I live on the Pacific Rim, in Portland, Oregon, which has a Chinese Gate, marking a long history of East-West traffic. The Dragon is pretty much a powerful creature but more often celebrated than demonized.
We have parallels in the west, but in Genesis the serpent plays a decidedly negative role. Although it’s a “dragon” that St. George would slay, I think that hearkened back to the generally negative connotations in the Bible. Serpents and dragons were one and the same.
These days, I do a lot of programming in Python, perhaps another reason to think ahead. True, Monty Python was more the target of Guido’s homage, but the familiarity of the Python, to Athena, really has to be remembered, which brings us full circle, as Athena reminds us of Eve, scholarship I needn’t duplicate here, check the Web.
In one re-telling of Genesis, in a limited edition art book called Tetrascroll, the authors of Genesis want to punish any knowing of the Dragon religions, older and further east.
That our world is actually a sphere, is also forbidden fruit.
Eve is like the name of a ship, feminine, where ships have ribs and convey Adam, around the world. They’d flip their ships over on shore back then: the ribs became eaves. Like I said, an art book, not real etymology.
My friend Sam Lanahan owns one.
He went to the Philippines with Bucky in Martial Law days, as a guest of the Marcos family. This was well before I knew him, even though I was in high school in Manila in those days. Our paths would cross later.
sam’s copy of tetrascroll
Posted: August 30, 1998
Modified: August 30, 1998
The Applewhites flew out to Portland for a visit with us in the summer of 1996. The details kept changing up until the last minute. At first it looked like maybe just Ed would come, but June decided to come along, despite health issues (the cane was new).
I’d visited with Ed in Georgetown a couple times, and Ed and June got to meet my parents briefly on another trip over Mexican food in DC. Ed and I also both attended the Synergetica conference hosted by the BFI in 1991, where Yashushi Kajikawa was the guest of honor. This was the Applewhites’ first trip out to the Pacific Northwest and it was our privilege to show them around.
Dawn, Tara and I drove Ed and June up to Crown Point in the Toyota. Alexia might have been working at the zoo that day — where Ed and I had dropped her off enroute to the Rose Garden on the first day of the Applewhites’ visit (Ed was waiting for us in front of the Arlington Club).
Ed was very taken with Portland’s architecture, doing lots of walking tours reminiscent of those he did of Washington DC for his
guide book. At Crown Point, he chatted a lot about the upcoming Nobel Prize for chemistry, much on his mind because it was to be for the discoverers-namers of buckminsterfullerne this time around.
June Applewhite Crown Point, Oregon
Interesting day for vehicles at Crown Point that day. I missed getting a shot of the classic Rolls Royce, but snapped these two of the sherrif’s van and this fleet of red cars around back.
Ed and June were most generous with their time, kind and gracious visitors to our neck of the woods. They took us out to dinner as a family at the Newport Bay restaurant on the Willamette, and got to meet my friends Matt Ryan, who had accompanied me on a visit with Ed in Georgetown some years earlier, and Harold Long, architect and student of Frank Lloyd Wright. Ed and June invited me to the Heathman for a lavish coat and tie dinner.
Ed and I also had steaks at The Ringside on West Burnside, after which I drove him up to Pittock Mansion for the view. Many fond memories.
Some other EJA references:
First Posted: Oct 3, 1997
Last Revised: Feb 9, 2010
The Oregon Math Summit, held on October 2nd, 1997, at Oregon State University in Corvallis, was certainly a valuable learning experience for me.
I took an active role, as a presenter of course, but also as the guy with the squeeky flip-chart felt pens who wrote down the points people made in the breakout session after the intro speeches by the heavy hitters. And I participated actively and constructively in Ralph Abraham’s chat session. For a grand finale, I got to run the microphone up and down and between the seats during the Q&A with Sir Roger Penrose (after first using my privileged position to get my own question in edgewise).
My Beyond Flatland presentation went OK. I packed a lot in, but it all self-reinforced, coming back to the same points from many directions. “This is starting to click” said one 5th grade teacher towards the end, after asking several good questions about the A module. Another teacher came up afterwards and mentioned having built a 3-frequency dome with his students long ago — said this material had clearly come a long way since, “lots of good geometry here” he added. A guy in front asked how long it took me to learn all this and I said about 10 years, but with all the work that’s been done to repackage synergetics in the interim, the next generation might get it down to more like 10 minutes.
My handouts contained pointers to Struck, Dome, Povray and VRML. No mention of my own site but I had that on the felt board from the end of the previous workshop in the same room, about math on the internet, wherein someone was asking about geometry websites — I was busy setting up my props for the talk and overheard the question.
I bought a new shirt, tie (computer chips motif), and green trousers for this event and stupidly left my wallet in the old pair (remembered everything but that). So come dinner I didn’t have any means and took a short nap in the front seat of the Subaru after ingesting some leftover symposium muffins. Turns out there’d been this dinner for all the invited speakers, with a name tag next to my plate and everything, but the organizers all assumed I knew about it so no one mentioned it to me (nothing in the speaker’s packet about it I don’t think). So I missed out on that one — then had to bum some bucks from Terry Bristol, ISEPP prez, to buy enough gas to make it home, that evening after the Penrose talk.
Keith Devlin showed us a video clip from a new PBS series on Mathematics making its debut in April of 1998. The show is about real people using mathematics. We saw a lot of glitzy, fast takes with a big M in each segment, finally getting to the word ‘Mathematics’ as a punch line. Ivars Peterson took this as a joke: people have this anti-math reflex so engrained in this culture that “Mathematics” is now “the M word” — we don’t dare truck it out before hooking the audience with MTV-style eye candy, or the viewers will scatter to other channels.
The consensus among the heavy hitters (each made a short presentation after drawing lots for sequence) was that computational skills and basic numeracy was something to be instilled by all teachers of all subjects (Keith was the most explicit on this), and we should recontextualize the rest, embedding “higher math” in a more humanities-style curriculum wherein students get the message that mathematics permeates every aspect of the technoculture, albiet somewhat invisibly and behind-the-scenes (given the slick interfaces) — and getting that message across is maybe more important than actually overdrilling in specific skills, which you’ll learn as you specialize, if you do, as some brand of engineer or whatever. In the meantime, we should mix in a lot of history and study mathematics through key personalities (including the teacher’s), with the basic “how to make change and read the newspaper bar graph” type skills for everyday living more diffused throughout the curriculum (Devlin again).
Of course the speakers had their differences with one another. Ralph Abraham is most into doing math in chronological order, in sync across all subjects, meaning you shouldn’t teach anything ahead of its proper time, out of sequence. His curriculum (prototyped on Long Island someplace) has first graders doing Stone Age math, up through Neolithic (emphasis on the garden — reinventing agriculture), with maybe third or fourth graders doing Babylonian clay tablet work, progressing through Egyptian, Greek, Middle Ages, Renaissance and so on up to the computer revolution in maybe eleventh grade — something like that. But Ralph is discouraged because of all the pressure to pass this or that benchmark test (e.g. the SAT), with parents crazy to have junior focus towards that goal. Plus you can’t find math teachers with all the credentials and still willing to relearn to the extent of knowing anything about Babylonian math or whatever — they come preprogrammed to teach the standard curriculum, not Ralph’s.
Personally speaking, the thought of spending my whole K-12 ontogeny slowly recapitulating math’s phylogeny sounded a bit tedious, though I’m sure high caliber teachers could make it worth my time. I’d like to have a time line on CD and the ability to access math in chrono-sequence as an option. Teachers could teach from the timeline in many lesson plans, but the whole curriculum wouldn’t be lockstep-synced to that ordering through a whole twelve year span (and beyond). Like, life’s just too short to waste a whole year pretending to be a Babylonian in my book, especially when you could be doing synergetics, which by Ralph’s standards shouldn’t be introduced until maybe the 1990s, well after we teach everything else on the time line (like in grad school or something). Actually I don’t think Ralph was considering synergetics at all as a part of his curriculum — though maybe he is a little bit since our informal meeting (I flipped through some of the dynamite transparencies I’d developed for my presentation, after establishing some credentials as a soul brother when it comes to respect for those geeky Greeks).
Sir Roger Penrose seemed into math for math’s sake more than Devlin, wanting kids to appreciate the sublime beauty of absolutely useless gizmos. But the gizmo he brought to show and tell about wasn’t all that useless. He’s been studying the Pythagorean scale of musical notes, putting the frequency ratios on a “circular slide rule” (a log scale bent around the octave), showing how you can visualize chords as angles and key changes by rotating the wheel.
But then he showed how these key changes get you these “commas” (small sinus gaps) if you go with the strictly Pythagorean frequencies, and how, ever since Bach, master of the well-behaved chord change, the piano has used an approximation (another circular transparency, this time with all pie-slices equal) wherein the ratios are all the same — it superimposes on the Pythogorean pretty well but not perfectly. Anyway, this was all useful and interesting music theory, with historical threads woven throughout. The audience was clearly entertained.
Ivars was worried about Sesame Street and MTV style fast cutting and topic hopping, noting in the breakout session that one of his kids just gets frustrated with that approach, preferring to drill down within a topic once hooked, vs. jumping from one thing to another. He’s a bit anxious about the giddy-minded approach which fast-cut TV seems to encourage.
The teachers at this event were somewhat in a tizzy about the new state standards, which get measured on a standard test. The test has just gotten a lot harder, as State Superintendent Norma Paulus reiterated in her post-lunch banquet hall speech (held in the fancy new OSU alumni center), and now most kids are likely to score rather poorly. Only half the kids in the state even take enough math in high school to have an opportunity to score well — and of those only a small percentage will actually make top marks. So the teachers are caught in a bind, getting mixed messages. The standard bearers are marshaling the rank and file to drill drill drill to score high benchmarks, while the assembled leadership from the mathematics department itself is suggesting a more Renaissance approach, with lots of team teaching and convergence of subject areas, “not trying to turn students into poor imitations of a $20 calculator” as Devlin puts it.
In the meantime, Norma Paulus (with whom Ralph was impressed) outlined her state-level coping strategy, which she’s developing in tandem with Governor Kitzhaber, to build a “firewall” around instructional time, aimed at creating an inviolate space wherein teacher-student communications might continue to develop and flourish. Plus she was encouraged by the literally truck-loads of hardware being offloaded by the corporations on the schools as they upgraded to newer equipment, which students were learning to cannibalize and reassemble into working systems. She encouraged all teachers to return to their schools and ask about all that staff development money we know is there (because of all the expense records) but which may not be getting used accountably enough from the point of view of the mathematics department.
My response, expressed in the breakout session (as the official note taker, I managed to interject twice — once about the wierdnesses in floating point math as implemented in computers, reason enough to learn the algorithms on paper as well) was that if teachers wanted to fight the standard bearers and take back control of the curriculum (they all agreed that these tests from on high were severely limiting their freedoms and creativity as teachers on the front lines) they could use the ammo I was supplying via Beyond Flatland. Here was basic, low level, primary school material that every kid should know, and yet isn’t part of the standard — clear evidence that the mathematics department knows relevant content far better than whatever officials charged with concocting these tests.
If teachers rally around obviously relevant curriculum that the standards people are oblivious about, they have a chance to convince parents that junior will get a better deal if the mathematics department is given more responsibility, not less — like I said to Ralph after his class, it’s folks like you that should be our chief curriculum designers, not committees of well-meaning bureaucrats who think they know what the curriculum should look like, and yet haven’t a clue about what Ralph Abraham is doing to enhance it.
I also emphasized “home schoolers” as a strong card in our hand, since we can always invoke this nebulous all-ages network of dedicated math students as not beholden to the state. Out here in the wilds of the web, we can experiment with whatever newfangled curricula we like, and some enlightened parents are going to clearly see that junior is getting a far better boost into a promising future from staying at home (or tuning in after hours) than by kow-towing to the text book gods and their stale recycling of whatever sold last year (octane-enhanced with whatever cosmetic improvements of course).
Alexander Graham Bell’s octet truss
Bob Burton, The OSU math prof facilitating the break-out session asked me for a specific example of what the standard curriculum is leaving out (he wondered if our math head speakers were more a “last gasp of the Renaissance?” — something I squeak-wrote on the flip-chart as a good leading question), at which point I inserted a plug for my afternoon session. “Come to Beyond Flatland and I’ll give you all the ammo you need” I announced — earlier I suggested we all use the internet to better organize and stay in touch between summits, after hours if necessary (lots of nods).
I started my talk off with ClockTet by Richard Hawkins (in association with myself), a four minute computer animation with the clock face as a central metaphor.
I told the class I’d show it again at the end, and they could measure their increased appreciation for its content by the new insights they’d have on second viewing — “an easy kind of test”. Then I got my 31 slides-on-tape rolling, 18 seconds per shot, running through three topics: Concentric Hierarchy, Sphere Packing and Frequency. I emphasized that most of this material was in the public domain, on the internet, and if their school had invested in a digital camera, they could duplicate my process for creating such slide shows (the handout gave more details). Then I went to the overheads and went over these same topics in more detail, every so often picking up a physical model (before class, as people were taking their seats, I passed a bunch of models around, including Russell’s high tech tensegrity coupler, plus some museum gift shop stuff) and going over the basic principles.
A couple of teachers were especially interested in the large styrofoam ball with toothpicks, green and red pipe cleaners, and black ribbon. The red pipe cleaners connect toothpick-vertices in an octahedron (volume 4), while the green make a cube (volume 3), and black ribbon outlines the 12 rhombic faces of the space-filling, sphere-containing dodecahedron (volume 6). They saw this as something they could replicate in class. I explained my process of looping rubber bands as three equators to give the XYZ framework for the octahedron — jabbing in toothpicks at the intersections and removing the rubber bands and adding red pipe cleaners. Then I basically eyeballed the octa-face centers for the cube toothpicks, and added the green pipe cleaners, topping the whole thing off with the black ribbon wrap.
I also went over Karl’s ingenious system of bent pipe cleaner joins for stuffing into thin cocktail straw edges — Russ had built several of those during his visit, sitting next to me at Megan’s Run, an annual 24 hour marathon wherein my friend Mike Hagmeier was a participant (I lap counted for four hours at Lincoln High while Russell amused himself with pipe cleaners — I later learned Mike ran 78 miles before collapsing at 3 AM). But I digress.
Ralph was just back from writing a text book in Italy — calculus with an eye towards laying the foundations of chaos theory (whatever that is, he seems deliberately fuzzy on that score). He hyped the European math curriculum quite a bit — was on sabbatical in France at some point years ago and helped his kids with their homework (in French of course). The Europeans never seemed to lose the visual link with Euclid — less cost-cutting in the graphics department, unlike in USA text books, wherein pictorial math is given short shrift. Ralph has recently put all of Euclid’s stuff on the web in the form of “dynapics” — Adobe Illustrator type GIFs taking you through all the constructions and proofs very explicitly. He’s very turned on by the Euclidean stuff — wondered if any of us knew how to construct a pentagon (takes 56 steps in Euclid Ralph told us). I said I thought I could, using two circles (I was thinking of a somewhat racist theosophy book I’d been recently perusing, L. Gordon Plummer’s The Mathematics of the Cosmic Mind Fig. 4 pg. 15 — that’s why I asked if Euclid’s solution involved constructing a decagon first).
Ralph told the teachers that more conservative minds, some of whom he much respected, had originally tried too hard to control chaos, like through the Office of Naval Research, narrowing its band of operations to something minimal and controllable. He offered this as an analogy as to what might be going on in Oregon around standards testing — too much rigidity in the face of inherently dynamical systems. Then he launched into a brief explication of what dynamical systems theory is all about (strong links to the calculus), getting around to his teaching in chronological sequence ideas.
I asked if maybe the chronological approach meant we should do more to present mathematics as a discontinuous zig zaggy evolution, with lots of dead ends and withering branches — more like the Kuhnian analysis of science’s checkered past, with all those upsetting paradigm shifts and oldsters going down with their ships and so on (that metaphor comes to me now — didn’t use it then, but did mention Kuhn). And wouldn’t this involve studying “the heresies” — like I’d just been reading Bishop Berkeley’s problems with ‘1/infinity’ as used to prove stuff in the calculus — people said they’d addressed his concerns after he died, but I thought he’d still be objecting today. “Mathematicians could sometimes be really mean dudes” I went on (thinking of those brutal dueling episodes, although the two such episodes I recall were with non-mathematicians) and math involves a lot of arguing, polemics, games with funding (like look what Kronecker did to Weierstrass and Cantor) — shouldn’t we share this with kids, instead of presenting the subject as a cold “everyone agrees” type thing? Ralph heartily agreed (he knows first hand what it means to wage an up-hill battle).
I also mentioned I thought the chaos people had a PR problem, in that the general public immediately thinks of fractals and recursive self-similarity, but Ralph hadn’t mentioned those at all. He agreed, and traced the problem to Gleick’s book, which went heavily for fractals because of all the pretty pictures. He didn’t begrudge James Gleick his million-dollar success with that book Chaos, acknowledging a narrow overlap of chaos theory with fractals in the region of strange attractors, but felt the general public had somewhat the wrong idea as a result (like, so what else is new when it comes to the general public, right?).
The evening Penrose talk was open to the general public (for a fee) with front rows roped off for us teachers and presenters. I got there early and bummed some bucks from Terry when he came in with Sir Roger, so I could get some gas for the Subaru (and a Big Mac as it turned out). Maggie Niess, OSU science-math ed chairperson and conference organizer (along with her secretary Rose) said she was sorry about the dinner snafu (I shrugged it off). Ralph took a front row seat across the aisle from me and we settled in to listen to Penrose do his thing.
Sir Penrose has added the “tribar” as something for which he and his father should be remembered. That’s that “impossible triangle” used in those famous Escher lithographs as a basis of the forever ascending staircase and descending waterfall. The toilet paper royalties infringement was also mentioned by OSU President Paul Risser as a part of his introduction, plus we all applauded state education head Norma Paulus for making the Math Summit a reality — thanks in part to Terry’s connections (my wife Dawn is the ISEPP bookkeeper, and she was quick to suggest I would be a relevant presenter at this event, and Terry immediately agreed — really quick thinking on her part).
Penrose is looking for physical phenomena of a sufficiently noncomputable nature to serve as an anchor for our mental world, wherein we have access to the Platonic realm (via mathematics especially). He emphasizes that computers don’t have access to the Platonic realm, because they simply follow rules whereas although humans accept rules, and recognize new truths as consistent with them, they then proceed to transcend them by accessing truths the rules simply don’t predict or anticipate. I was thinking of Kant’s “synthetic a priori judgements” and Stuart Kaufmann’s synergetic “exaptations” as relevant in this connection.
To prove how computers can be trapped into revealing their innate stupidity by their own rule-following limitations, he had some chess problems which show at a glance how white can force a draw, but which computers, including Deep Thought and Deep Blue, all bungle, by taking the rook as bait, breaking the wall of pawns that is clearly the white king’s only defense. He went on to discuss other brilliant insights that mathematicians have had (proof of their transcendent access to the Platonic realm) which leave computers in the dust, as far as manifesting any creative intelligence is concerned.
So where in the brain do we localize the phenomena which make humans so special vis-a-vis their robot analogues? The title of the talk was The Large, The Small and the Human Mind — also the title of the upcoming book. The idea here is that we have fairly clear and deterministic models of the very large and the very small, but they don’t converge in mediophase very successfully, suggesting some new physics is needed, which will incorporate the human mind as a bridging phenomenon. The new physics will replace the conventional view that quantum mechanics goes to probabilities in mediophase, suggesting instead that it goes to certainties in short time frames with quantum gravity perhaps explaining what forces the wave function to collapse, leaving the cat dead or alive, not in some superpositional quasi-state.
Penrose is focusing on microtubes in the neurons as possibly sensitively constructed enough to register superpositions of quantum states for very short time periods, thereby allowing human consciousness make use of quantum entanglements to escape any rigid determinism and to make these leaps to higher Platonic levels even while still remaining brain-embedded in the physical world — as are the stupid computers. The difference between brains and computers is that brains are sensitive enough to register the effects of quantum gravity (whatever “gravity” is — and whatever “awareness” is, a key term Penrose was reluctant to define, but which he suggested we all know from experience).
Although he mentioned philosophy as feeding into this talk in places, Penrose made it clear that mathematics and physics were his focus areas. My question was clearly coming out of the philosophy department however — at the end I quoted Wittgenstein saying something like “if ‘consciousness’ has any meaning, it must be a humble one, like ‘table’ ‘lamp’ or ‘chair’.” I said I understood from his presentation of Gödel that we could keep enlarging the sphere of truth, consistently with accepted rules (i.e. without breaking any) but in ways not predicted by them either (thinking of Kaufmann again, though I didn’t try to stuff his “exaptations” into my question). Now that physics was feeling ambitious enough to absorb “consciouness” as another one of its key terms, I wondered if maybe the evolving usage patterns were changing its meaning, such that “consciousness” out the other end of the Penrose syllabus had a different “spin” than it had going in — a sort of moving target in other words, “a kind of zero.”
Penrose didn’t agree of course, as he sees consciousness as an objective, static thing which we all experience through qualia and experiences of emotion and understanding and so on — a view shared by the general public by and large. The goal is to discover its physical basis while consciousness itself “holds still” as we solve the surrounding puzzles. By analogy I suggested that if Newton returned he’d need some considerable reschooling to catch up on what we mean by “gravity” in this day and age (e.g. all that “shape of spacetime” stuff) and again Penrose disagreed, seeing “gravity” as a fixed point of reference as familiarly Newton’s today as ever. Obviously my view that key terms have wordmeaning trajectories within fields created from usage patterns, and do not map to reality as representational so much as operate within it as codefinitional, is not one with which Penrose feels comfortable.
I introduced my question by saying I would then follow Terry’s suggestion and run the cordless mike to others in the audience wishing to ask questions, and this I proceeded to do, although some had such booming voices I judged no amplification necessary. Audience questions focused on whether our orderly and comprehending awareness were really evidence of access to a Platonic realm or more a reflection of physical principles embedded in the design of nature, and hence the brain. Perhaps our intelligence is of local views only, with a greater universe not penetrating our thick skulls, because of no survival value (this from an Asian gentleman with a thick accent — Penrose seemed to have a hard time with it).
Towards the end of the Q&A Penrose willingly superimposed his three worlds (physical, mental and Platonic), apparently acknowledging that the mysteries associated with their interrelationships were perhaps collapsible into a single mystery of being.
People seemed honestly intrigued and engaged by the talk, although of course a lot of it went right over our heads (Ralph seemed to be nodding off in a few places I noticed, but laughed at the cartoon Penrose projected, which showed how nature probably doesn’t select for pure mathematicians, the early human kneeling in the dust with his gizmos, oblivious to his surroundings, being almost certain tiger-bait, compared to his compatriots busy doing more “useful and meaningful” applied technology type things).
I bought some gas (Arco) for the Subaru and and a Big Mac (Mickey D’s) for me and headed north to Portland on 99W, arriving home after midnight, and falling into my dreamworld while reading Feynman’s introduction to QED.
For further reading:
maintained by Kirby Urner
So… does the Grunch mean the noble ideals of Freedom and Democracy, fundamental to the rhetoric of USA institutions, are destined to fade away? Humanity cannot afford this option.
An army of literally soulless corporations, endowed with immortality, human rights, and the power to limit the liability of their shareholder creators, stalk the planet consuming its resources with little thought for the future.
People with the power to challenge (even while staffing) these goliaths form one global network, under God, indivisible, with liberty and justice for all.
USA iconography and the republican ideals for which it stands carries over into cyberspace for use by the pioneers of a newly democratic operating system — or call it USA OS.
Log in, and ask not what cyberspace can do for you, but what you can do for cyberspace.
For further reading:
On Virtualizing Government: A Design Science Approach
So… will current economic theory transform the world’s starving into global university students, sorely in need of Food Services, living in dilapidated dorms without plumbing or email? How long will administrators let these living conditions persist? Will access to resources, both physical and metaphysical, be channeled through curriculum circuits, rewarding starving students with meal tickets, plane tickets, and other university catalog items in exchange for the hard work of learning a living through work/study in oft-times harsh campus environments? Not likely.
Economics is too comfortable with its monopoly status, as the one discipline qualified to study resource distribution, to fundamentally recast its worldview — and yet economics itself teaches that monopoly is unhealthy.
In order to end Econ’s unhealthy monopoly, the global university curriculum is phasing in General Systems Theory (GST) as a competing discipline, rhetorically positioned to take away a lot of Econ 101’s market share. If you want to work for the goliaths of the future, better to have GST on your resume than a lot of worthless economics.
I expand on some of these ideas in:
Shape is defined by angles irrespective of scale. A regular tetrahedron has unchanging surface and central angles regardless of its size — and so for any object, no matter how complex. This angular aspect of experience links metaphorically to the idea of eternal principles. The ideally angular is only fleetingly grasped through intuition however. Our actual communications of eternal verities inevitably appear aberrational — idiosyncratic even — doomed to become nonsense in the long (or short?) run.
The concept of Frequency identifies the cyclic aspect of experience. An object persists in space and time as an aggregate of repeating energy events. The electromagnetic spectrum separates angular identities according to their different energy involvements. With the addition of frequency, the hitherto only-angle-defined becomes special case, temporal, mortal.
Or, borrowing computer terminology, we could say that eternal principles define the template class hierarchy, and special case events represent instantiations of these templates, as application objects with a finite life span, measurable in clock cycles.
4D in synergetics does not allude to 3D-space plus 1D-time. Space is experientially volumetric in synergetics: no Euclidean objects of zero, one or two dimensions (e.g. points, lines, or planes) are defined as such i.e. “planes” have thickness. Fuller argued that volume (space) is intuitively four dimensional (4D) because the tetrahedron is the simplest shape, the most primitive model of space. The tetrahedron has four faces, four corners, six edges. The “4D” tag derives from the tetrahedron’s inherent 4ness.
Synergetics also defines time/size as a unitary concept — persistence in time is not inherently separable from extension in space. Timeless, sizeless shapes, defined solely by their angles, take on the added attribute of frequency once made real in the form of special case experiences.The Cartesian XYZ axes do not delineate three independent dimensions — height, width and depth are not mutually separable aspects of experience. The mutually orthogonal arrangement of 3 positive and 3 negative vectors penetrate the 6 mid-edges of a tetrahedron — the “3” in “3D” relates to either open-triangle “zig-zag” from which a tetrahedron is comprised.
In place of the 90-degree-based XYZ lattice of Cartesian geometry, synergetics invests in a primarily 60-degree-based isotropic vector matrix (IVM), a lattice defined by the centers of closest packed spheres of equal radius.
Western science originally portrayed race and class as characteristics of a person’s blood which, as such, could be subdivided in proportion to a person’s ancestry, “blood” being treated as a mathematical quantity, contributed in equal proportions by one’s parents. Hence such terms as “octamaroon” (one eighth black). Whereas “class” is no longer regarded as a genetic entity, “race” has remained a popular concept for grouping genetic characteristics, even if the link to blood is no longer made. Like anthropologist Ashley Montague, Fuller felt the concept of “race” had outlived its usefulness, that the cross-breeding of the world’s people, especially evident in North America, was exposing the old racial categories as mere snap-shots of genetic traits thrown together by the exigencies of time, but available in any number of permutations from that vast grab bag of traits known as the human gene pool. In the Fuller lexicon, a racist is perhaps most straightforwardly defined as someone who believes in races.
Structure by Kirk Van Allyn
Fuller conceived of Design Science as an application for synergetic geometry: the geodesic dome and Fuller Projection (a world map) were examples of putting general principles to work as a source of practical artifacts. Design Science, which draws on many sources for its content, including software engineering, architecture, and the visual arts, is an approach to problem solving which looks at redesigning processes and/or inventing new tools, rather than trying to convince people to change their beliefs. “Don’t change people, change their environment” was one of Fuller’s favorite encapsulations of this approach. Fuller hoped that by means of a design science revolution, humans would be able to dramatically improve their living standards while coming into a sustainable long-term relationship with their ecosystem context. He insisted this revolution would need to be bloodless and artifact-centered, as opposed to violently political.
Letter from Kiyoshi Kuromiya, Oct 9, 1994, Fuller’s adjuvant on Critical Path, Editor of the posthumous Cosmography, to Mitch C. Amiano, participant on Geodesic list (used with permission)
Dear Mitch:
I was at EPCOT in Orlando, Florida, earlier today. I was attending/meetings nearby. The Geosphere ride called “Spaceship Earth” was closed. The sign read: “Spaceship Earth is being refurbished for your future enjoyment.” Besides lifting nearly intact Bucky Fuller’s themes for this “theme” park, they feature his inventions and ideas freely around the park and even in new projects like “Innoventions” without properly crediting Bucky.
At a book signing in 1982 in Beverly Hills, Bucky met Ray Bradbury for the first time. Ray Bradbury rather sheepishly admitted that he wrote the script for the Spaceship Earth ride.
Bucky Fuller was not invited to attend the Grand Opening of EPCOT, although it was known to the company that Bucky was working in nearby Deland, FL during that period of time.
Adding even more insult to injury, Bucky had been hired by Disney’s creative department to lecture Disney artists and designers in Burbank for three days in 1977 or 78.
Although Bucky was granted nearly 30 patents during his lifetime, he almost voluntarily cooperated with efforts to implement his ideas. Disney however took the ball and ran with it. Bucky was not one to resort to litigation (see Legally Piggily, in Critical Path), which does not mean his feelings were not hurt in such instances of ripoffs of creative properties.
As I once proposed in 1981, if Disney conceded sales of Dymaxion Maps at EPCOT, all of Bucky’s work present and future could have been funded.
–Kiyoshi Kuromiya
Synergetics takes up the subject of spheres packed tightly together. Mathematicians have not yet reached consensus on a proof that a Barlow packing, including the face-centered cubic (fcc) and hexagonal (hcp) is actually the densest possible, although Gauss proved the fcc’s density of approximately 0.74 optimal for a lattice (any denser arrangement would have to be more random).
The fcc packing is easily described in terms of adding successive layers of spheres to a tetrahedron. Canon balls and fruits in the grocery store are typically stacked in this fashion. A single orange nestles in the “valley” formed by three below it. A triangular layer of six oranges underlies that one and so on. Each layer adds a certain number of oranges, which may be expressed as a function of the growing tetrahedral shape’s ‘frequency’. The frequency is equivalent to the number of intervals between oranges along the tetrahedron’s edge.
* * * * * * * * * * * * * * * * * * * * Fig. 1. Tri-ville Packing (or Pool Ball)
Kepler studied sphere packing pretty intensely and knew that you get the same fcc packing if you start with a layer of spheres packed in a square arrangement and nest the next layer in the valleys so formed. If you taper off as you go upwards, this looks kind of like a Mayan Temple, so I call it “Mayan temple packing”.
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Fig. 2. Squares-ville Packing (or Mayan Temple)
As Jim Morrissett pointed out to me during an IRC chat one morning, the Mayan temple packing forces the fcc, whereas the pool ball packing does not. This is because a “squares-ville” layer presents only one set of valleys for the next layer of spheres, whereas a “tri-ville” layer presents twice as many valleys as we will find usable — presents an alternative, allowing us to go for hcp instead of fcc or some other Barlow packing.
The table below has a slightly different focus from Fuller’s in Synergetics in Synergetics Principles (section 220.00). Fuller’s 2nd power derivations were of the form 2 p ff + 2 where 2 p is the number of non-polar vertices (V-2). This expresses the number of spheres in the outer layer of a shape as a function of frequency. Below, the focus is on the number of spheres added as a shape grows in size, versus the number of spheres exposed on a surface. For example, a tetrahedron starts with a nuclear sphere, and then an expanding base of 3, 6, 10… new spheres per layer.
Sphere Packing Equations (from Synergetics or derived by the author) The octahedron and cuboctahedron are other shapes which grow through successive frequencies. For example, the cuboctahedron begins at Frequency zero with a single sphere. When twelve spheres close-pack around it, we get a one-frequency cuboctahedron. With frequency-2, we find 42 more spheres get added, then 92 and so on. Again, the number of spheres added per frequency is expressible in terms of a mathematical formula, as well as the cumulative total number of spheres in the growing cuboctahedron.
Spheres packed in an icosahedral conformation have the same number of spheres as in a shell of the cuboctahedron of the same frequency, although not cumulatively. This fact has proved useful in virology, where many viruses have icosahedral encasements composed of capsomeres.
The half-octahedron is like an Egyptian pyramid in shape. It’s base consists of spheres arranged in a square n-by-n pattern. Each layer above the base is one sphere less along each edge. In other words, a five-layer half-octa consists of 25+9+4+1 spheres, or 39 total. Since the number of layers is one more than the frequency, our equations start with one for F=0. When F increases to one, four spheres are added, giving a total of five spheres etc. The octahedron is the same as a half-octa but with layers building from the base in both directions. Counting all the spheres in the half-octa twice, with the exception of those in the “middle” layer (formerly the base), gives us a way to derive an equation for the number of spheres added per frequency, along with a cumulative total equation.
All of the packings described above are equivalent to the face-centered cubic. Connecting the centers of adjacent spheres with rods, and allowing the spheres to fade from view, is what gives us the isotropic vector matrix in synergetics.
As Robert W. Gray pointed out in a posting to Synergetics-L, the 2nd column of formulae in the above chart derive easily from those in the first column, given that the sum of successive 2nd powers (1+4+9+…) is given by n(n+1)(2n+1)/6 (n=number of terms), which formula may be proved by mathematical induction. Jakob Bernoulli derived the general formula for the sum of successive n-powered terms in about 1690.
See A Study in Sphere Packing for a more computational approach to sphere packing in a cuboctahedral conformation (IVM-style).
Relevant readings:
Synergetics on the Web
maintained by Kirby Urner
12-around-1 graphic by Richard Hawkins using
Alias Animator V5.1 on an SGI Indigo2 Extreme
Equations developed and depicted using MathCad 6.0
Thanks to Kevin Brown for email re Bernoulli, and to Dr. John Conway
for a lot of historical information and up-to-date nomenclature.
“Climaxing Bell’s architectural experiments with tetrahedral structures was an observation tower at Beinn Bhreagh, his summer estate near Baddeck, Nova Scotia. Each unit for this tower consisted of six 4-foot pieces of ordinary galvanized iron pipe and four connecting nuts; the units, themselves, were riveted together in the field by unskilled labor. Upon its completion in September 1907, the tower stood nearly 80 feet high.”
“It was a characteristic of Dr. Bell’s inventive genius that he was able to apply the discoveries in one field to another, entirely different, discipline. He once said on the subject of discovery and invention: ‘We are all too much inclined, I think, to walk through life with our eyes shut. There are things all round us and right at our very feet that we have never seen, because we have never really looked.’ The time has come to take a real look at Dr. Bell’s work more than 50 years ago.”
FORUM wishes to thank the Bell family and the National Geographic Society for their generosity in making the material available and this article possible. All photographs used herein have been copyrighted by the National Geographic Society.
Synergetics on the Web
maintained by Kirby Urner
All you Richard Hawkins fans will be pleased to know you can now visit the new Richard Hawkins’ Digital Archive — a more complete collection of Richard’s high quality graphics, both moving and still.
Animated GIF (right-click to copy)
Synergetics on the Web
maintained by Kirby Urner
Synergetics, departing from convention, replaces the cube with the regular tetrahedron as its principal unit of volume. The four-sided tetrahedron is the simplest possible enclosure — which is why mathematicians call it a “simplex”. Drawn as a cage, or wire frame, it has four windows, four corners and six edges. No space-enclosing network has fewer windows (facets) than four. The cube (or hexahedron), by contrast, has six facets, eight corners, and twelve edges, whereas a geodesic sphere may consist of hundreds or even thousands of facets!
Given the status of the simplex as “simplest space-enclosing network”, the decision to use its regular form as a unit of volume makes some sense. As a consequence of this decision, we obtain whole number volumes for other familiar shapes (including for the cube). This aesthetically pleasing and streamlining result provides additional reinforcement for those taking the time to learn this alternative (yet logical) approach to spatial geometry.
The tetrahedron has another interesting and unique property: its dual is also the tetrahedron. We get a shape’s dual by switching faces for corners and vice versa, while keeping the number of edges unchanged. The cube and octahedron are duals, as are the cuboctahedron and rhombic dodecahedron.
The dual of a shape need not be any specific size, however we may specify that the edges of the two shapes intersect at right angles. By combining duals in this way, we define the vertices of additional shapes. For example, the regular tetrahedron and its dual (also a tetrahedron) intersect to form a cube. Likewise, when we intersect the edges of a cube with its dual, the octahedron, we get the vertices of a rhombic dodecahedron.
The volumes for the shapes mentioned and displayed so far are as follows:
Shape Volume Tetrahedron 1 Duo-Tet Cube 3 Octahedron 4 Rhombic Dodecahedron 6
The cube and rhombic dodecahedron are both space-fillers, meaning they fill space without gaps. The tetrahedron and octahedron fill space in complement (by working together) with twice as many tetrahedra as octahedra.
We can put spheres inside each rhombic dodecahedron of volume six, sized to touch its twelve rhombic face centers. When these volume 6 dodecahedra pack together to fill space, the spheres will be tangent to one another at these 12 “kissing points”. Each sphere will be tangent to 12 neighbors (unless at the border of the packing arrangement).
These spheres we define to be of unit radius. The unit volume tetrahedron is formed by four such spheres and has edges of 2 radii, or one diameter in length.
The 12 neighbors to each nuclear sphere in the above packing will be at the vertices of a cuboctahedron. This cuboctahedron has volume 20 relative to our sphere-diameter-edged tetrahedron of volume one. The cuboctahedron’s 12 radial (center-to-corner) and 24 circumferential (corner-to-corner) edges are all the same length: one sphere diameter (the same as the tetrahedron’s).
In Part II of The Synergetics Hierarchy of Concentric Polyhedra, the Jitterbug Transformation will be introduced as a conceptual bridge from the above family of shapes to another family with five-fold rotational symmetry.
For further reading:
Synergetics on the Web
maintained by Kirby Urner