:: a mini-course for teens ::
Sam and I served as co-teachers for our own kids last night, walking our talk as geometry teachers. Where does Flextegrity fit in, what's its context?
We pulled up the website on an XO (One Laptop per Child) to set the theme: design science, livingry (the opposite of killingry), making the world a better place.
One of the teens brought his high school geometry book to class. Was R. Buckminster Fuller anywhere mentioned? He checked the index. Yes, Fuller is given sidebar treatment on a page with a tetrahedron in one of the exercises (coincidence?).
We learn that Fuller is the "poet and philosopher" behind the geodesic dome, and yet there's zero information about his synergetic geometry, what he'd consider most relevant to teenage geometry students.
I covered some basics of space-filling, "tessellations in 3D", focusing on the rhombic dodecahedron of volume 6 (relative to our unit tetrahedron). Yes, the cube (volume 3 in our hierarchy), is likewise a space-filler. Regular tetrahedra (volume 1) fill space in complement with octahedra (volume 4) to form the octet-truss (a space frame or lattice). Our minimal tetrahedral space-filler, the Mite (volume 1/8) may be used to assemble all of the above (except the unit tetrahedron is just A modules, volume 1/24).
These whole and simple fractional volumes help kids tune in and stay connected, help them track. Mnemonics matter, provide a mental glue for cohering concepts. The concentric hierarchy is a pedagogical breakthrough, a life line, yet has almost no footprint in today's mass produced paper textbooks (why?). This explains our compulsion to supplement. We're an ethnic minority, a subculture, hoping to keep our flame alive and pass the torch to future generations.
The aforementioned octet-truss, variously named in the literature (CCP, FCC or IVM per Fuller) is a focus in many sciences. Flextegrity uses it as a backdrop, extending a purely geometric concept with real world materials (the difference between pre-frequency and frequency in Fuller's philosophy). There's a background / foreground relationship. Alexander Graham Bell was another original pioneer in this domain.
Sam covered tension and compression as co-occurring in any structure. He wanted to communicate the load-bearing responsibilities of real materials at different scales. He had assembled a tensegrity for this purpose, given how it isolates tensile and compressive elements. Kenneth Snelson of Oregon, Black Mountain College... he wove the historical tapestry, adding threads from his own life where appropriate (he'd studied under Joe Clinton while a poor student in New Jersey -- I'd forgotten that, even though by coincidence I had one of Joe's CDs in my bag).
Our kids were not expected to sit passively through all this. They ran the camera, interjected observations. Sam and I plied them with questions, solicited feedback. Our time was limited though, so no hands-on modeling, computer programming, video editing -- maybe some other time. This wasn't part of some grand scheme or plan, was more a spontaneous get-together. It's not like we have a private school on the side.
How might we encourage more teachers to join us in adding a more user-friendly spatial geometry to their toolkit? Here's a golden opportunity to augment and surpass stagnant standards, to innovate and pioneer. Oregon aspires to grow its high tech economy. Effective pedagogy is critical to realizing this dream. How might we signal a green light then? What wheels need to turn?
My other props included a Fuller Projection, CubeIt and Ball of Whacks. The latter two are useful for discussing further modular dissections of space-filling polyhedra. The concepts aren't difficult with the right mix of graphical and lexical. Going back and forth, between "left and right brain" is another aspect of load-balancing. Mathematics is imaginative.
Speaking of imaginative, the teens had all seen Avatar, more than once in some cases. We agreed that our tetrahedral geometry might be considered Na'Vi, the ethnicity under siege in that movie. That would go with our "geometry of nature" theme. Or just call us Earthlings, hoping to save our ship.
My thanks to crew member LaJean for a healthy vegan meal and insights regarding the power of hidden persuaders. She has a doctorate in nutrition and health, is also an authority on textiles and fabric, is able to sew together anything from a tent to a tuxedo. As I perused 4D Timelock after class (Fuller's early barrage of shelter-related writings), Sam and LaJean discussed Fuller's possible over-reliance on rationality, on reason, as a chief motivator.
Marshall McLuhan, Vance Packard... TV provides a window into the collective unconscious, sews the seeds of self-fulfilling prophecies. You'll find lots of readable tea leaves on TV, with even more on the Internet. Stay tuned.