Wednesday, June 23, 2010

End the Qyooban Embargo!

For quite awhile now, I've been using the word "Qyoob" to parody the word "Cube". "Qyoob" has a kind of Persian or Arabian Nights look and feel (Al Qyoob), which creates some valuable artistic distance for some readers.

Looking back at the Euro cultures, we see "four square" and such inter-twining attributes as "upright" "normal" "orthodox" and "upstanding" -- not to mention "regular guy" and "square deal".

Sound familiar?

Remember, the "beatniks" didn't want to be "squares" in an age of hyper-dimensional "cubism" -- were perhaps prototypical of the "buckaneers" in that way.[0]

"So what?" you ask, "what has any of this to do with Philosophical Investigations?"

Well, when undertaking a philosophical investigation, in the Wittgensteinian tradition, it pays to take in the lay of the land, to map out the surrounding grammar.

Connotations matter, not just denotations.

The goal, to fight the bewitchment of our intelligence by means of grammar, will be more effectively reached if we've actually done some homework.[1]

To get how these key vocabulary words weave their meanings, one actually needs to study them in action, not just assume one "already knows" as a kind of gedanken experiment.

To assume one "already knows" is to bleep over the whole point of Wittgenstein's philosophy, which is to get you to actually investigate, not take "it" -- whatever meaning -- for granted.

Back to Al Qyoob: the prevailing caliphate starts drilling children early on to think of "squaring" and "cubing" as mathematical equivalents of "2nd powering" and "3rd powering".

Some rather subtle mathematics inheres in this difference twixt saying "squared" and "to the second power", as the former presumes a shape, whereas the latter leaves the door open to different shapes, such as a triangle -- or a tetrahedron in the case of 3rd powering. [2]

My recent essay Aristotle was Right! is an opening salvo from the newest NeoPlatonists, one might say, as it challenges philosophers to revisit their ancient ties to the Polyhedra (and their inter-relationships).

Geometrical thinking is a primitive form of logical reasoning.[3] Before we had modal logic or a theory of logical types (ala Russell), we had the golden ratio, vesica pices, and the accepted methods of construction and proof.

Architecture was accomplished by these processes. Real work got done.

The story going forward, starting from Aristotle's claim that pyramids fill space (dismissed as bogus in superficial treatments), wends its way to recent times, to the geometrical explorations of a lone geometer named Sommerville.[4]

Sommerville circles a certain tri-rectangular tetrahedron as the simplest candidate that follows some specific rules in his mathematical language game, namely that:
(a) to count as "space filling", the tetrahedra must face bond (as is typical in such studies) and

(b) the space-fillers in question must not rely on mirror images of one another (i.e. sometimes a left and a right fill space, but not something alone).
Said tri-rectangular tetrahedron is found in Regular Polytopes by Coxeter (Wittgenstein's student, contributor of a venue for the Blue / Brown Book notes I think it was), where it is referenced by page number from our NeoPlatonist work. [5] There it is dubbed the Mite, or Minimum Tetrahedron, because of its space-filling role (no need of mirrors -- nor smoke for that matter).

In this referring work (dedicated to H.S.M. "King of Infinite Space" Coxeter), said Sommerville shape is in turn dissected, into the two As and 1 B particle, also described and depicted by Robert Williams in his groundbreaking work The Geometrical Foundation of Natural Structure. [6]

The Qyoobans have been embargoing this information (hence the term "Qyooban Embargo"), as evidenced by Math World, which on the page about space-filling polyhedra pointedly eschews mentioning any of the tetrahedra meeting Sommerville's rules.[7] These would be the Mite and two Sytes in current parlance, a tetragonal disphenoid (Rite) and another one (Bite), all three illustrated in the chart by Guy Inchbald.[8]

Furthermore, the Qyoobans censor and/or forbid sharing about the alternative model of 3rd powering, whereby said Mite and Sytes have volume 1/8 and 1/4 respectively, vis-a-vis a cube of volume 3 (no longer unity). Their public school systems deny their own citizens access to this heritage. Write to Congress?

Having their beloved Qyoob (a hexahedron) booted from His Eternal Throne is tantamount to heresy for them, hence the Embargo or Boycott, easy to document for the history books. The sense of entitlement among these Qyoobans (not to be confused with Cubans) is legendary. Talk about road hogs!

Those philosophers paying any attention to this little tempest in a teapot (not!), are probably aware that the dike is breaking, and that Qyooban philosophers are soon to be held accountable for their intentional and insistent dumbing down of our common heritage (logic).

Their overly high price tags may be questioned, even as their bigoted and ethnocentric behaviors fall into disrepute. All math is ethno-math.

I'm thinking Wittgenstein's philo will come out smelling like roses though. His investigatory style has proved a real asset. Hooray for the linguistic turn.

To summarize:

If you want the benefit of a liberating gestalt or enlightenment (as the fly escapes the fly bottle), then you'll need to find all those pesky little strings the Lilliputians have used, to tie down their Gulliver (our hero), to trap him with their small-minded meanings.

"Snip snip" go the Philosopher's Scissors (similar to Occam's Razor). Don't let those lab-coated Qyoobans nab you in their straitjacketed way of thinking.

Qyoobism tends to be overly-confining, awkwardly unimaginative, some say a symptom of its insecurity in its logico-authoritative role (arrogated).

Cubes are unstable unless triangulated, as any honest architect-engineer will tell ya.

Lets end the Qyooban Embargo.


[0] Of beatniks vs. squares, also buckaneers:

[1] from a recent journal post:
Lots of comparing notes occurs among the spin doctors, with secondary sources echoing primary ones in disseminating the latest bewitchments (the great Austrian philosopher Ludwig Wittgenstein used this term "bewitch" in a technical sense, in the context of investigating the hypnotic powers of "language games").

[2] "Triangling and Tetrahedroning"

[3] Aristotle was Right! (remember the Mite)

[4] D. M. Y. Sommerville (1879-1934), see:
Which Tetrahedra Fill Space? by Marjorie Senechal, Mathematics Magazine, Vol. 54, No. 5 (Nov., 1981), pp. 227-243.

[5] One of several portals to said Neoplatonist philo:

[6] He changes the meaning of B-particle from its original source, but is otherwise on target:
R. Williams, The Geometrical Foundation of Natural Structure, Dover, New York (1976), see footnote 20 on pg. 136.

[7] No mention of Mites or Sytes (as of June 23, 2010):

[8] The Archimedean honeycomb duals by Guy Inchbald, The Mathematical Gazette 81, July 1997, p.p. 213-219.

Kirby Urner is a RadMath teacher living in Portland, Oregon.

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