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. > >Wayne. 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 reason.[2] 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! Kirby Curriculum writer Oregon Curriculum Network http://www.4dsolutions.com/ocn/ Notes [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 http://news.bbc.co.uk/hi/english/sci/tech/newsid_148000/148645.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:
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