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Lattice Gallery

Adventures in Math Teaching

In the months leading up to the outbreak, through the transition to schools closing, I had some duties in a Hillsdale middle school.

In my coordinate system, there’s a “Hillsdale” nestled in some nearby hills, a small dale, so aptly named. Quakers have a worship group in there somewhere, however my work was with a private company with permission to work in public schools. We also have a Hillsboro, somewhat further away from my current 97214 zip code.

My duty this time was to serve as co-teacher, a format I favor. With two experts in the room, students have the benefit of their peer-to-peer banter. Sports broadcasts conventionally feature two commentators in dialog. Why not teach the same way?

Two teachers in a classroom halves the student to teacher ratio, say from 30:1 (students:faculty), to 30:2 (same as 15:1), and all with no need for more floor space. We call these ratios by the way, and find them at the roots of rationality, and yes, you guessed it, I’m a math teacher of sorts.

The curriculum I’m using is one I fused together from company repositories and my own, on Github and elsewhere. Source code is involved, yet I consider my topics mathematical in nature. I don’t see the need to artificially run a fence between lambda and delta calculus.

Earlier on the timeline, even longer before the covid outbreak, I was helping staff an art gallery, Lattice Gallery, on SW Broadway in downtown Portland.

The gallery’s existence was timed to coincide with a sculptors’ conference, even though these exhibits featured assembled constructions rather than material chipped at or whittled away, as with stone, or with woodwork.

The term “sculpture” is by now more an umbrella term for any and all “three dee” (3D) spatio-temporal works of art, whether chiseled, shaped from clay, or otherwise assembled from a variety of components.

The corresponding “two dee” (2D) genre is any work on a screen or canvas, which would include photography and film.

The criteria have to do with dimensionality or “D-ness”.

Writing might be considered 1D or linear, even though against the backdrop of a 2D canvas (the page), in turn embedded in 3D sculpture-space (the book).

Kenneth Snelson is considered one of the great sculptors, yet a Tensegrity is assembled, from tension and compression members.

Lattice Gallery featured Flextegrity, a concept closely related to tensegrity, and invented by Sam Lanahan, an early student of Tensegrity as well.

Sam’s several versions of Flextegrity have culminated most recently in C6XTY, which I exhibit in front of my Portland house as well.

The structures are definitely eye-catching, and their assembly requires interesting motor skills involving tool use.

I’ve used C6XTY in my curriculum quite a bit, as an example of a load-bearing matrix or lattice, also understood in purely Platonic space (de-energized, purely imaginary) as juxtaposably consistent with the CCP, that honeycomb of rhombic dodecahedral voronoi cells known to Kepler.

In Synergetics, we use the ratios 6:4:3:1 to describe the volumes of this rhombic dodecahedron (6), comprised of an intersecting octahedron (4) and cube (3), with the cube in turn comprised of a tetrahedron (1) and its dual.

If you’re thinking C6XTY sounds a lot like C60 or carbon-60, or buckminsterfullerene, that’s astute of you, and yes the connection is by design.

Each C6XTY ball has a C60 soccer ball appearance and indeed, some of our gallery visitors were Adidas people, curious to check out what maybe at first appeared suspiciously like competition. Was Nike somehow involved? True: a C6XTY sculpture would be a good fit for a sports bar or soccer stadium.

Sam had a small installation, a spiraling tower, two doors down, at Stone Soup, the eatery where Lattice Gallery would entertain, and where Sam gave his main lecture to the conference attendees, an event professionally video recorded for posterity.

Twelve balls around one define the vertexes of a cuboctahedron when you do it the right way in C6XTY, then comes a next layer with 42 balls, then 92 and so on, in principle filling all of space (picture a swimming pool sized tank or aquarium).

You might overlap this ball packing with any CAD-created sculpture and keep only the balls inside.

Depending on the size of these balls (think “voxels”) and their fixed inter-distance, one might be getting very fine sandcastle-like resolutions, or something more coarsely sketched in terms of baseballs. The scalability of these designs suggest applications along the spectrum, from micro to macro extremes.

C6XTY as cast in plastic, and assembled in the gallery, is of a specific size and shape. However the other versions of Flextegrity all around the visitor, along with the poster art around the walls, would remind the viewer that the “IVM” is not size-specific; it is frequency-modulated or “FM”.

“IVM” is a term from American literature, well post the Edgar Allen Poe chapter and Eureka. What’s the history here?

A New England boy from the late 1800s, born with bad eyesight (later corrected), had been expelled from Harvard for irresponsible flamboyance, and by 1927 had come into young adulthood resolved to reinvent his inherited language under the guidance of intuition.

As a returning vet from the navy, he had attempted to adjust to civilian life as an architect-engineer, only to fail at business and finding himself struggling to support a new family. He became suicidal after the death of his first daughter, and wracked with guilt.

His background before serving shipboard had included training at the Annapolis naval academy, which he’d preferred to Harvard. All his writings going forward would have a maritime flavor, mixed with the navy’s big picture view.

No, I’m not talking about L. Ron Hubbard here, the Scientology founder. I’m talking about the American Transcendentalist for whom buckminsterfullerene was named, grand nephew of Margaret Fuller Ossoli, a first editor of The Dial magazine, an early publisher of both Emerson and Thoreau.

Buckminster Fuller, celebrated by artists and playwrights alike, still has a lot of street cred in the humanities, even to this day.

The nod in his direction from STEM, across the C.P. Snow chasm, in the naming of buckminsterfullerene, had further helped bridge that cultural divide, between Hugh Kenner types, and engineers.

Sam knew Bucky personally, and visited the Philippines with the guy. I was unaware of that visit at the time, yet was in the Philippines contemporaneously, enjoying high school at International School of Manila (ISM) under the martial law government of Ferdinand and Imelda.

I would later wind up in an anthropology class at Princeton, with their daughter Imee.

I would connect more with Bucky later, after Princeton, when working in a Catholic academy in Jersey City, not that far from the World Trade Center and Statue of Liberty. I could hop on a PATH train and visit the Macmillan Company easily, procuring some hard to find titles (Synergetics 2 for example).

Later, I would work for nearby McGraw Hill in Rockefeller Center, having quit my teaching job and taken up babysitting for a working couple I knew down the street, Ray and Bonnie Simon.

Ray, the dad, was really into Synchronicity and later wrote a book on the topic, subtitled The Pursuit of Happenstance. He was a skilled office worker who schlepped into Manhattan (across the Hudson) on temp assignments. He knew a lot about show business and was an L. Ron Hubbard fan.

Bonnie, the mom, was a registered nurse who later died young in an earlier epidemic, the same one that we think killed Jim Henson, of Muppets fame.

I had been greatly influenced by Sesame Street, starting in the Philippines, in terms of the promise it showed for educational television. How could these breakthroughs be adapted for older audiences?

Jim Henson faced similar questions, as did Walt Disney. Andragogy was still in its infancy back then.

I caught wind of an RFP (request for proposals) the Children’s Television Workshop was sending around and wrote The Videogrammatron as my amateur contribution, basically outlining a future Youtube. This was in the mid 1980s. I wrote up the idea more specifically for my 4D Solutions website in 1999.

Ray and Bonnie had a new baby, Julie, whom I took care of, after completing my two year stint as a high school math teacher. I could walk to their place from my rental house behind Loews.

I was also trying to form a startup, somewhat involving my high school friend and current housemate. I had big dreams but little in the way of capital.

My friend would later move on to film school at Temple University, and make award-winning films, whereas I would end up moving back to Portland, Oregon, the city of my early childhood.

Even while babysitting for Ray and Bonnie, I printed up DIADEM Center for Creative Networking business cards, somewhat inspired by how “networking” was taking off in the jargon.

I was already enough influenced by reading the Bucky stuff that I was agitating for curriculum reform. I had EPCOT as my focus, given the Spaceship Earth buckyball and the upbeat utopian promises of its technological Tomorrowland.

Perhaps we could have a bulb-lit Dymaxion Map billboard on the back of Loews Theater? Advertisers could rent time on the thing. My house was right there.

I also had my eye on the Stanley Theater a few blocks away, then abandoned. Why not show IMAX movies for credit i.e. get NYU students out of their shells and over to the Jersey side for required viewings?

The idea seemed intelligent enough at the time. I wrote some letters. The IMAX company was encouraging.

The light pollution from the billboard Dymaxion would have no doubt been obnoxious, but I was really into such a thing, seeing it as a way to promote some promising futurism, as well as global airlines showing off their route networks etc.

Back to terminology, “IVM” stands for “Isotropic Vector Matrix” which Bucky associated with an ideal gas per the Avogadro model.

The lattice is everywhere about the same, uniform, isotropic, with every center connected to twelve surrounding centers, by means of vectors, to form a matrix.

How might we sculpt and/or model and/or project a uniform distribution of matter (mass) in the sense of “average molecule positions” in an ideal gas?

Let’s pick a noble gas, such as argon or neon, at standard temperature and pressure. What should our textbook renderings look like, if we freeze the motions in a single frame?

In a kinetic model, the molecules are incessantly moving at high speed of course, with higher speed connecting to higher temperature. However gas density is pretty much uniform.

This concept of a uniform distribution, represented by spherical masses, connects us to our Platonic conception of what J. H. Conway termed a “Barlow packing” (for the crystallographer William Barlow).

Crystallographers had found the same pattern or packing in various solids (in crystals).

The IVM is but one of infinitely many Barlow Packings of sphere-to-space density ~0.74 (74%), another ratio.

The IVM follows the pattern known as the CCP or Cubic Close Packing. C6XTY follows the same pattern.

These were curriculum ideas I could share with kids, as well as a adults. Sometimes I taught CAD (3D), other times Codesters (2D).

My company (the one I worked for, both online and around Portland) looked for cloud-based applications with individual accounts, so that students could save their work and progress up our Coding Ladder.

MIT Scratch, Codesters,, was roughly the curriculum sequence back then.

Once the covid outbreak started, I would continue up the ladder with Jupyter Notebooks and straight Python.

Lattice Gallery closed around the end of December of 2019, having hosted some wintery First Thursdays, when Portlanders would go out to see galleries. With social distancing the new normal, those evenings may soon resume, I don’t know for sure.

In any case, the gallery itself, between the Pendleton Building and Union Station (roughly speaking), was just a popup, and is today mostly memorialized in pixels.

The C6XTY sculpture was disassembled and trucked away, for later reuse, perhaps in teacher workshops, should our Silicon Forest curriculum catch on a little more.

One of the maths I tend to share, off the beaten track as it were, is what I’ve branded Martian Math, part of a four part heuristic crafted with the needs of our Silicon Forest in mind.

We’re big into nanotechnology and microcircuits in our bioregion. Buckminsterfullerene, nanotubes, and graphenes more generally, have been an ongoing topic for excitement and scientific research.

Exposing students to such ideas as the CCP and Barlow Packings makes a lot of sense for our region.

Meanwhile, on the literature side, the metaphysical artifacts (mental assets) provided by Synergetics, have continued to inform my curriculum writing.

My Jupyter Notebook on the Flextegrity Lattice weaves in the Quadray Coordinates we’ve been evolving, as a kind of literary comment on the dominant paradigm, the XYZ orthodoxy of the rectilinear algebras.

In basing itself around the tetrahedron, versus the cube, quadrays give us ways to exercise computer language skills.

I’ve so far developed them in Python (most maturely), Clojure, and Rust. The Silicon Forest is heavily into programming, especially around free and open source.

Portland hosts OSCON for example, O’Reilly’s flagship Open Source Convention. I would later work for O’Reilly too, well after winding it up at McGraw-Hill. In between, I’d gotten married and raised a family.

Teenagers continuing their educations during the covid crisis might stumble across some of these threads and follow them. Online sleuthing has become something of a hobby for those born since the invention of the Web.

The fact that conventional high schools are closed, yet life goes on, has given rise to some more serious study habits taking shape, for adults as well.

Fuller’s Education Automation: Freeing the Scholar to Return to His Studies, about the concept of UBI and work-study, from the perspective of the 1960s, was starting to seem like prescient futurism, which it was.

As a futurist, Fuller had a pretty good track record, prognosticating much of what has come about, and encouraging a World Game mentality going forward, to continue making more sense of it all.

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