Friday, August 29, 2014

Koski: Visiting Artist Series (continued)

From math-teach @ Math Forum:

Re: should students watch this video series?: Dimensions (originally in French)
Posted: Aug 27, 2014 1:53 PM

On Mon, Aug 25, 2014 at 3:04 PM, kirby urner wrote:

Of course my answer is "yes", and I was just again watching it over pizza, no beer (this was my lunch break). The series is Made in France but was translated / dubbed into Brit English. Portland Cable TV, bless its little heart, broadcast the whole series and one of my neighbors ordered the DVD, which is why I got to rewatch some of it today on break:

Today on break I'm watching this video by my friend D. Koski, with whom I've collaborated a lot over the years, including on an in-person pilgrimage / visit to Magnus Wenninger, a grand-daddy of polyhedrons in our age.
Dave's stuff is solidly three dimensional yet still comes off as somewhat alien given he has adopted the Fuller School's unit of mensuration, the tetrahedron, and here compares the volume of an enneacontahedron inscribed in a rhombic triacontahedron ("NCLB Polyhedron") in turn compared with a sphere.

But his sphere's volume is (sqrt 2)(pi) instead of (4/3)(pi) for radius = 1. That's owing to our different interpretation of L^3 i.e. 3rd powering is a tetrahedron for us, when represented geometrically. You'll remember I've talked about "our branch" in the tree of living mathematics.

Also: getting into David's stuff more deeply requires making room for yet another meaning of '4D'.

Let me explain...

The Dimensions TV show cited above (it aired on Portland Cable Television) is what I might call Coxeter.4D in flavor, where I use a proper name as a "namespace" and use "dot notation" to show '4D' "belongs to" that namespace (GSC take note).

However, in some of the movie's narrative, we seem to bleed over into Einstein.4D wherein "time is the fourth dimension" with x, y, z for three "spatial" dimensions.

Coxeter himself is at pains to distinguish between these two meanings of 4D in his Regular Polytopes i.e. "the tesseract" and "the time machine" are two different animals, much as science fiction writers might want to conflate them in the popular imagination as a way to drive their plots.

These are two of the great schools of thought that survived the early 1900s "shake out" re 4D as a meme. Linda Dalrymple Henderson has written a finebook on this topic.

David's stuff comes from a third school (which Dr. Henderson also traces), less well known, that associates '4D' with the "four directions" of the regular tetrahedron, i.e. four points and four faces, carving space into four quadrants instead of the eight octants of the XYZ / Cartesian apparatus.

Lets call that Fuller.4D.

So we have three meanings of 4D to stay aware of, each anchored in a different namespace:

polytope R^N geometry as in Regular Polytopes and n-dimensional sphere packing ala Conway

Minkowski space, Relativity, three spatial dimensions, one of time

four directional tetrahedron as volume unit and model of 3rd powering

More reading on this topic of namespaces in mathematics:

OK, back to work.



For further reading:
Has GST been applied effectively to promulgate Fuller's work?
Action plans to spread the word about Synergetics (retrospective) 
Various flavors of GST...