I've been encouraging teachers to get into the Radical Math business.
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...
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:
Students want to know about relevance. We're talking about a "mental 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.
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.
This workshop will improve as we develop more short video clips, geometry cartoons such as we already find on Youtube. Writing about 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.
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 that radical. Like "so what?" about the whole number volumes and 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 might, nevertheless. Donald Coxeter also links to this radome 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).
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 Rad Math teacher eh?
(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 was posting about recently. More about anxiousness, as it's really unfortunate for everyone that these breakthroughs in pedagogy have been shelved by the big name publishing companies. We're all much poorer as a result, both mentally and physically. That's why I'm out there recruiting, have no desire to hog all the limelight, believe you me.