You very likely had a high school introduction to the triangle and the trigonometry that rests on it as a mode of perception. And you know some of the consistencies in, say, the small or so-called “flat” triangle. A main one is that 180° total of the three angles. It’s sort of magical. The figuring always works out this way. And few seem bothered by this not meaning anything. By that, I mean we can’t “see” that half-turn or 180° turn. The angles considered are the “interior” angles, and most think these are the only angles that exist. Bucky Fuller pointed out that, however tiny the triangle on our globe, there’s still an outside angle that’s 360° - inside angle. That doesn’t help much in our quest for some meaningful relation or action for the constancy to rest on.
If you ride the light beam, of course, you find a third set of angles. The interior and exterior angles are not part of the triangle system. They are fallout angles embedded in a larger modeling involving the center of the walked off (closed) path and the “out there” that expands and, then, contracts to an opposite pole. Those angles don’t, of themselves, tell us much.
Mark a starting point with your heel (or put your pencil tip to the paper). Face down the first leg of your path. And walk to the end of it. Now, you want to walk down a second leg. Say it’s an equiangular triangle, with three 60° angles inside. So, do you now turn 60° and start walking. If so, you head out into the unknown with a centrifugal tug. Your turn is 120°. And you will do it twice more, each time to your same side, to end up standing facing down that first leg.
Those three turns do not add up to 180°. They add up to 360°. And that describes what you did. You turned completely around, as surely as if you spun on your heel. But, ...you took a more circuitous route. An interesting thing is that this is true not only for a triangle, but for any closed chordal path. So we’d say that this constancy means something. We can understand it as against simply memorizing it. You could say that with an interior angle of 60° and an exterior angle of 300°, we’ve a measure of the cramping or tightness of the turn.
If you’ve gotten here from Kirby Urner’s Synergetics pages and 4D Solutions, you’ve likely expected me to get off into “systalk.” That’s my reworking of the names for the polyhedra and polygons. And the whole underlying vision rests on ...well, edges. The names bequeathed us by the elder Greeks mislead, blind. and impose a sense of solidity on us. The names are built by counting “faces.” There are three surface characteristics that Euler tied together with a constancy. Faces (or bases), vertices, and edges. If bases are the base reality, so to speak, ...then the edges are edges of the bases and the count is off by half. In a tetrahedron, there are twelve, not six, edges. However smooth the bend in the surface, ...two edges are “up against” one another. More nitpicking? Like the angles in a triangle? But look what treasures are nestled among the nits, eh?
If you don’t whittle your tetrahedron, but build one as we build things today, with frames and skins, if we want apparent solidity, you’ve a system again, as with that triangle. Now, you’ve a real six, the six members that the triangular openings edge up against. In fact, those structural members, energies, or trajectories are key to tying all the polyhedra (or important sets of them) together. You can read a sketch of my systalk from a few years back in a “Guest Editorial” I wrote for Kirby’s Synergetics pages in The Synergetics Mind. What matters in this context is that the almost unseen boundary conditions of those faces turn out to be where the action is. Holding them apart ...creates them, in a sense.
Nothing in the names of the polyhedra tells you anything much about them, certainly don’t suggest anything, hint at any relations among them. The name of the syses (systems) are as loaded as XML-defined tags. 4Hedron, 6Hedron, 8Hedron, ...and so forth. But, 6sys3 (where 3(fig) is the triangle), 12sys3, 12sys4, 18sys3, 24sys3,4, 30sys3, 30sys5.... See what I mean? It c’n be eery.
If you read Jeff Duntemann’s Visual Developer Magazine regularly and look forward, as I do, to his editorial columns (“The vision thing”) you might remember one of these columns from a few issues back (I’m lousy about remembering citations), he introduced “edges” as another way of seeing “interfaces.” It shows up again in his Foreward to one of the Explorer series of books, or maybe it’s the High Performance series.
In fact, the maverick thinkers who draw my attending seem always, sooner or later, to have a passage or several in their thinkings out loud that rest on some altered sense of these ...well, edges. Remember at the top of this page, when I mentioned Bucky Fuller’s exterior angles around the triangle? Well, he was, of course, thinking about the edges of the original isolated (no context) triangle. XMuLl that over. Edges are only contours ...until you etch them. And pry them (they’re always in pairs) apart.
At the front edge of writing, now, we cast frame punctuation forward and write into the pool, up to that cast closing tag. And we can write all sorts of things, in all sorts of layouts, into that etext.