Original posted to AMTE listserv, April 10, 2000. Some typos fixed, italics added, hyperlinks activated. For the original thread, and more context, click here.

Date: Mon, 10 Apr 2000 11:42:15 -0700
To: amte@esunix.emporia.edu
From: Kirby Urner <pdx4d@teleport.com>
Subject: Re: The count down continues (less than a week to go)

>This motivated me to inquire of my son, who is finishing his Ph.D. 
>in Chemistry at Princeton this summer, as to whether he thought 
>this premise as ridiculous as do I.  He'll be the lead author on 
>an article in Nature in a few months but I doubt if it mentions 
>buckyballs so feel free to discount it in advance. Here is his 
>response with attribution authorized.  Seems that Greg had
>professional insight.

Thanks for sharing Tony's analysis.  He confirms that buckyballs 
were a huge breakthrough in chemistry -- not something I'd dispute
of course.  Then he passes the buck back to you, saying he 
doesn't really see the tie-backs to math ed.  

Greg and yourself are similarly clueless I'm sorry to say (it's 
more forgivable in Tony's case, as he's not a math ed specialist).  
That's about where we started, although we now know that your 
son is heavy into chemistry at Princeton (my alma mater as 
well -- philo department, Rorty as thesis advisor, Class of 1980).

What I said in my more recent post is that I consider buckyballs
something of an indicator species ("canary in a mine shaft").  
They represent such a golden opportunity to tie together 
important math concepts (K-12 level) and I'm suspicious of 
1990s texts which do nothing to exploit them ("asleep at 
the switch" is a phrase I've used elsewhere).

I think we can safely assume that texts which do nothing with 
buckyballs probably bleep over the concentric hierarchy as well.  
Whereas not including the former is perhaps acceptable (though 
hard to fathom, given the golden opportunities), not sharing 
the latter in early math education is competely unacceptable, 
carries a curriculum below any reasonable minimum standard.  
No A and B modules?  Throw it in the river.

Of course buckyballs are so-named because they resemble the
geospheres explored by Bucky Fuller (USA Medal of Freedom --
see USA History 101 curriculum [1]). Some people call that 
golf-ball looking thing at EPCOT the "BuckyBall" for the same 

The microarchitecture of the virus is likewise connected, as 
discussed by Coxeter (geometer) after an earlier chapter in 
which Casper-Klug approached Fuller with their new X-ray 
diffraction data asking for an explanation for capsomere 
counts.  Fuller came back with 10 ff + 2 -- where f = number 
of intervals between capsomeres along an icosahedral edge.  
This formula was published on the front page of the New York 
Herald Tribune at the time.[3]

10 ff + 2 represents a series.  Starting with f=2, it goes
12, 42, 92... giving the number of spheres in each concentric
layer, as we close-pack spheres around a nuclear sphere in
a cuboctahedral conformation.  This is the so-called face-
centered cubic packing, and is what Kepler conjectured is
the densest possible (Gauss proved it's the densest _lattice_
possible, but what about some more "random" packing?)[4].

That's all fine and good, I suppose you're thinking, but
sphere packing isn't something we normally take up in the 
early grades.  Exactly my point.  Why not?  Why does early 
math ed usually just start with the cube, pyramid (meaning 
half-octahedron), cone, cylinder, rectangular and triangular 
prisms, and "three sided pyramid" (tetrahedron). "For no
good reason" is the answer.  Some early math mentions 
the Platonic Five.  Does the California Standard require 
that kids know what these are I wonder?  Just an aside.

Four spheres tightly packed together define a tetrahedron.
Suppose we make this a unit of volume, as well as a model
of 3rd powering (most math texts entirely skip mentioning
that using the square and cube as 2nd and 3rd powering
models is not the only way to go -- missing an important
opportunity to keep minds limber and open to different
conventions (what helps keep math from becoming mere 
dogma, with teachers harping on the "one right way" to
think about anything)).

The tetrahedron divides evenly into a cube with the same
diagonal (volume 3), octahedron (same edge, volume 4), rhombic 
dodecahedron (same long diagonals = octahedron, cube = 
short diagonals, volume 6), cuboctahedron (volume 20).  
Let me show that again, in a table:

   Shape             Volume
   Tetrahedron         1
   Cube                3
   Octahedron          4
   Rh Dodecahedron     6
   Cuboctahedron      20

Hey, pretty simple.  This is part of what I'm calling the
concentric hierarchy.[5] These shapes develop in a sphere 
packing matrix (lattice).  That volume 20 cuboctahedron is 
the same we get from packing 12 spheres around a central one 
(12, 42, 92...).  The lattice so-defined consists of tetrahedral 
and octahedra voids (this is the octet truss, pioneered by 
Alexander Graham Bell in the form of towers and kites [6] 
-- more USA History 101) of relative volume 1:4 and relative 
frequency of 2:1 (twice as many tetrahedra). 

And notice the duals: tetrahedron is self-dual, cube and 
octahedron are dual, rh dodeca and cubocta are dual.  Plus 
the tetra and its dual define the cube, the cube and it's 
dual define the rh dodeca.  The fact the our rhombic 
dodeca is a space-filler, and contains the unit radius
spheres we used to build our lattice in the first place,
was very important to Kepler (of "Kepler's Laws" fame).
Does the California Standard require that kids know 
anything about space-fillers I wonder?  If so, does it
include the primitive irregular tetrahedron which fills
space.  What is this called?[7]

The tetrahedron is the topologically minimum enclosure
that we can make using match sticks, is the "box with the 
fewest sides".  Calling it a "three sided pyramid" is dumb.  
Why is one of 3 identical sides a "base" and why is 
"pyramid" defaulted to a pentahedron, with the tetrahedron 
only filed as a special case of the above (i.e. "three 
sided").  Higher math calls it a "simplex" with good reason.

Look at this again:

   Shape             Volume
   Tetrahedron         1
   Cube                3
   Octahedron          4
   Rh Dodecahedron     6
   Cuboctahedron      20

Is this college level?  I don't think so. I call it "baby
math" (what anyone should know).  I shared it successfully 
with pre-first graders at the local Montesorri[8], have 
field tested plenty (others too).  Really super-basic stuff.  
Plus you can build all of the above out of two tetrahedral 
modules called the A and B, both with identical volume 
of 1/24 the tetrahedron.[9]  Very simple math (useful 
trig:  figure out the plane nets).  

2 As and 1 B make the minimal tetrahedral space filler 
I was talking about. Kids should know that (many of them 
do -- not like we just started teaching this stuff 
yesterday).  Pretty basic (don't wait for them to 
encounter page 71 in Coxeter's 'Regular Polytopes' -- 
it probably won't happen).

What does this have to do with buckyballs?  Remember 
10 ff + 2 gives the number of spheres in the outer layer
of an icosahedron (your typical virus is icosa shaped),
the same as the number of spheres in the corresponding
layer of the cuboctahedron.[10]  This is easy to 
demonstrate visually, and our DVD jukeboxes will have 
lots of clips about this.

A 3-frequency icosasphere (10 ff + 2, f=3, so S = 92) is 
a buckyball with the hexagons and pentagons omnitriangulated.  
Given buckyball = C60, i.e. has 60 carbon atoms, we're 
adding 92 - 60 = 32 spheres (atoms).  Given the buckyball 
has 12 pentagons (same as any icosasphere), it must have 
30 hexagons (as we've added a sphere to the center of 
each face). 

Euler's Law comes in here:  V + F = E + 2 (every 4th
grader should know it -- part of the California Standard?
-- 3rd graders had no problem with it in Lesotho or 
Bhutan (where I've tested a lot of these concepts)). 
Basic exercise: Figure out how many edges in the 3-frequency 
icosa, how many carbon-carbon bonds in the buckyball. The 
tie-in was 10 ff + 2 (and other fullerenes may have a higher 
f, i.e. more atoms than 60).[11]

Is this high level math?  Not at all.  The series 2,12,42,92... 
is just another number series (icosa shell numbers) and 
should be introduced right along with triangular numbers 
1,3,6,10... and square numbers 1,4,9,25...  and tetrahedral 
numbers 1,4,10,20,35.... There's a simple visual proof that 
every square is the sum of two triangular numbers:

* * * * * *            * * * * * *     *   * * * * *   *   * * * * *
* * * * * *          * * * * * *     * *   * * * *    * *   * * * *
* * * * * *        * * * * * *     * * *   * * *     * * *   * * *
* * * * * *  =   * * * * * *  =  * * * * + * *   =  * * * * + * * 
* * * * * *    * * * * * *     * * * * *   *       * * * * *   *
* * * * * *  * * * * * *     * * * * * *          * * * * * *

 SQR(N)                                          TRI(N) + TRI(N-1)

That's 4th grade stuff.  Gauss was able to prove the that 
the sum of consecutive integers 1..N = triangular number 
N = N(N+1)/2 when he was 7 years old.  Kids should go over 
his thinking here -- not hard to follow.[12]

Using triangular and square numbers, you can prove 10 ff + 2 -- 
given the cuboctahedron consists of nothing but squares and 
triangles -- or just use triangular numbers, given the 
squares each morph into triangles, as per the transformation 
I was mentioning (in the DVD jukebox).  

Coxeter himself (probably greatest geometer of the 20th century) 
said any high schooler should be able to prove it, given 
some clues.  Definitely this is a part of the standard, 
given how important is 10 ff + 2 both in the sphere packing 
world (metal clusters, crystallography), and in the geodesic 
architecture world -- which includes naturally occurring 
microarchitectures, like the fullerenes and the virus.[13]

Show how both triangular and tetrahedral numbers appear as 
columns of Pascal's Triangle (Math Forum has a great applet 
about this).[14]  From Pascal's Triangle, you link to 
Newton's Binomial Theorem (which he tackled as a young boy), 
to the expansion of (a+b)^n, to random walks ala Pachinko 
(gives the Gaussian distribution -- a well-known science 
museum exhibit), to the Fibonacci numbers (also found in 
Pascal's triangle), to Phi (as the limit of Fib(n+1)/Fib(n)), 
to the geometry of phi, which is five-fold symmetric e.g.
the icosa.  

In fact, going from the cubocta to the icosa is a matter 
of seeing 3 mutually orthogonal root(2) x root(2) squares 
morphing into 1 x phi rectangles (more vid clips).  Note 
that Pascal's Triangle is visually a triangle, developed 
from a triangular packing of spheres -- easy segue.

The above hyperlinks are already standard in many math texts.  
I'm just hooking up an existing web to the concentric 
hierarchy (which has been missing).  Given the whole number
volumes and how easy it is to pack spheres (physically -
all this stuff modelable, manipulable, experiential, 
hands-on) none of this has to wait until college -- very
poor design to put it off that long (oh wait, they don't
usually teach the concentric hierarchy in college either).

Note that all of this spatial sense was so far developed 
without coordinate algebra (no XYZ, no vectors).  But once 
we get to vectors (in 7th/8th grade), we have ample tie-ins, 
and lots of computer work lined up (including ray tracing).  
Plus we introduce quadrays (before the end of high school), 
to keep student minds limber.[15]

When will all this happen?  It's happening now, with the 
web as curriculum driver.  Of course it's not what's in 
the early math ed text books.  Too bad for the text books, 
as this is clearly a healthier diet of inter-related concepts 
and computer skills (I think so as a parent too).  

The geometry is "beyond flatland" but still requires the 
Euclidean-style reasoning.  Trig and algebra have natural 
segues to spatial geometry and back.  The polyhedra are
restored to front and center (instead of "back of the 
book, probably never get to 'em"), plus we have the 
opportunity to introduce them as objects (in the data + 
methods sense), to develop matrix and vector algebra in 
the context of rotating and translating them (a core topic 
in math, far more core than just solving "simultaneous 
equations" -- which often accounts for the bulk of matrix 
algebra in high school, if they're used at all (so lame)).[16]

We should not ask kids to pay the price for adult sluggishness
when it comes to overcoming inertia and poor design in the
culture.  We should simply bypass those adults and give 
kids access to the "right stuff" in a hurry.  That's what 
my corporate sponsors and I are up to.  There's no time left 
to wait for text book publishers to get a clue.

Time's up!

Curriculum writer
Oregon Curriculum Network


[1]  http://www.teleport.com/~pdx4d/grunch.html
[2]  http://www.teleport.com/~pdx4d/domegeo.html
[3]  http://www.teleport.com/~pdx4d/virus.html
[4]  http://news.bbc.co.uk/hi/english/sci/tech/newsid_670000/670429.stm
[5]  http://www.teleport.com/~pdx4d/volumes.html
[6]  http://www.teleport.com/~pdx4d/bell.html
[7]  http://mathforum.com/epigone/geometry-research/swelddwunnan
[8]  http://www.egroups.com/message/synergeo/236
[9]  http://www.teleport.com/~pdx4d/modules.html
[10] http://www.4dsolutions.com/ocn/numeracy0.html#jitterbug2
[11] http://mathforum.com/epigone/k12.ed.math/stitulveh
     (note: 65 million, not 650 million years before the present)
[12] http://www.4dsolutions.com/ocn/numeracy0.html#gauss
[13] http://www.4dsolutions.com/ocn/xtals101.html
[14] http://forum.swarthmore.edu/workshops/usi/pascal/mo.pascal.html
[15] http://www.teleport.com/~pdx4d/quadintro.html
[16] http://www.4dsolutions.com/ocn/trends2000.html

For further reading:

Synergetics on the Web
maintained by Kirby Urner