted" category.
I had a morning meeting with Glenn, who then got to scrubbing some glyphs off of Julian's AlphaHelix, then got to work on Synergetics 101, depicted above.
Starting at the XYZ origin (0,0,0) we embed three polygons in their respective XY, XZ and YZ planes: two equal diamonds and a square.
Note our "equal" doesn't mean "same object" i.e. isn't Python's "is" so much as its "==", but they do have the same angles and share a hinge, the short diagonal. The square (3rd cross-section) connects each of these triangular tips.
This Coupler so formed has unit volume in Synergetics (yes, just like the tetrahedron) in addition to being a space-filler.
Google Earth isn't that clear about the Columbia River Correctional Institution, right next to the country club, where the Safeway Open goes on, and Dignity Village, in a triangle with Babecko's Marina and PDX airport. It's not hard to stick in those XML thumbtacks though, and swap them around.
Omnitriangulation may start with something prefrequency, like a Coupler, but we're quick to return to the geographic applications, as one expects of a place-based curriculum.
The Columbia Gorge is one of our anchoring spaces, given Outdoor School, the many tourist attractions.
The Columbia Scenic Highway connects The Dalles to Crown Point. Sam Hill enters the picture. Students remember the field trip, reinforcing the more abstract trigonometry.
Back to the Coupler, you'll get a lot from Richard Hawkins and the Clocktet approach. Use bisecting body diagonals of the cube, unity-two face diagonals, or make all 12 edges 1 if you prefer, you'll get the same angles.
The sharper point of the rhomb, where four faces meet, is the V to the cube center, between any two consecutive vertices along any edge of our (volume 3) cube.
Yo! off to Laughing Planet. See ya 'round.