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Topic: should students watch this video series?: Dimensions (originally in French)
Replies: 1   Last Post: Aug 28, 2014 12:58 AM

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kirby urner

Posts: 2,015
Registered: 11/29/05
Re: should students watch this video series?: Dimensions (originally in French)
Posted: Aug 27, 2014 1:53 PM
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On Mon, Aug 25, 2014 at 3:04 PM, kirby urner <kirby.urner@gmail.com> 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:
>
> http://www.dimensions-math.org/Dim_E.htm
>


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.

https://www.youtube.com/watch?v=ycADhMVa_q0

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 fine
book on this topic:

http://www.amazon.com/Fourth-Dimension-Non-Euclidean-Geometry-Leonardo/dp/0262582449/

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:

Coxeter.4D : polytope R^N geometry as in Regular Polytopes and
n-dimensional sphere packing ala Conway

Einstein.4D : Minkowski space, Relativity, three spatial dimensions, one
of time

Fuller.4D: four directional tetrahedron as volume unit and model of 3rd
powering

More reading on this topic of namespaces in mathematics:
http://controlroom.blogspot.com/2006/08/more-dimension-talk.html
http://controlroom.blogspot.com/2009/02/dimension-talk.html
http://mybizmo.blogspot.com/2006/07/practicing-multiculturalism.html

OK, back to work.

Kirby


Message was edited by: kirby urner



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