I’m actually not about to launch into some TED talk (or TEDx) about high voltage direct current, used in both hemispheres. Pacific Intertie starts along the Columbia River, a first vista for serious landscape photographers.
Film hitherto had been mostly for portraits and cameos, nor could the framed picture be any larger than the negative itself. Intrepid pioneers of the landscape genre knew there would be a public for large framed photographs of Celilo Falls, where a major dam churns out high voltage today. They hauled film developing chemicals and dark room tents right to the site, as in those days one needed to develop the film right after taking the picture.
Several companies have gigantic data centers parked nearby, to better avail of the copious wattage, along with lower cost land. A data center at the heart of a city would displace its downtown, though many skyscrapers do host some rack space connected by optical fiber.
Thyristors do their magic, and California gets the juice. These two states have a mostly peaceful relationship, in spite of some rivalry, twixt a Valley and a Forest, both considered Silicon. Historically each was at the end of a fork in the Oregon Trail. Speculator venture capitalists headed south, following rumors of gold, whereas others sought the green pastures of the Willamette Valley, with an agricultural lifestyle in mind.
However, my subject today is more purely Platonic, concerned with geometry, practically devoid of empirical content besides the generic classroom props (“manipulatives”) of an elementary school.
We don’t study Latin and Greek language and culture the way some of our ancestors did in the United Kingdom, but we learn enough to speak of “roots” (same root as “radical”) and recognize our “cognates” on one Standardized Assessment Test (“SAT words”), as administered by the Global U’s testing service (there’s a more proper name for it I’m sure).
Many of us first learn about the Greek root “poly” (as in “polymath” and “polyamory”) in some early grade. We may also learn about “hedron” as in “go wash your hedron” (actually no one ever says that). “Hedron” means “face” as in “please wash your face” and so “polyhedron” means “many-faced” or “multi-faceted”.
Then come the numbers: one, two, three…. Very Sesame Street (a television program for young children and stay-at-home parents, other guardians, featuring puppets and animations, by the Children’s Television Workshop).
However, when it comes to Polyhedra, if the game is to enclose volume, to slice out an interior (carve a cavity) then the tetrahedron takes the stage as our Number One, with a starring role.
One facet is insufficient for any sort of box or container, as is two, whereas six (as in “hexahedron”) is really more than we need. Why not four? Like, three facets enclose a space how?
You have only facets (F), vertexes (V) and edges (E) to play with. The tetrahedron has six edges, four vertexes, four facets (V + F = E + 2). That’s our home base.
Now we jump to Lord Kelvin and his bubbles. He was studying foam and what, today, we consider part of complexity theory, as promulgated by the Santa Fe Institute. In complexity theory, we learn about power laws, which relate three dimensions of subdivision: linear, areal and volumetric.
As one subdivides the Tetrahedron into sub-volumes, linear and areal intervals are likewise affected.
Frequency increases as intervals shrink. But how exactly? What relates these different aspects of subdividing?
Whole books have been written, but a simple rule is: “as 1:2:3” in terms of the three exponents needed.
Subdivide edges by two, and see the number of areas increased by four, and the volume by eight.
Divide edges by three, and see areas increased by nine, volume by twenty-seven. That’s the arithmetic you need for now.
Peter Sloterdijk, who writes in the German language, has already graphed the arc of culture, from shared macrocosm in the Middle Ages, to bubble to foam in Berlin today. We don’t have as much of a consensus reality perhaps. But then don’t bubbles in the form of foam still have great interdependency and the potential to work as one?
Add plasma and nuclei, various signaling systems, and you’ve got cell cultures and the most advanced eukaryotic life forms that we know about.
Now for our first graphic, by David Koski, a colleague in the State of Minnesota, which has no common border, geographically, with either Oregon or California.
In the Noosphere though, in a different data layer, one sees parallel patterns in both Portland and Minneapolis. Shall we add Santa Clara and Austin, Texas? Those who study neighborhoods using zip codes see these more fine grained commonalities. Kombucha. Micro-breweries.
Santa Clara, south of San Francisco, is at the heart of the Silicon Valley, whereas Austin sometimes refers to itself as the Silicon Hills. Portland is in the Silicon Forest.
Silicon is a plentiful element, not unlike carbon, which facilitates our neural networks with electronic prosthetics, helping us with handicaps we didn’t know we had, until putting on those VR helmets, or watching TED talks stored in data centers. We have lots of ways to participate in Cyberia. We even engage in steering, in the sense of governance.
The cell-silicon hybrid we have become, as a planetary nervous system, gives humanity greater potential for self awareness, not just as an individual, but as a people. Satellite pictures have likewise bolstered our global awareness. Truckers are less likely to get lost, and are therefore more willing to try new routes.
What we see in Figure 1, are some twelve sided polyhedrons (also “polyhedra” in the plural) which fascinated Kepler, we’re told, by Arthur Koestler, in a biography. But why?
Well, for one thing, they fill space without gaps, like many hexahedra do (e.g. cubes) and some tetrahedra (not the regular ones).
Put a ball (sphere) in each twelve-faceted cell, and see them “kiss” at the diamond face centers. Every ball is surrounded by exactly twelve other balls. Space seems everywhere about the same in this lattice. All very pretty, and very dense. The ratio of space filled by balls, versus left empty, comes to about 0.74, or 74%. Crystallographers know this packing well, as the CCP. Many physicists call it the FCC.
Then notice the outline: the truncated octahedron Lord Kelvin considered. The shape has fourteen faces, as does the cuboctahedron, and happens to fill space (it makes a foam with itself).
David, myself, and Casey House, the artist behind the next graphic, a draft for a final poster, were musing about the truncated octahedron and wondering how to conceptualize it in terms of other shapes.
All three of us were raised on that House of Tomorrow philosophy, to some extent, insofar as we know what “isotropic vector matrix” means. About the same as CCP and FCC.
We think of “octet truss” as another synonym, and then immediately flash on Bell’s “kites” (that’s Alexander Graham Bell, who worked on these things after inventing the telephone and the Bell System, progenitor of Bell Labs, a source of UNIX, which inspired GNU and then Linux (GNU is not UNIX)).
This discussion twixt the three of us was on Facebook, in a public thread, and at some point we were intruded upon by a jealous rival who has a habit of using the same shoptalk we do, but more in an effort to expose its supposedly fatal flaws. We’re sometimes impressed that a dismissive outsider would work that hard to adopt our terminology. That’s somewhat flattering.
Lots of jealous rivalries crowd into this esoteric branch of the literature. Sometimes mathematics is portrayed as this serene and sedate vista, with monks busily getting on with church business, all on the same page, singing from the same hymnal.
Unbeknownst to the layman, the general public, are some of the religious rivalries that bedevil our “higher” culture. Witches get involved, upsetting the priests and monks, as they may be better at some of the spells.
Cosmologies vie for attention and respectability. Everyone wants a Big TOE as Arnold Mindell puts it.
Arny is a local physics-informed shaman, a psychotherapist in the Jungian lineage. He and Amy lead workshops at the Process Work Institute downtown, on SW Hoyt Street. TOE means Theory of Everything. Portland is pretty deep into Jungian psychology. We even have a Jungian Society. I used to attend the meet ups, in a First Methodist Church on SW Jefferson.
The cuboctahedron and octahedron will fill space as a duo, and we also recognize the octahedron as the cube’s dual. I have a Youtube channel devoted to what I call a Genesis Story, wherein dual pairs “beget” additional family members. For example the icosahedron (twenty faces) and pentagonal dodecahedron (its dual) beget the rhombic triacontahedron of thirty faces.
The cube and octahedron combined, edges crisscrossed, give us the vertexes of our space-filling diamond faced shape, including the ones below by David Koski, built using silicon circuitry running vZome.
I should explain that vZome is a “virtual ZomeTool” which is in turn a marketing name for Zome.
Zome is somewhat a descendant from the Bell kites, a ball and rod tinker toy or “manipulative” as maths educators call toys of this genre. They’re “educational toys” like LOGO and MIT Scratch. The Zome hub is quite a work of art.
Scott Vorthmann developed vZome to give students a way to construct spatial geometries using Zome, but in virtual reality, on a screen. Maybe someday we’ll see a more immersive VR version. Stella is another popular one, used by Magnus Wenninger and others, and then of course there’s Mathematica. The latter was used by Russell Towle in his Zonohedron studies.
Here in the Silicon Forest, we tend to teach our kids MIT Scratch. I have a star second grader adding to my studio all the time. He’s amazingly good at what he does, and isn’t the only genius in that school. I’ve been working in many schools around town.
Silicon Forest may stretch as far northward as Redmond, Bellevue and Seattle for all I know. My company (one I work for, sharing MIT Scratch in schools) has recently started a new gig teaching at Amazon. Geek parents tend to encourage Learning to Code for their kids, a branded movement.
Oregon and Washington States also have a kind of rivalry but their fates are closely knit. States can’t just pack their bags and drive off. We exchange humans (bodily fluids) daily, across our busy I-5 and I-205 arterials, the two jugulars of our regional commuter economy. Amtrak goes through here too. There’s even talk of higher speed rail, like in Asia.
Sometimes I think of Portland as being in Asia, as a Pacific Rim gateway. Many Japanese tourists love coming here, and take it easy riding our much slower trains. The Coast Starlight goes all the way to San Diego, and up to Vancouver in British Columbia (there’s another Vancouver just north of us in Washington).
Long time readers might expect me to segue into Truckers for Peace at this point, now that I’m talking about the “I-net” our North American freeway system. I do need to keep contributing to that genre, I agree. My own trip through Persia is still science fiction, as is my “bizmo” (business mobile) fleet. We use science fiction to explore possible futures in General Systems Theory.
Look for more stories on Medium about all that, or in my Bizmo Diaries (a blog).
However it’s getting late and the fire is starting to burn low. So at this point let me wind up the geometry story and remind readers of some important volume ratios.
The octahedron and cuboctahedron relate as 4 : 20 (same edge lengths), whereas the tetrahedron-octahedron grid (the octet truss) consists of polyhedrons related as 1 : 4, volume-wise. The space-filling dodecahedron (begotten by two Platonics) ratios with our tetrahedron as 6 : 1, the cube and octahedron edges still visible in each diamond. They crisscross at the kissing points.
Take a volume four tetrahedron, subdivide by three, and we get twenty seven times the original volume or four times twenty seven, or 108.
Now lop off six half octahedrons (see Figure 2), subtracting 6 x 2 or 12 from 108 to give 96.
So, to make a long story short, Ninety six is the number of unit volume tetrahedrons (shown as voids above) in a truncated octahedron (these tetrahedrons are regular, with edges equal to sphere diameters).
By this reasoning Lord Kelvin’s space-filling solid weighs in. Then we may re-size it however we like, to any frequency, according to the aforementioned power rule of 1:2:3. Shrink it down, expand it out. Make copies of many sizes.
Or was his tetrakaidecahedron something else, in terms of shape?
The literature suggests a great number of shapes may lay claim to that same moniker, with tetradecahedron a synonym (the “kai” is optional).
Rivals might take my unwitting conflation of distinct shapes as evidence that our truckers have a weak grasp of spatial geometry, and by extension, reality itself. They will try to use this perceived weakness to derail our intercontinental HVDC plans (back to the top paragraph), a strategy to conserve peak oil and make trucking more electrified.
So I try to cover my bases as best I can, meanwhile admitting in advance that my Silicon Forest brand Digital Mathematics (as I’ve called it, as a Silicon Forester and founder of 4D Solutions) has its shortcomings. I’m not planning to claim infallibility. Checks and balances work for me. Arnold Mindell calls his process work Deep Democracy. That works for me as well.