Re: [Math 2.0] Re: Tessellations Again
Bradford Hansen-Smith
>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.
The problem I have with the Mite is that there is no Mite. The tetrahedron is pattern without scale, and yes, can be seen as a thing, has for centuries. My experience tells me it is always in dual form to itself in a functional inside/outside, right/left, or positive /negative context. It happens when folding the circle in half and then all way round in both directions; four points show twelve relationships defining eight triangles, four open and four solid in a reciprocal spherical movement. Without context of maximum, the minimum does not exist. This leads me to your next statement with which I fully agree
>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).
Quantum physics put an end to solid geometry with everything being relative, leaving little room for anything absolute (what is one without the other?). To regain importance of geometry without old baggage it must be conceptual aggressive in a contemporary perspective, more congruent with present experience and higher levels of understanding. The pattern and forms of functionality have not changed but interpretations have fallen into error through misinformation, meaning most of the contextual information that gives meaning is missing. We have lost it. Fuller was always looking ahead and reevaluating inheritance in light of birthright, what we presently know, tied to ideals of a distance vision that must by necessity start right now. On more than one occasion he said that if we can make a decision now, we then have all the time in the world to bring it about.
>> 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
We have been led to believe the tetrahedron is a static object sitting on a flat surface with a given set of properties, not as a dynamic 7-axial spatial pattern of movement ordering the arrangements foundational to the development of all other symmetries. It is now seen as an object that has been isolated from any context and just sits there as a three sided pyramid.
Brad
Bradford Hansen-Smith
Wholemovement
4606 N. Elston #3
Chicago Il 60630
www.wholemovement.com
From: kirby urner <kirby...@gmail.com>
To: mathf...@googlegroups.com
Sent: Fri, June 11, 2010 1:51:34 PM
Subject: Re: [Math 2.0] Re: Tessellations Again
On Fri, Jun 11, 2010 at 7:25 AM, Bradford Hansen-Smith
<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
>
>
> Bradford Hansen-Smith
> Wholemovement
> 4606 N. Elston #3
> Chicago Il 60630
> www.wholemovement.com
>
> ________________________________
> From: kirby <kirby...@gmail.com>
> To: MathFuture <mathf...@googlegroups.com>
> Sent: Fri, June 11, 2010 1:56:33 AM
> Subject: [Math 2.0] Re: Tessellations Again
>
> On Jun 10, 5:56 pm, kirby <kirby.ur...@gmail.com> wrote:
>> 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
>>
>> """
>
> Here is the screen shot again:
>
> 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
>
> --
> You received this message because you are subscribed to the Google Groups
> "MathFuture" group.
> To post to this group, send email to mathf...@googlegroups.com.
> To unsubscribe from this group, send email to
> mathfuture+unsub...@googlegroups.com.
> For more options, visit this group at
> http://groups.google.com/group/mathfuture?hl=en.
>
> --
> You received this message because you are subscribed to the Google Groups
> "MathFuture" group.
> To post to this group, send email to mathf...@googlegroups.com.
> To unsubscribe from this group, send email to
> mathfuture+unsub...@googlegroups.com.
> For more options, visit this group at
> http://groups.google.com/group/mathfuture?hl=en.
>
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To post to this group, send email to mathf...@googlegroups.com.
To unsubscribe from this group, send email to mathfuture+unsub...@googlegroups.com.
For more options, visit this group at http://groups.google.com/group/mathfuture?hl=en.
kirby urner
<wholem...@sbcglobal.net> wrote:
> Kirby,
>
>>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.
>
> The problem I have with the Mite is that there is no Mite. The tetrahedron
> is pattern without scale, and yes, can be seen as a thing, has for
> centuries. My experience tells me it is always in dual form to itself in a
> functional inside/outside, right/left, or positive /negative context. It
> happens when folding the circle in half and then all way round in both
> directions; four points show twelve relationships defining eight triangles,
> four open and four solid in a reciprocal spherical movement. Without
> context of maximum, the minimum does not exist. This leads me to your next
> statement with which I fully agree
>
You'll find said Mite diagrammed as Figure 10 in here:
http://tetrahedraverse.com/tverse/packings/temp2/tetraspace.pdf
Sommerville is studying the issue, of whether Aristotle was right
about a three-sided pyramid being a space-filler, not just the cube
(nevermind some other shapes we could mention). Aristotle was
indeed correct in this assertion, as I spell out in this journal entry:
http://controlroom.blogspot.com/2010/06/more-stress-tests.html
>>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).
>
> Quantum physics put an end to solid geometry with everything being relative,
> leaving little room for anything absolute (what is one without the other?).
> To regain importance of geometry without old baggage it must be conceptual
> aggressive in a contemporary perspective, more congruent with present
> experience and higher levels of understanding. The pattern and forms of
> functionality have not changed but interpretations have fallen into error
> through misinformation, meaning most of the contextual information that
> gives meaning is missing. We have lost it. Fuller was always looking ahead
> and reevaluating inheritance in light of birthright, what we presently know,
> tied to ideals of a distance vision that must by necessity start right now.
Cromwell in his 'Polyhedra' is good at documenting how the contemporary
notion of a polyhedron has drifted away from "solid" over the centuries to
something more gossamer and Web like (consistent with graph theory,
topology, and a "more with less" aesthetic).
By the time we reach Fuller in the late 20th century, a "tetrahedron" is
any four experiences chosen at random from one's life, and their six
inter-relationships.
Not that we can't still have those metal ones (which aren't "solid" either,
in light of atomic theory -- Fuller consciously aligns with Democritus
over Euclid when it comes to postulating "continua", as do many
writers in the digital / quantum age).
> On more than one occasion he said that if we can make a decision now, we
> then have all the time in the world to bring it about.
>
>>> 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
>
> We have been led to believe the tetrahedron is a static object sitting on a
> flat surface with a given set of properties, not as a dynamic 7-axial
> spatial pattern of movement ordering the arrangements foundational to the
> development of all other symmetries. It is now seen as an object that has
> been isolated from any context and just sits there as a three sided pyramid.
>
Koski, a geometer I work with, calls these static images "mug shots".
Later treatments often include, rather than replace, older ones (as Einstein's
equations contain Newtonian mechanics, differ in corner cases (like when we
start going fast)).
I'll trim the tail on this donkey, as the Google group preserves the thread.
Kirby
kirby
On Jun 12, 12:45 pm, kirby urner <kirby.ur...@gmail.com> wrote:
<< SNIP >>
> Sommerville is studying the issue, of whether Aristotle was right
> about a three-sided pyramid being a space-filler, not just the cube
> (nevermind some other shapes we could mention). Aristotle was
> indeed correct in this assertion, as I spell out in this journal entry:
>
> http://controlroom.blogspot.com/2010/06/more-stress-tests.html
>
essay where I mention mathfuture, but don't actually provide
a link directly to this thread. Others might want to put these
links together in something related -- lots of curriculum writing
still needed in this area.
Here's the little essay, for which I've already received some
positive feedback:
Aristotle was Right!
http://mathforum.org/kb/thread.jspa?threadID=2084375&tstart=0
Kirby
kirby
> Kirby,
>
> >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.
>
> The problem I have with the Mite is that there is no Mite.
> From: kirby urner <kirby.ur...@gmail.com>
> Sent: Fri, June 11, 2010 1:51:34 PM
> Subject: Re: [Math 2.0] Re: Tessellations Again
>
> On Fri, Jun 11, 2010 at 7:25 AM, Bradford Hansen-Smith
> > 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)
>
> I'm impressed by your cube necklace. Reminds me of Yoshimoto Cube.
> Well done.
Sytes (= Mite + Mite) in a necklace.
"Home position" is a cube of 12 Sytes, each a pair of Mites
(emphasized
by the color scheme):
http://www.flickr.com/photos/17157315@N00/4709499223/in/photostream/
Unfold into a necklace and involute the torus....
http://www.flickr.com/photos/17157315@N00/4710141824/in/photostream
and you get a rhombic dodecahedron (Kepler's polyhedron):
http://www.flickr.com/photos/17157315@N00/4709509937/in/photostream/
In our spanking new volumes table, said Mite has volume 1/8,
Syte 1/4, Cube (home position) volume 3, and rhombic dodecahedron
volume 6.
Most curriculum writing isn't up on this rational / whole numbers
approach, which depends on using a tetrahedron as a model of
3rd powering (versus a cube).
I've been in touch with editors of the Encyclopedia of Mathematics
and Society, thinking what a scandal it might be if their article
on Polyhedra bleeps over this core innovation in pedagogy.
We shall see.
Kirby