Sunday, July 11, 2010

Filling the RITE

by Dave Koski with vZome

Whereas I've made much about the Mite, as dissected by Fuller into A & B modules, as the most primitive space-filling tetrahedron without overt handedness, we do have another contender, just by those simple criteria (tetrahedron, no handedness).

Because of its all-isosceles triangular facets, the Rite's component quarter tetrahedra (apex at the center of gravity) may all be rotated into one another, and so are minus overt chirality.

Assembled from two Mites, the Rite is itself a space-filler, as is the Bite (the other of two tetrahedral Sytes). These quarter Rites would have a volume of 1/16, same as a half-Mite (or Smite we sometimes say, or "characteristic tetrahedron"), also same as a K-module or 1/120th of a 7.5 volumed rhombic triacontahedron.

D.M.Y. Sommerville's 1923 paper, Space-filling Tetrahedra narrows it down to the three Fuller yaks about: Mite, Bite and Rite, plus this fourth one.

This fourth one, a quarter Rite, is believed by Sommerville to round out the complete list of Euclidean space-fillers.

Whereas the Mite will assemble the Bite and the Rite, this final tetrahedral space-filler is not filled by the Mite.

Saturday, July 10, 2010

On Wittgenstein and Fuller

Reprinted from the Wittgenstein Discussion Board, hyperlinks and graphics added.
"Mathematics hasn't been philosophy for 1000's of years Kirby, and in preparation to my reponse to Richard, Physics hasn't been philosophy for 100's of years." -- Robert Hansen
The context of the above remark is my ongoing thread about exploring alternative "foundations" in the realm of mathematics, drawing on philosophical work of the late 20th century.

Long timers on this list may recognize this as my pet topic. I am interested in language games that "revector" (change the meaning of) such basic words as "dimension" and "volume" -- not for all time for all people, but within one more sandbox or sandcastle on the beaches of possibility, another way to think and compute, design, get results, with a place in the sun.

We're talking about "forms of life" in other words, or call them "ethnicities" (the anthropological dimension is apropos when yakking about Wittgenstein).

The underlying plot line (you need one of those, to sustain interest and commitment) is that civilization got off on the wrong foot to some extent, in becoming so enamored with the cube as its favorite space-filling polyhedron.

We consider volume to be the product of three mutually perpendicular edge lengths multiplied together and assign a cube the role of "unit volume" as a result of this mindset.

To turn a 90 degree corner is to access a whole new dimension, making 90 degrees a somewhat mystical angle. Right angles are "normal" and to be orthogonal is to be orthodox (to believe all the right things). The cube is an ultimate bastion of conservatism, and to question its primacy is indeed to engage in a radical operation (or philosophical investigation).

This rectilinear beginning is now thoroughly taken for granted of course, gets passed along essentially unquestioned by one generation after another. It would take a rather willful and obstreperous youth with privileged access to education (e.g. Harvard) and a commitment to make a name for himself, to ever buck this trend. One in a million or billion might try this. Most would be quickly overwhelmed by the seeming hopelessness of their calling.

So yeah, along came R. Buckminster Fuller, born in the late 1800s, lived until 1983. He developed a philosophy which gave primacy not to the cube, but to the tetrahedron instead.

The latter is topologically simpler in having fewer edges, vertexes and windows (thinking of it more like a network than a "solid"). It's known as a "simplex" for this reason.

You may likewise use it to anchor a notion of 3rd powering i.e. volumetric growing and shrinking relative to linear growing and shrinking of its edges. Triangles may be used for 2nd powering the same way.

There's no logical reason that 3rd powering has to be called "cubing" and modeled as such. You need to get back to your mathematical foundations to "see" this -- as I've endeavored to do on several occasions on this list, in the spirit of Remarks on the Foundations of Mathematics and Philosophical Investigations.

Assign the role of "unit volume" to a tetrahedron (a regular one) and some magical things start to happen. This shape plays well with others and although it does not fill space alone, it does in complement with an octahedron of precisely four times the volume.

Another tetrahedron, known as the Mite (volume 1/8) does fill space without complements (Aristotle was right, remember the mite -- a new slogan).

The cube is reintroduced in this language game, but with a volume of three this time.

The rhombic dodecahedron, which embraces the octahedron of volume 4, the cube of volume 3, has a volume of 6.

Simple whole number beginnings, not shared with any students in US elementary schools because the Way of the Cube is considered the only way, best way, and it's "my way or the highway" when it comes teaching math's foundations (the life form in question is totalitarian in that respect, Borg-like ("resistance is futile")) -- partly why math is often such a turn-off to those who think freely and creatively (e.g. artists).

It needn't stay this way, were "philosophy for children" to open more doors, challenge the dogmatists.

Yes, the current foundations are primarily dogmatic in their delivery, like a catechism. Philosophers could be chipping away here, restoring some mental flexibility, freeing us from overly straitjacketed thinking, a kind of paralysis that keeps us stuck, awkwardly trapped.

A few of us are doing that work -- a real uphill slog given how people glaze over at the slightest mention of anything "mathy". We obviously could use some more help. Consider me a recruiter for the cause then, looking for allies.

A favorite way to keep "tetrahedral mensuration" from making any headway in the current "devo" context (ultra dumbed down, ethically in the toilet) is to dismiss it as "trivially true" i.e. the mathematics is well above the threshold of "false" (cannot be falsified) but then it's just too easy and simple to merit the attention of high level guru-geniuses, the caliphate as it were.

Elementary school kids who might benefit from earlier exposure to spatial geometry with this newfangled approach, never get the opportunity. They don't even have a clue what they're missing.

I call this "verboten math" therefore, because people such as myself, Amy Edmondson (Harvard Business School), Ed Applewhite (CIA), David Koski etc., who put many years of work into this project, encounter mostly resistance and put downs, transparent delay tactics.

Fuller is still a frequent target of character assassinations, even though he's dead (especially because he's dead?). Once on the cover of TIME, he was more recently ridiculed in the same magazine for his ugly "lemon" of a car (so how many philosophers do you know who invented a car? -- another reason he can't be a real philosopher, like Hegel or Marx: he had patents and inventions (a huge dis-qualifier, by today's academic standards)).

Amy wrote a book (A Fuller Explanation) which Branko Grunbaum nastily panned, Ed collaborated on Fuller's magnum opus (wrote Cosmic Fishing about the experience (talk about uphill slog!)) and David Koski has mapped all of the Archimedean honeycomb duals to Fuller's more simply named and volumed "modules" or "cells" (among many other achievements).

I've posted a table of David's results to the Math Forum thread above, fingers crossed it gets through. My previous response appears not to have made it past the censor, was perhaps too vituperative in tone (par for the course on that list, but I'm held to a higher standard perhaps).

So where does Wittgenstein fit in again?

I think one meaning of "show" stemming from Tractatus days, relates to what in psychology we might call a "gestalt switch" -- except sometimes that gets too narrowly interpreted as a merely visual phenomenon, such as in the case of the duckrabbit, Necker Cube and such.

When LW talks about the world waxing and waning, from the perspective of the subjective viewpoint (in some sense synonymous with the world itself because that perspective or angle colors everything), he's talking about how meaning is "orthogonal" to facticity i.e. to the world of facts (of true and false). A gestalt switch or new way of seeing (feeling, being) may leave everything as it as, factually speaking, and yet the world has changed its meaning in some way. This relates to what we mean by Zeitgeist, as many people seem to come to similar realizations, or call them "currents in the collective unconscious" (lots of ways to talk).

I realize talk of "many people" may not sound solipsistic enough to fit the mood of the TLP, but by the time of the PI, I think we're looking for people who "breathe a different air" (to understand what's presaged).

We could connect to William James at this point, or any of a myriad number of writers more cogent than I on this topic. My core thesis about Philosophical Investigations and LW's philo more generally, was that it aimed to catalyze or induce precisely such gestalt switches (aspect changes). Hence: philosophy leaves everything as it is, contains no theses (except of a tautological nature), is about liberation from reflexive, unexamined habits of thought. It's an ethical work in other words (ethics = aesthetics, per TLP) and therefore religious in some dimension.

In the Fullerian world view, human beings have crossed a threshold in their ability to leverage eternal principles (nature's "rules of the road") such that they have the option to take care of themselves at a pretty high living standard, though that doesn't mean simply amplifying the wasteful and resource-intensive lifestyles of North Americans etc.

This ability to mitigate human suffering on a vast scale is actually within reach, from an engineering perspective, but at the level of perception and conditioned reflex, our language is keeping us imprisoned (would be the view -- a tough one to stomach, as so much unnecessary evil appears humanly contrived -- not so easy to blame the gods then, nor even "politicians").

We're still enslaved by the thought patterns of darker ages, and cling to older dogmas out of habit and a need for security. People don't like having their cages rattled. The idea that we actually could eliminate death by starvation from the planet is a huge threat to business as usual, which is entirely premised on 'never enough to go around' or 'enough is never enough' as they say in Over the Hedge (a fun cartoon).

Again in the Fullerian world view, getting more air time for tetrahedral mensuration was a "foot in the door" that would get the sciences and the humanities to open more of a dialog. He aimed to bridge that C.P. Snow chasm, seeking a common language for both sides to invest in. Literary critics want to read texts on many levels, not get too mired in "the one literal truth" (per Norman O. Brown). Fuller's text does not disappoint in this regard, yet the ability to grab literal meanings from his fish tank is still very much there.

Fluency with sciences and maths on the humanities side has the potential to skyrocket, given a philosophical language well stocked with core memes from those disciplines, organized according to some broad heuristics centered around syntropy and entropy as the countervailing tendencies.

Yes, you could see this as just one more metaphysics, a Neoplatonism we could say, but then doesn't every age need to keep upgrading and updating? Are such language games entirely dispensable, now that we've gone through a linguistic turn?

I would argue that "new ways of thinking" remain as relevant as ever, and that a lot hinges on our ability to remain flexible and non-dogmatic, not overly reliant on inherited mental habits.