Tessellations Again
kirby
Here's a link to a Math Forum post from earlier today:
http://mathforum.org/kb/message.jspa?messageID=7094310&tstart=0
Note the link to ConceptNet, a cool (and free) MIT resource.
However (a sign of the times?), the assertion about the tetrahedron is
false.
And I quote:
"""
Speaking of our geometry junkyard, check out this
screen shot from ConceptNet out of MIT. Do you see
the error (two of them actually):
http://www.flickr.com/photos/17157315@N00/4680326861/
http://csc.media.mit.edu/docs/conceptnet/overview.html
"""
Ask a student today how many sides a cube has, and you might
get "four" as an answer.
At the very time when screens are getting bigger and brighter,
some turning to stereo, we allow spatial geometry to go out of
style.
Rhombic Dodecahedra anyone?
http://mybizmo.blogspot.com/2010/06/outlining.html (Martian Math)
Kirby
RadMath!
usa.or.pdx
http://www.flickr.com/photos/17157315@N00/4585728237/in/photostream/
Bradford Hansen-Smith
>Yeah, need to shed this image that spatial geometry is
>just for uber-geeks. Rather, it's more experiential and
>close-to-home than pretty much any other math, because
>we live and breath in a spatial setting defined by corner-like,
>edge-like and facet-like features (V, E, F). It's "the math
>we live within". Polyhedra are everywhere (starting with
>our own bodies).
Yes we do live in a spatial setting. Geometry is the experiential base for all math, though little acknowledged. No our bodies are not polyherda. From the DNA on up we are all ordered to the tetrahedron as the only stable and most compliant structural pattern there is; otherwise there would be no accommodation for dynamic changes and interactions necessary for a living, growing bio-electrochemical organization of consciousness through generations of biological human development. As for polyhedra, are bodies are not, but through various patterns of movement of the body we can assume any number of discernible geometric positional arrangements, and of course the observation of ratios and proportions of body parts within a generalized envelope.
Error is often not so much about understanding, but of misunderstanding, and language is sloppy by nature and our off-handed way of making connections leads to sloppy thinking.
As for your passion about Geometry, I love it.
Brad
Wholemovement
4606 N. Elston #3
Chicago Il 60630
www.wholemovement.com
From: kirby <kirby...@gmail.com>
To: MathFuture <mathf...@googlegroups.com>
Sent: Thu, June 10, 2010 7:56:42 PM
Subject: [Math 2.0] Tessellations Again
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kirby urner
you find that offensive. The grammar hasn't been nailed down, don't
have strict rules (wouldn't call it "sloppy" necessarily, to have leeway
-- more of that flexibility you tout (it extends to our human expressions)).
However, of all the math objects I'm familiar with, polyhedra would come
closest. Certainly an artist could sculpt a multifaceted object, that,
even when animated, had the "polyhedral" look at feel -- those "wire
mesh" people from the computer world (e.g. 'Tron'). But I'm comfortable
calling that "an analogy" (not a sloppy on, suitable for many a precise
model).
Pure Platonists will say things like "no polyhedra really exist in nature
because polyhedra are eternal forms and have no specialcase material
existence". That's trudging into philosophy at that point. I'm somewhat
a post linguistic turn Wittgenstein guy in that case (did my thesis on
his later thinking at Princeton, Rorty an advisor -- much more posted
to Sean's Wittgenstein list).
Thanks for writing. Sure, I get passionate about geometry. As we
seem to agree, it's "where we live".
Kirby
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kirby
> Here's a link to a Math Forum post from earlier today:
>
> http://mathforum.org/kb/message.jspa?messageID=7094310&tstart=0
>
> Note the link to ConceptNet, a cool (and free) MIT resource.
>
> However (a sign of the times?), the assertion about the tetrahedron is
> false.
>
> And I quote:
>
> """
>
> Speaking of our geometry junkyard, check out this
> screen shot from ConceptNet out of MIT. Do you see
> the error (two of them actually):
>
> http://www.flickr.com/photos/17157315@N00/4680326861/http://csc.media.mit.edu/docs/conceptnet/overview.html
>
> """
http://controlroom.blogspot.com/2010/06/more-stress-tests.html
Re: IsA(tetrahedron, four polygon equal)
Answer key:
Error 1: a tetrahedron is not a polygon
although it is the only polyhedron comprised
of just four polygonal facets.
Error 2: these facets need not be equal.
Now maybe this IsA(tetrahedron, four polygon equal) is saying
a tetrahedron is comprised of four equal polygons. That'd be
only half the mistake then ("equal" still too restrictive).
I'd need to see what it says for a Cube -- although in that
case more than one polyhedron consists of six equal polygons
(there's a dipyramid...).
We find very few associations for Tetrahedron in general in
ConceptNet. No "simplex". Kinda scary, to think this
might be a mirror of our collective semantic web. Our
spatial geometry is so weak!
However, in Math 2.0, we could build our own ConceptNets,
and purposely invest them with lots of associations,
connotations. The Web is already playing that role at
some level, but it's fun to use more specialized tools
sometimes, such as Maria knows about.
My own geometrical subculture has tetrahedra called
Mite and Sytes (Syte = Mite + Mite), also A, B, T and
some others (Mite = AAB) -- canonical "blocks" in a
language game of assemblies and dissections, relative
volumes. I'm alluding to published literature, not just
some ultra-esoteric board game or science fiction.
Practically no one teaches this stuff though, outside
of certain math circles. We live as outlaws.
The Cuisenaire rods of Caleb Gattegno fame, far better
known, likewise give that "right brain" experience (spatial,
graphical). It's so important to find those bridges, twixt
lexical and graphical content. Those colored rods are
far more rectilinear than ours however, and reinforce
the dominant ideas of "box" and/or "blocks".
I've cast our more tetrahedrally based ethnicity in the
form of so-called Martian Math (science fiction flavored).
I've got it linked to some school Web pages here and
there. For example:
http://worlduniversity.wikia.com/wiki/Mathematics#Select_Wikis
http://worlduniversity.wikia.com/wiki/Quakers_-_Religious_Society_of_Friends#Select_Wikis
Sometimes we make fun of how the mainstream culture
seems so ignorant of the Tetrahedron, can't bring itself to
use the word, has to say "three sided pyramid". Even
NASA does this.
Kirby
Bradford Hansen-Smith
As you stated....
>Aristotle was right by the way, tetrahedra *do* fill
>space. According to this recent source, he never
>said "regular":
> >
> > "Which Tetrahedra Fill Space?"
> > author Marjorie Senechal,
> > Mathematics Magazine, Vol.54, No.5, Nov. 1981
> > (pp 227-243)
"... Aristotle did not state explicitly that he meant regular
tetrahedra..."
I do not understand the importance of all-filling space objects (a static concept) when space is necessary for anything to grow, move, develop, change, etc. People have a strange idea that space needs to be filled. It goes along with thinking of space as commodity to buy and sell. Where is the sense of all this?
Attached you will find pics of a 1985 model I made to demonstrate one way of dividing the cube into a tetrahedra necklace. You might find it interesting. This is not an easy puzzle to reassemble, but educational when thinking about space-filling.
Brad
Bradford Hansen-Smith
Subject: [Math 2.0] Re: Tessellations Again
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kirby urner
<wholem...@sbcglobal.net> wrote:
> Kirby,
>
> As you stated....
>
>>Aristotle was right by the way, tetrahedra *do* fill
>>space. According to this recent source, he never
>>said "regular":
>
>> >
>> > "Which Tetrahedra Fill Space?"
>> > author Marjorie Senechal,
>> > Mathematics Magazine, Vol.54, No.5, Nov. 1981
>> > (pp 227-243)
>
> "... Aristotle did not state explicitly that he meant regular
> tetrahedra..."
>
> I do not understand the importance of all-filling space objects (a static
> concept) when space is necessary for anything to grow, move, develop,
> change, etc. People have a strange idea that space needs to be filled. It
> goes along with thinking of space as commodity to buy and sell. Where is the
> sense of all this?
Hi Brad --
I'm impressed by your cube necklace. Reminds me of Yoshimoto Cube.
Well done.
As to all-space-filling objects, they're a part of spatial logic and even
nature has lots of frozen static geological forms (many of which are
bought and sold, I can't deny it -- was happening long before I got here).
I've been harping on this Mite (minimum tetrahedron) in the last
few weeks, having held a contest, timed with annual Portland's
Rose Festival, for a Minimum Space-filler.
I was looking for the minimal shape that'd fill space alone, without
mirror images (left and right versions). The Mite was already the
answer I had in mind but maybe some judges would come out of
the woodwork and tip the scales another way? We had some
lengthy debates on another geometry list.
Having been to a scientific lecture on the naming and categorization
of planets by the Vatican's chief astronomer some years ago, I know
that nomenclature is an important business -- the Vatican takes it
very seriously.
Mathematics, touting itself as a "universal language" is nevertheless
full of nooks and crannies where esoteric nomenclature pertains.
In this tiny subculture that cares about polyhedra, from several
angles, it makes a difference if we call our minimum space-filler
a Mite or not.
Mites make Sytes which make Kites (not the same as the
Penrose kite but there's room for qualified meanings).
Although the LA Times says there's "no magic bullet" for fixing
math education **, I'd like to propose that the Mite is our magic bullet.
I'm only being somewhat tongue in cheek, as I think spatial
geometry should and could be a part of an education Renaissance,
should we decide to pull ourselves together and plan for the future
more seriously (understood that many have given up on doing
that).
What makes the Mite / Syte / Kite nomenclature attractive is
it comes with a more complete set of polyhedra organized in
a table with many more whole number and/or rational volumes
than are usually conveyed, streamlining the whole subject
immeasurably.
For example, our Mite has volume 1/8 while our rhombic dodeca-
hedron (another space-filler) has volume 6, and our cube
(another space-filler) volume 3 and so on.
The regular tetrahedron, which you alluded to going back a
post, has a volume of unity. That's part of what's innovative
about the Montessori-compatible set of shapes...
From my point of view, we're sitting on a gold mine of
mostly unshared spatial geometric lore, part of our
collective heritage. That's probably a metaphor you
don't like though (gold mine) as it connotes buying and
selling. I understand your qualms.
** http://articles.latimes.com/2010/may/30/opinion/la-ed-eval-20100530
>
> Attached you will find pics of a 1985 model I made to demonstrate one way of
> dividing the cube into a tetrahedra necklace. You might find it interesting.
> This is not an easy puzzle to reassemble, but educational when thinking
> about space-filling.
>
> Brad
Looks to be. Pondering those pictures...
Kirby
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