A Portal for Silicon Foresters
from the Oregon Curriculum network, sponsored by 4D Solutions
What goes on in my little office, my personal workspace (PWS per GST)? It’s IRS compliant in terms of having no bed, and yes I’ve claimed it some years, its portion of heat and electric, internet and miles, all business related expenses. And I’m not lying.
The work is ongoing. Let me give you a window.
4D Solutions is the DBA (“doing business as” alias) and has been serving mostly Oregon-based establishments since the 1990s. When I say establishments I mean mostly NGOs (non-governmental organizations), otherwise known as non-profits, an example being ISEPP, but then I’ll throw in a few GOs (governmental organizations) and a few for-profits, to round out my client portfolio (resume online).
Since starting out as a high school mathematics teacher in the 1980s, I’ve been one to develop curriculum, an arc which took me through McGraw-Hill (textbook publishing, New York City) where I was on contract as a contributing editor, then onward to the Center for Urban Education in Portland, Oregon, with lots of trainings our team authored, then branching out on my own, in partnership with my wife to be, her Dawn Wicca and Associates being the company for which my 4D Solutions was a DBA. When she passed away in 2007, DWA became a sole proprietorship.
Going way back to St. Dominic Academy in Gotham, near the Statue of Liberty for cues, I’ve been looking for ways to leave some bread crumbs regarding a widely recognized yet oft unread hero of mine, one Buckminster Fuller, referred to today as a “popularizer” of the geodesic dome, and if you’re deeper into it, of tensegrity.
You may have seen the floating chair? You maybe saw Kenneth Snelson’s Needle Tower at the Hirshorn on the Mall in the District of Columbia (DC)? Geodesic domes dot many playgrounds; you may have grown up climbing around on one.
Getting more of this guy’s (Bucky Fuller’s) mathematics into the mainstream became a career-long theme, and I was to make a lot of headway between then (1980s) and now (2020s). We were to pick up “quadray coordinates” somewhere along the way, another “teaching tool” of high andragogical / pedagogical value.
With all that as background, let me relay some news from this morning, which featured my usual weekly meetup with an Active Inference Institute board member, and PhD level expert in the secret lives of ants.
Like my middle school friend Hayden’s dad, DAF is something of an entomologist, which gives him access to a huge corpus of ant-related writings that might someday be and / or actually has been fed to an LLM. He’s enabling AI to become more bug-eyed and bug-friendly so to speak.
He’s one of my go to guys for AI skills.
We were going over some of his new AI art this morning, and some of mine also, although I’m more of a dabbler and meme maker, more run of the mill in terms of output.
My memes of this morning were continuing in the Sam Hill vein, connecting his replica of Stonehenge on the north bank of the Columbia River, to the ongoing ecumenism of the Quakers.
What I was updating DAF about was local progress I’m making on a lesson plan featuring the Tetrabook as a kind of portal, a liminal device for transporting us, we who fully experience this lesson, from one world to a next.
We get to come back though. We’re only visitors in Narnia.
Here’s a Tetrabook in a nutshell:
Two triangular “book covers” (in this case yellow), lie flat against the table, while the one page (in this version, colored orange) is free to swivel around the Tetrabook spine.
In the straight up position (left panel), two complementary right tetrahedra are defined. They’re “right” in the sense of having a dihedral or between-face angle of 90 degrees, not that any of the corners are right angles.
In the slanted position (right panel), the corners all become 60-60-60 degree angles where three triangles meet, with the mentally added edge having the same length as all the others, but for the elongated edge of the complementary tetrahedron of same volume (same altitude, same base).
This gizmo operates like a light switch.
Flip the page to vertical and maybe call that “right tetrahedron” so-defined your “Sapien” unit or quantum. As hominids with much experience in land-based living, our orthodoxy is rectilinear. Our cubic unit is all 90-90-90 degree cornered.
Flip the page to slanted and your unit is “ET” for extra-terrestrial. The ET’s point of view is as if “from outside” meaning from an alien vantage point. The sapien approach to space & volume looks somewhat strange to them. The experience is mutual.
ETs or ETUs (equiangular tetrahedral units) seem unusual volumes to us. ETs are weird.
Why the vertical page configuration is associated with Sapiens relates to the fact that it has a volume equivalent to that of a Cube with half-size edges, half the TetraBook’s spine, and the length of all excepting the two elastic edges, connecting page tip to the tips of the two book covers.
Consider the Cube with edges R, R for Radius, while the regular tetrahedron, at the vertical page setting, has all but one of its edges D, D for Diameter, D = 2R.
These two shapes have the same volume. This is what we’re calling the Sapien.
Consider now the tetrahedron of all edges D. This is the ET’s volumetric unit and it’s smaller than the Sapien by a fixed ratio.
The ratio of Sapien to ET is about 1.06066, which is also known as S3. It’s the 2nd root of 9/8.
By “throwing the switch” (from 90 to 60 degrees), we enter a world wherein “closing the lid” characterizes multiplication rather generally.
We use equiangular triangles and tetrahedra respectively, for flatspace and roundspace, with both spaces considered inside the same 4-directional container, a tetrahedron, or icon for volume, where volume means space.
We leave the shoptalk of Hilbert Space and mutual orthogonals back in squaresville. But we need only trip in Narnia for a few minutes, just long enough to appreciate “closing the lid” to get our respective multiplicative products. That’s a different mental picture for sure.
And now we’re safely back in Sapienville, having maybe grokked for an instant the ETs’ exotic volumes table.
See Wikipedia under Synergetics (Fuller’s).
Now that we know how to go back and forth through this TetraBook portal, we’re free to return to ET world for additional studies, “ET world” being an alternative namespace or discourse, featuring the unit-volumed D-edged tetrahedron, and these alternative models of 2nd and 3rd powering, based on the triangle and tetrahedron instead of the square and cube.
Sapiens have only started showing up here recently, some claiming to have been abducted.
That’s some of the story anyway. A memorable mathematics curriculum comes with easy to remember memes and narratives and ours is no different in that regard.
Earthlings and ETs have taken up working on a hydroelectric dam together, our excuse to start getting into the physics of AC vs DC electricity. How do HVDC lines work and where are they? Google Earth oughta know.
Building a dam requires pouring concrete and here is where those Tetrabook computations come in handy.
By the way, DAF was the one to remind us, awhile back, that Ed Applewhite included a whole chapter of the life of ants, their intelligence in community, in his Paradise Mislaid.
Ed was both a collaborator on the ET stuff (Explorations in the Geometry of Thinking was the subtitle), and one of those Mockingbird guys (CIA).
We talk about him quite a bit, along with his place in the network, something of a “Hugh Kenner type” (a worldy writer, see The Pound Era) as he would prove when writing independently of Fuller (the geodesic dome guy) vs. as a co-conspirator.
So why not learn that this door was opened? Even if we didn’t go through it (or only Bucky did). This is all part of the historical record so give it a few minutes at least. If your high school mathematics teachers don’t want to touch this material, how about your history teachers? Invite them to take a page from the Silicon Forest, by introducing the TetraBook.