I live in a neighborhood at least some Episcopalians refer to as “the Buddhist Ghetto”. I was visiting St. David of Wales, friendly to Food Not Bombs, when I learned this, and I emphasize the term “Buddhist Ghetto” was said with some pride, by a church member Portlander glad for neighborhood exotica. Gold Door. Linus Pauling House.
My trajectory has been through many countries, even kingdoms, rare as monarchies have become, some demographically mostly Christian (like the Vatican or Lesotho), others more Islamic (Libya, Egypt), others Buddhist, like Bhutan.
I should emphasize that, whereas I’ve toured the Vatican (not just the museum), our family lived in Rome at the time (EUR at first, then Viale Parioli), so that dad could commute to Libya.
Mom & dad (sometimes with sis) went on to live in Cairo, Thimphu, Maseru, whereas I’d visit from North America, sometimes for months, and do volunteer work (computer programming, teaching etc., or just wander around).
Bhutan was the most Buddhist of all those countries, however in this Portland neighborhood (Richmond-Sunnyside) it’s the Newar region within Nepal that is best represented, in the form of a temple around the corner from the Linus Pauling House. My former housemate Lindsey ended up diving in to that tradition and has an apartment in Kathmandu. She’s a scholar, like me.
However, I myself was already into Buddhism well before I entered the university (Princeton, Class of 1980), having become a huge fan of Alan Watts and by extension Zen, in my formative years in the Philippines and elsewhere.
Although raised a Quaker, I was flirting with Buddhism almost from conception. Portland is highly Asia-fied. When tourists take me for their guide, I sometimes intone this is a gateway city to Asia, already in Asia really, this being the Pacific Rim (shared with Fukushima).
A doctrine within Buddhism that most fascinated me was that of prajna paramita, or co-dependent origination.
The idea that “height, width, depth” fall apart into three independent linear dimensions always seemed somewhat sleight-of-handy to me, and I resonated with New England Transcendentalism when I found a like-minded skeptic, one R. Buckminster Fuller.
He saw volume or space in terms of a simple skeletal wireframe, that of four corners, four windows, and six edges. How many dimensions is that, then?
In Western Civ, we learn the difference between primary and secondary characteristics. In Physics, mass or energy is still primary, however by Philosophy even the physical is demoted to “special case”, as what “circle” refers to is “the form” and that lives independently in some aloof realm (the so-called “Platonic Realm” the way some say it).
Fuller was employing the same idea, using “special” versus “general” to distinguish secondary from primary. The smell and color of a cube are definitely special case and need not count as “dimensions” in the primary sense.
We’re left with X, Y and Z, the three dimensions of Cartesian space. Each axis becomes a real number line, with positive and negative numbers, the basis for addressing all pointal subjects of the realm (res extensa) by means of xyz 3-tuples.
Three of these six radii are christened “basis” while their three mirrors are considered “inverted” and/or “negated” relative to their positive peers. Together, this team of six spokes creates our XYZ Jack of Creation. That we use 3-tuples (x, y, z) is taken as prima facie evidence that “space is 3D”.
The tetrahedron, cube, octahedron, icosahedron, pentagonal dodecahedron — the Five Platonics — all live within this same space of 3Dness.
That dogma used to go entirely unchallenged as the prevailing orthodoxy. Then Synergetics came along, by some punk and/or maverick, and championed by my friend E. J. Applewhite of CIA fame. What’s this?
In going up against this 3D windmill dogma so directly, our Don “Bucky Fuller” Quixote draws attention to himself. Is he heroic or quixotic or somewhere in between? He colludes with Japanese, both revealing their sea-based empire in Fortune magazine, and working with men of Japanese heritage: Isamu Noguchi, Shoji Sadao, Kiyoshi Kuromiya. Was there a Buddhist connection even then?
Fuller’s trademark logo was 4D (along with Dymaxion) and although he was happy to go along with Einstein, he wasn’t saying 3D + Time, per the latter’s non-Euclidean Minkowski space. He was saying “4D as in tetrahedron”, emphasizing its inherently “pre-frequency” fourness — and then it may get to be special case beyond that (4D+), adding yet more spatio-temporal dimensions. With “frequency” comes specific scale, directed energy or mass i.e. more than just a logical shape. With “frequency” comes existence.
He’s deliberately putting his 4D Tetrahedron next to the 3D Cube, while contrasting his “IVM” (the octet-truss) with XYZ. Is this a cockfight or what? Anthropologists take note.
Actually, this boldly challenging maneuver goes mostly unnoticed outside any but a few philosophical narratives, as who has time for Don Quixote these days? Journalistic accounts are impressively minimal.
This idea that each “dimension” might not exist so independently after all, got a boost in Georg Cantor’s studies, perhaps more of a boost than most popularizations let on. I’ll let my colleagues tell that story, if not too busy shoring up the status quo.
Then, in a next chapter, in the wake of Cantor Dust we got to fractals of “fractional dimension” such as the Julia and Mandelbrot sets. I’m moving quickly here, through general systems theory.
Dimension theorist Karl Menger did some of the most overlooked yet foundational thinking in this area, proposing a “geometry of lumps” that did away with the “3 dimensional” genesis story per some new axiomatic beginning. My Coffee Shops Network blog picks up on that story, for those of you seeking deeper background.
To review the prevailing cosmology: a mathematical “big bang” narrative unfolds in almost every elementary school textbook, wherein zero dimensional points beget lines, and from lines we get planes (log rafts), and from planes: everyday space (a log cabin in the woods, a cave), perhaps with a crackling fire (+ Frequency). Oregonians may learn a little differently (lots of logs) but the ideas are the same.
Then comes a confusing chapter wherein we keep adding perpendiculars (more than three?), begetting more and more dimensions, providing the “n-dimensional” playground for the polytopes, their vector spaces. We call this “extended Euclideanism” or maybe “Descartes on Steroids” and acknowledge its power to pack data. Lots of programming tools depend on n-D spaces.
The layman may be forgiven for sometimes confusing 3D + Time (4D) with the 4D of n-D polytopes, but in reality these two namespaces are far apart.
To master them is to not confuse them, unless, as Coxeter put it in Regular Polytopes (page 119, Dover Edition) you’re intending to write science fiction (or perhaps start a new religion, I might add). There’s a difference between “extended” and “non-” Euclidean geometries.
Finally, enter stage right, the “4D” avatar of Bucky Fuller, the Tetrahedron, strutting to center stage and daring to question the ultimate authority of the Cube, the avatar of Rectilinearity, the Rector of Rectitude, the very embodiment of Unquestioned Orthodoxy.
Right angles are called “right” for a reason.
The sheer melodrama and suspense of it all suggests almost tabloid treatment, in self parody.
[ By the way, political aside: I never considered it fair that the “right wing” got to capitalize on being “right” whereas “left” connotes “sinister” almost by definition. What a handicap. Talk about bullying, to insist on playing in those self-serving terms. Solution: reverse the labels every four years or so? Seems to be working pretty well, with so-called lefty-liberals snuggling up to their deep state USSA KGB… OK, back to our regularly scheduled programming… ]
In Bucky’s 4D rants (sutras) we get the Buddhist notion that height-width-depth is manifestly a study in co-dependence, not in linear independence as the XYZers would have us all agree.
The primitive concept, of a concept per se (something with an inside?), is not of any one of these three dimensions persisting alone, nor of no dimensions at all, but of that proverbial waxy blob, that res extensa, of primary characteristics only, namely “shape” in pure principle. Conceptualization begins with containerization.
So then here is the Karl Menger “lump” we might start with, the clay of our claymation and program station for synergetic geometry cartoons. Lumps are bigger or smaller than other lumps. They translate, rotate, and morph in accordance with whatever laws they obey (whatever physics engine). They may be shaped like a line (as thin as thread) or a plane (rolled out like a pancake), but with no special need to be infinite, nor somehow “fewer dimensional” i.e. 2D or 1D. They’re all “lumps”.
Fewer dimensional than what, pray tell? Why start with three? What was the reasoning again? Something about the corner of the room featuring three right angles? Or was it the roof of a tent and three sixty degree angles? How is it that we have no more corners again? What happened to the rest of the tent? Doesn’t “observational” imply “spatial”? Self-other distance implies space, no?
We’ve been informed “no, observational needn’t be of anything spatial” by a political satirist, one Edwin Abbott. He wrote Flatland in 1884 to make political points but mainly fueled the view that observational beings need not start with any Kantian a priori experience of space or volume. Time maybe, but not space of three dimensions. One or two were enough. We could conceive of ourselves as lower dimensional beings. Perfectly flat spaces exist, and by extension, spaces of higher dimension than ours. The new religion was established.
All lumps are of equal dimensionhood in Menger’s proposal, whether playing the role of point, line, plane or polyhedron, cubic cabin or pointy tent. How many dimensions is that?
[ The ray tracer world (e.g. POV-Ray) is pretty lump-like, as what good is a point if zero-D? How will we even know it’s there? I suppose light sources and cameras (points of view) might get modeled that way, more as subjects than objects. ]
If we take a “blob in space” as our conceptual beginning, why go along with any Kantian argument that said blob is “3D” a priori? Says whom? Why is that narrative given monopoly powers? Isn’t “3D” cultural? Lets ask the anthropologists.
Don’t we at least have room to notice the tetrahedron’s four arrowhead corners, shooting away from a common origin, dividing space into four quadrants? The caltrop has an economy the jack just doesn’t have: four spokes instead of six, dividing space into four quadrants instead of eight octants.
Four corners, four quadrants, four faces… what’s so “three” about all this, unless we mean the three-vector zig-zags (like two spirals) that define the 6 edges of a tetrahedron?
Oh, you meant the corner of a cube, a cabin — of six faces. Three axes and hexahedral symmetry rule the day, day in and day out. Yet the Cube in Chief keeps showing signs of Imposter Syndrome. Expect defensive defiance, at the mere idea of sharing the road.
If a triangle signifies a plane, the tetrahedron a volume, and all objects are lumps (occupy space), then why not say these lumps are “4D” in honor of the four arrowheads, none the creature of the others?
In Quadray notation we point the four arrows from a common origin of (0,0,0,0) through address locations (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1). Any point may be addressed with 4-tuples, using conventional linear combinations of vectors, with scale factors. Vectors add and subtract as usual.
The spherical tetrahedron of four radii is a less complex contraption than any spherical cube, of six (to the face centers) or eight (to the corners) radii, no?
So lets enshrine the Caltrop over the Jack of Creation. Heresy!
OK, so how about “live and let live”? Vive la différence?
I’ll conclude by pointing out the prajna paramita (co-dependent, mutually definitional situation) obtaining among, by now, at least three meanings of 4D:
- the n-D meaning in Regular Polytopes (its author a former student of Wittgenstein’s), n = 4 (extended Euclidean);
- the 4-D meaning in Special Relativity (Einstein et al) wherein “time” is the “fourth dimension” (more into complex numbers);
- the 4D of Fuller’s Synergetics (dedicated to the author of Regular Polytopes), more philosophical than mathematical, but not entirely bereft of math facts, oft pointing towards new discoveries.
What I’ll keep recommending in the Buddhist Ghetto (Linus Pauling House etc.) is that we continue carving ye olde block along these lines, as a way of tracking a lot of dots on our radar. No need for unnecessary confusion. More with less.
If yet another meaning of 4D comes along, that won’t be surprising. Indeed, art history is packed with lore about the creative ferment first occasioned by dreams of dimensions higher than three, and how thinker-artists from C.H. Hinton to Claude Bragdon to P.D. Ouspensky took off with this idea.
It’s from the perspective of the early 21st century that we have the benefit of this vantage point, as spelled out above.
Synergetics was published in the late 1970s, early 1980s, not long before Fuller died. Without undertaking an exhaustive review of it here, I can attest to its sparking further innovation, promising more co-dependent origination to come.