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 kirby urner Posts: 2,578 Registered: 11/29/05
Posted: Apr 5, 2010 6:40 PM

These could be continuing education courses. One wouldn't have
to use that title, or the same trademarks, but it might be fun to get with
the program and do as I do sometimes.

One may also take it directly to high schoolers (or younger).

As one might expect, "radical" relates to the "radical sign" or surd symbol,
used to indicate the root of a number.

By default, the root is typically called a "square root" when it's a
2nd root, however we have some reasons for challenging that, as
a triangle with n intervals on a side ends up with (n x n) similar triangles
for area, when cross-hatched with a 3-way weave.

I've projected these many times, but for those new to this list...

http://www.rwgrayprojects.com/synergetics/s01/figs/f00412.html (4 x 4 = 16)
http://www.rwgrayprojects.com/synergetics/s01/figs/f00413.html (ditto)
http://www.rwgrayprojects.com/synergetics/s01/figs/f00412.html (ditto)
http://www.rwgrayprojects.com/synergetics/s01/figs/f00415.html (ditto)

Skew quadrilaterals and parallelopipeds don't stay self-similar upon
subdivision, not as cool:

http://www.rwgrayprojects.com/synergetics/s01/figs/f00301.html (boo hexahedra!)

Summary of "verboten math" (what rad math teachers aren't afraid to share):

http://www.rwgrayprojects.com/synergetics/s09/figs/f9001.html

That's part of what makes this math Radical [tm] (pun): we question
established dogmas, help students think in new ways. Most people think
of math as the epitome of conservative, which is why Radical Math
gets people queuing up around the block: they've never heard of
such a thing! Like a circus! (is Britney a sponsor?).

The signature content in a Radical Math teacher's bag of tricks is of
course a Concentric Hierarchy of Polyhedra (CHP). That comes across
as esoteric, a selling point, as mainstream K-12 currently avoids spatial
geometry almost entirely.

Our CHP is so-called because the polyhedra nest, one inside the other,
somewhat like Russian dolls although that's maybe not the best analogy.

We start with a tetrahedron as the innermost shape and intersect it with
itself to get the Stella Octangula (advert: 4dsolutions.net). Connecting
the eight tips begets us a Cube. The Cubes dual, the Octahedron, with
edges made to intersect the cube's, begets us another shape: the
Rhombic Dodecahedron.

Now here's the kicker (what makes this stuff radical). The volumes table
we use for this shapes is ultra simple, way more streamlined than anything
used in most colleges, which is why we have this market niche:

Tetrahedron: 1
Cube: 3
Octahedron: 4
Rhombic Dodecahedron: 6

But wait, there's more.

geometry" to go with "mental arithmetic": something to carry around in
your head and apply, any time spatial geometry might be what's up.
This could be in chemistry, architecture, art, other science, some
engineering application... we have lots of concrete examples. The idea
though, is what you're getting in this workshop is simple enough to
commit to the imagination and store there, as a permanent asset.

That's where storytelling comes in, as many people aren't familiar with
what's been happening in chemistry, since Linus Pauling especially.
Nor do they have but the vaguest ideas about nanotechnology. The
workshop might contain illuminating material in this regard.

There's stuff about sphere packing you'll want to share. Those rhombic
dodecahedra, studied intently by Kepler in the late 1500s, early 1600s,
are space-filling. If you put a "ghost sphere" in each one, such that
this touch at the 12 diamond face centers, then you'll have what
mathematicians call a Cubic Close Packing or CCP. Every ball is
surrounded by 12 others. What's more, these 12 others are centered
at the vertexes of what's called a Cuboctahedron. What's more is that
this Cuboctahedron weighs in with a volume of precisely 20.

http://www.rwgrayprojects.com/synergetics/s02/figs/f2230.html (20)
http://www.rwgrayprojects.com/synergetics/s02/figs/f2201.html (ball packing)
http://www.rwgrayprojects.com/synergetics/s04/figs/f1301.html (2nd
powering again)

Tetrahedron: 1
Cube: 3
Octahedron: 4
Rhombic Dodecahedron: 6
Cuboctahedron: 20

Going in the other direction, towards the smaller instead of the larger,
we smash the Cube into 24 outwardly identical pieces of volume 1/8.
We call this our "minimum tetrahedron" because (a) it's a space-filler
and (b) tetrahedra are minimal and (c) this shape requires no complement,
no left and right handed editions. It's fundamental in that sense.

http://mathforum.org/kb/message.jspa?messageID=7028526&tstart=0

This workshop will improve as we develop more short video clips,
spatial geometry in a purely lexical medium, such as here, tends to be
somewhat mind-boggling to many. The lexical needs to be abetted
by the graphical.

I have lots of graphics corresponding to all of the above, plus there's
lots more on the web.

What's so radical about any of this? Well, we need to compute some
edges and angles, do some trig. That surd symbol will definitely be the
there quite a bit. For example, the ratio of the short to long diagonal
on the rhombic dodecahedron's faces is 1:radical(2). The ratio of the
face diagonal to edge on our volume 3 cube is likewise radical(2):2
(same ratio).

What's also radical is we do some storytelling and that in itself marks
a radical departure from what goes on in most math classes today,
especially in K-12. History, the time axis, has been divorced from
technical content, meaning we're losing the lore. Math teaching today
encourages amnesia. Is this a best practice? I think not.

http://mathforum.org/kb/message.jspa?messageID=6990127&tstart=0

For example, even though Euler is credited for discovering V + F = E + 2
for polyhedra (vertexes + faces = edges + 2), we now know that
Rene Descartes made the same discovery, but was afraid of the
Inquisition, so instead of publishing his work he encrypted it.

Gottfried Leibniz later got hold of Descartes' notebook, and both
transcribed it and decoded it, but then the original notebook got
lost and all of this work got overlooked in the huge library Leibniz
bequeathed to posterity... until the mid 1800s.

Even then, only the transcribed notebook was discovered, not the
decoding.

Not until 1987 (!) do we finally learn that V + F = E + 2 was something
Rene Descartes (the "cogito guy") was also into, well before
Leonhard Euler.

Imagine how different history might have been, had Descartes not
so feared the wrath of Rome.

Source: Descartes' Secret Notebook by Amir D. Aczel, 2005

Speaking of the Inquisition, Rad Math teachers have the option to
bring in a computer language. I favor Python (as readers here well
know), but one might use Scheme or Ruby or Java or Perl. Python
was named for Monty Python, the comedy troupe, which did some
skits around the Spanish Inquisition (if you were wondering about
the segue here).

Going by this route will tend to turn your one day workshop (with a
lunch break) into an entire course. But sometimes that's what you
want, right? Use the workshop as a sampler, a smorgasbord,
and then go more deeply (spiral) in subsequent meet ups.

This may sound somewhat interesting and doable to math teachers
looking for new opportunities, but probably still doesn't sound all
easy fractions. "So what?" about the connection to the CCP (= FCC)
and thereby to the mineral kingdom (crystal lattices).

Well, I'd need to do more storytelling to convince you this is really
a radical math. Storytelling is itself pretty radical, but then what are
the stories?

Lets move closer to the present and talk about the Cold War. Could
we use our Radical Math to percolate out more of our shared history
since president Eisenhower?

What about that geodesic dome in Kabul, Afghanistan the Premier
Khrushchev liked so much? Why did USIS turn down the global data
display idea for Montreal '67? How do radomes fit in?

Returning to mathematicians more specifically, what about
Donald Coxeter, the King of Infinite Space, where does he fit in?
Yes, he studied with Wittgenstein in Cambridge but what of it?

Do Wittgenstein's investigations into mathematical foundations
allow us to question "squaring" and "cubing" as the only
interpretations allowed for 2nd and 3rd powering? I don't
think appealing to Wittgenstein is necessary here, but one
thread, and the dome in Montreal.

As you can see, we're into esoterica, which is why I've been
suggesting our more advanced teachers go for the cruise ship
milieu. People look for scintillating conversation on cruise ships,
want their cocktail party patter to be up to snuff. If you know a lot
of good stories, know how to lace them with math, then you gain
entre into these circles, are encouraged to mingle, give workshops,
project video clips.

However, fun as that may sound, the priority should be helping
front lines math teachers get a better handle on their discipline,
and this workshop is going to help with that. "Mental geometry"
is an idea whose time has come. You'll have an edge when it
comes to plane geometry too, and trigonometry, even graph
theory and topology (V + F = E + 2 is considered one of topology's
great beginnings, along with Descartes' angular deficit of one
tetrahedron's worth of degrees).

http://www.neubert.net/DESCarte.html

Remember, you heard it here first on math-teach, one of the
more avant-garde lists on the web (thanks to me etc.). Or maybe
you already know all this stuff, in which my case hat's off to ya (if I
weren't a Quaker (joke)) -- bet you're having fun as another

Kirby

(to be continued -- didn't get into the 5-fold symmetric polys
enough in this post, which we often get to by way of the
Jitterbug Transformation).

Also: there aren't really a lot of us, would be more fun if there
were. Not sure if this is about "anger and hatred" as Groves