> But just the concentric hierarchy piece:  you've got these few polyhedra,
> they organize this way, and the relative volumes are like this.  It's not
> that big -- something we can share in K-12 no problem (and I do so).

So I walk into this classroom with a box of polyhedra.  Stiff paperboard
jobbers with one face missing, so they serve as mixing bowls, measuring cups
(a pre-K girl called 'em that, and it works).  We're in the kitchen,
measuring for a recipe (boys too).

In the bottom of my plastic carrying case:  about 3 inches of fava beans (I
think that's what they are, dried white beans -- I tried corn meal but that
was a disaster, got everywhere, in the floorboards even).

I pick up my tetrahedron (black electrical tape on the edges, good
contrast), and say "Measuring cup!" (I probably say more).  Then I scoop up
a boat load of beans.  Now it's full.  In my other hand, Cube.  How many
tetrahedron cups to fill my Cube?  Guesses, kids calling out.  Well, let's
see... one.... two.... three.  Done.  It's brimming with beans, no room for
more.  Is the ratio exact?  You betcha.

So stop right here.  How can this be?  There's the hidden (wrong in this
case) assumption that the cube edge and tetra edge must be the same length.
But that's not the only logical relationship.  You can intersect two
tetrahedra to get what Kepler called his Stella Octangula (a star with eight
points).  Connect those 8 points and you have a cube.  The tets inscribe as
criss-crossing face diagonals.  *That's* the relationship we want to burn
in.   *That's* how we get our 1:3 ratio.

So, did anyone ever teach you, in like 3rd grade, about this simple 1:3
relationship?  Do you need to study Klein or Coxeter to get that?  Do *they*
even tell you?  It's not that they wouldn't agree, but this is just simple
baby stuff, so why bother?  Except why don't we tell the babies even?

Moving on:  here's a rhombic dodecahedron.  Kepler loved it.  Did you know
it fills space without gaps?  Plato's pentagonal dodeca is more famous.  The
Platonic Five.  Heck, this rhombic guy ain't even an Archimedian.  But look,
trace the short diagonals (of its 12 diamond faces) with your finger:  you
get a cube.  Let's make that the *same* cube of volume 3, the *duo-tet*
cube, as Fuller called it.  So what's the volume of this guy then, this
other space-filler?  Pouring a cube into the rhombic dodeca.  One... two...
Done.  Cube was 3, so rh dodeca is 6.  Simple, easy, memorable -- and never
shared in any K-12 geometry book I've ever come across.

But wait, there's more.  Trace the *long* diagonals of our rhombic dodeca
with your finger.  An octahedron.  Dual to the cube (we'll explain what that
means later).  Let's find out the volume of this very octahedron.  We'll go
back to our tetrahedral measuring cup (the unit).  One... two... three...
four.  Done.  Four.  So what do we have so far:  

Tetrahedron  = 1
Duo-Tet Cube = 3
Octahedron   = 4
Rh Dodeca    = 6

But wait, there's more.  Remember Rh Dodecas fill space?  Imagine a sphere
or ball inside each one, perfectly encased (ball touches each diamond face
center -- where the cube and octa criss-cross).  Pack those babies together.
Hey, that's CCP (= FCC).  Kepler's Conjecture:  densest possible (finally
proved, 300+ years later).  CCP is a branch point to a whole other set of
cool topics.  Another time.  But hey we're there, ready to rumble.

OK, so 12 rhombic dodecas snuggling around a central one, corresponds to
some shape.  Which one?  Cuboctahedron.  I've lost you ASCII readers, but in
the classroom, this is all visual and/or tactile.  I've got ping pong balls,
I've got hypertoons (written in Python + VPython -- improved yet again as of
late last night).  And what's the volume of this 12-around-1 CCP embedded
cuboctahedron, dual to the rhombic dodeca?  Pour beans (your choice as to
which combo of measuring cups).  Answer:  20.  So...

Tetrahedron  =  1
Duo-Tet Cube =  3
Octahedron   =  4
Rh Dodeca    =  6
Cubocta      = 20

And yes, there's more.  I've got a simple transformation of the cubocta that
takes my to the icosahedron (incommensurable volume, and that's OK), and on
to the octahedron (volume 4).  Links to viruses, geodesic domes.  Pentagonal
dodeca comes back (yay Plato) as the dual of the icosa.  And the rhombic
triacontahedron, so very very close to volume 5 when shrink-wrapped around a
CCP sphere -- but that's for another day, another grade maybe.  Plus we can
break the tetra and octa into more basic slivers of equal volume and
assemble the rest of the not five-fold (other mods for them).

My point:  this is a very memorable spatial construct.  It could be in there
between the ears, along with the alphabet, the multiplication tables, and a
gazillion other factoids we ask kids to remember, to learn by heart.  But
this one is visual, right brained, spatial, geometric.  It addresses an
imbalance (so much of what we remember are lists, algebraic rules,
identities, with no pictures, especially not *spatial* pictures).  

If you commit this concentric hierarchy to memory, learn it in your bones,
you'll have something rational, wholesome, and fun to build on.  The
concepts embedded here connect everywhichway to other curriculum topics
(Pascal's triangle, great circles, crystals, octet truss, Kepler, Alexander
Graham Bell, cartography, navigation, spinning, Euler, coordinate systems,

>From my point of view, it's mind-boggling that we don't share the above
ultra basic information with kids, at any level in their academic careers.
Sure, *I* do, but why just me and a few others?  It's just shocking,
crazy-making.  The best retort mathematicians seem to come up with is:
trivial (sniff).  Well yeah, duh, it *is* trivial.  So is:  there's a tiger
behind you, ready to jump!  Why bother sharing that if it's so obvious,


Hypertoon about the concentric hierarchy (required viewing, K-12): (new improved version:
John Zelle made it sense if you're using 2.3 instead of 2.4, and implement
the necessary workarounds, plus defaults to not-fullscreen so is now
friendlier in Linux).

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