[Edu-sig] python satacad: class 3

[Kirby Urner]

OK, regarding my 3rd class in this Saturday Academy thread, I'm gonna be
lazy, sort of, and paste something from another list (with a closed archive,
which is why I'm not *just* putting a link).

http://groups.yahoo.com/group/math-learn/message/7177

Note this is a *math* class (per catalog description), not just a class
about learning to program, so a lot of what follows is very mathematical in
nature.  

It also has the usual conversational layering.  I'm following up on a post
of my own, so zero and one pointer (>) is me, whereas two pointers (>>) is
someone else.

===============

From:  "Kirby Urner" <urnerk at q...>
Date:  Sat Jan 29, 2005  8:57 pm
Subject:  Re: Inching along


--- In math-learn at yahoogroups.com, "Kirby Urner" <urnerk at q...> wrote:
>
> --- In math-learn at yahoogroups.com, "Marie Bahlert" <mbahlert at m...>
> wrote:
>
> > OK - Uncle!!!
> >
> > What is 'octet truss' and 'concentric hierarchy'?
> >
> > I can't shake the feeling that once I do know what they are it
> > won't bother that I didn't.
> >
> > Marie
>
> Octet truss is a space frame used in architecture (once you know
> what to look for, you'll see it everywhere), but is also a spatial
> grid, like the XYZ cubes, and corresponds to a sphere-packing
> arrangement known as the CCP and/or FCC. That sounds too technical
> for middle schoolers of course, but (a) it's easily graspable using
> animations or cartoons and (b) there're a lot of fun off-shoots,
> e.g. into architecture as a mentioned, where Alexander Graham Bell,
> the telephone guy, was one of its chief pioneers (also
> crystallography, virology, and polyhedral numbers ala 'The Book of
> Numbers' by Conway and Guy).

OK, so some hours have passed, and in the meantime I covered a lot of
this material in a class for 9th and 10th graders. My room has a
ceiling mounted projector, so it was no problem to dial up some of the
relevant content e.g. at:

http://www.4dsolutions.net/ocn/numeracy0.html

we have the triangular numbers, phasing to the tetrahedral numbers,
then a discussion of the cuboctahedral and icosahedral numbers (same).

The tetrahedral and cuboctahedral packings are both FCC/CCP, so the
link to Bell's octet scaffolding was fairly clear. I pulled up my
bell.html and pointed out the date: 1907 -- hey, look at that horse
and buggy. This stuff is *old*.

Plus I had, as manipulatives, a set of properly proportioned
polyhedra, ping pong balls, glued in a set of 6-around-1, 3 and 3 (to
give the CCP or HCP seeds). Other stuff.

Plus we had a Python command line, so we could build on last week's
excursion into core Python syntax, and start generating sequences of
these figurate numbers. I showed 'em list comprehension syntax.

Plus we had Sloane's Encyclopedia of Integer Sequences on the web,
giving us an easy way to dial in some of our Python output, and pull
up interesting background materials.

To take an example:

>>> def cubocta(n):
........if n==1: return 1
........return 10*(n-1)**2 + 2

(the periods shouldn't be there, but I'm trying to foil the Yahoo web
interface, which is cavalier with significant whitespace).

>>> cubocta(1)
1
>>> cubocta(2)
12
>>> [cubocta(n) for n in range(1,21)]
[1, 12, 42, 92, 162, 252, 362, 492, 642, 812, 1002, 1212, 1442, 1692,
1962, 2252, 2562, 2892, 3242, 3612]

OK, so let's paste that sequence into the encyclopedia:
http://www.research.att.com/~njas/sequences/

And here it is (takes a few seconds); I'm reformatting a bit to
accommodate the Yahoo interface:

=========================
ID Number: A005901 (Formerly M4834)
URL: http://www.research.att.com/projects/OEIS?Anum=A005901
Sequence: 1,12,42,92,162,252,362,492,642,812,1002,1212,
1442,1692,1962,2252,2562,2892,3242,3612,4002,4412,4842,
5292,5762,6252,6762,7292,7842,8412,9002,9612,10242,10892,
11562,12252,12962,13692,14442,15212,16002

Name: Points on surface of cuboctahedron (or icosahedron):
a(0) = 1, for n > 0, a(n) = 10n^2 + 2 (coordination sequence
for f.c.c. lattice).

References

H. S. M. Coxeter, ``Polyhedral numbers,'' in R. S.
Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974,
pp. 25-35.

Gmelin Handbook of Inorg. and Organomet. Chem., 8th Ed., 1994, TYPIX
search code (225) cF4

R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster
Fuller. Anchor, NY, 1973, p. 46.

S. Rosen, Wizard of the Dome: R. Buckminster Fuller; Designer for
the Future. Little, Brown, Boston, 1969, p. 109.

B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and
polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

Links: Index entries for sequences related to f.c.c. lattice

J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices
VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997),
2369-2389 (Abstract, pdf, ps).

R. W. Grosse-Kunstleve, G. O. Brunner and N. J. A. Sloane, Algebraic
Description of Coordination Sequences and Exact Topological Densities
for Zeolites, Acta Cryst., A52 (1996), pp. 879-889.

R. W. Grosse-Kunstleve, Coordination Sequences and Encyclopedia of
Integer Sequences

N. J. A. Sloane, A portion of the f.c.c. lattice packing.

K. Urner, Microarchitecture of the Virus

Program: (PARI) a(n)=if(n<0,0,10*n^2+1+(n>0))

See also: Cf. A004015.
Adjacent sequences: A005898 A005899 A005900 this_sequence
A005902 A005903 A005904

Sequence in context: A022672 A085798 A045945 this_sequence
A090554 A009948 A007586

Keywords: nonn,easy,nice
Offset: 0
Author(s): njas, R. Vaughan
=========================

Hey, did you catch my name in all that? There's a link to one of my
web essays. I'm proud to be in such august company, and as a
Princeton philosophy alum at that (philo overlaps with math of course,
e.g. I'm pretty up on my Wittgenstein, author of 'Remarks on the
Foundations of Mathematics' and related works).

> The concentric hierarchy fits into the octet truss, in that each
> ball in the CCP may be considered nested inside a space-filling
> rhombic dodecahedron of volume 6.

I usually plug Kepler at this point. We also got into Euler's Law for
Polyhedra (V + F = E + 2), same as I do with the 2nd and 3rd graders.
We counted up these topological features and did a chart on the
whiteboard -- standard Fuller School content.

> Relative to this volume, the octahedral and tetrahedral voids
> defined by CCP spheres have volumes 4 and 1 respectively -- nice
> whole number volumes (you can get the XYZ cubes into this too,
> with volumes 3).

Of course in XYZ we want said cubes to have volume 1, not 3, but
really, given the face diagonals are 2, we want root(2) to the 3rd
power for these volume 3 guys, hence a candidate conversion constant
(as between the royal and metric systems) -- not the kind of detail I
needed to get into in my class today, but a few words on math-learn
can't hurt.

> Finally, the cuboctahedron, defined by 12 balls packed around a
> nuclear one (the basis of the CCP -- from there you may keep packing
> outwardly: 1, 12, 42, 92, 162 balls etc.)

Yeah, we just did this. Thanks to the Jitterbug Transformation, we
have a way of showing how cuboctahedral and icosahedral shells contain
the same number of balls, per aforementioned encyclopedia entry.

> has a volume of precisely 20. I show all this by pouring beans
> between polyhedral mixing cups; in front of 2nd and 3rd graders (or
> pre-school Montessori).
>

Didn't get to the beans this time. Maybe next week.

> You may think this all sounds very college, very liberal arts, and
> you'd be right, but we're dropping ladders to middle schoolers, so
> they can start climbing past their clueless NCTM teachers. This is
> the 21st Century after all.
>

But you wouldn't know it from randomly dropping in on our nation's
math classes. Very 20th Century, or maybe 19th?

Hey, did you know that Pascal's Triangle fits in here really well?
You've got both the triangular and tetrahedral numbers going down
adjacent columns. That's an opportunity to link in some of these
combinatorics we've been discussing, ala the bionomial theorem. Cite
'The Book of Numbers' again. Then there's Pascal's Tetrahedron and
Half-Octahedron.

> Here're a link, apropos the above, to a page about Bell:
> http://www.grunch.net/synergetics/bell.html
>
> Kirby
>
> PS: I'll actually be going over much of this material with 9th and
> 10th graders later today, in the course of an extracurricular
> enrichment class that's also heavy into programming. I have a
> whole box of manipulatives, and plenty of graphics (included
> animated ones) to project off the web, so I don't expect them to
> be unclear about any of this, a few short hours from now.

Well, I've showed you some of the web pages I projected (did you catch
that animated GIF of the growing FCC lattice -- I hope your web
browser hasn't been crippled to block GIF animations).

But what about my manipulatives, I haven't shown you those. What I'll
do right now, is go to my living room, spread my manipulatives out on
the floor, and take a digital picture. I might get my dog in it too.

I'll upload the photo to the Photos section, and those of you who
know how to navigate the Yahoo interface can check it out!

<< pause while I do said activity >>

<< tarnation! this FujiFilm FinePix 2600 has been going down hill for
awhile, battery door held on with tape, but tonight she appears to
have lost all utility -- I'll need to rescue that SmartCard as it has
lots of good pix of Julian's sculptures at Gallery 500. Oh well, some
other time then -- I'll let ya'll know >>

> And yet, back at their high schools, their teachers'll likely just
> give 'em blank stares if they try to talk up any of what I'm
> showing 'em.

> That's why we're looking to TV: it's too late to keep worrying
> about the teachers at this point, plus they've had over 30 years to
> catch up on this stuff.

Yep, lots of TV queuing up. We'll be encouraging students to stay
home and get an education, or maybe go to school and use the same
projector we use for the web, to screen vid clips.

Kids'll help make these clips too, cuz in the 21st Century, it's part
of basic literacy to know how to make video (storyboarding, editing
and like that).

Speaking of which, I also showed 'em a 24 minute video made with
student assistance at Yorktown High in Virginia. It's a pro-Python
propaganda piece: a lot of talking heads interspersed in a hokey love
story about this boy geek with a crush on a girl geek. She knows
Python, he's just learning. She learned it from her dad, a 60s hippie
(now oldster) who calls it his "secret weapon" (check it out!).

http://www.ibiblio.org/obp/pyBiblio/pythonvideo.php
(big file, you need bandwidth).

The rest of my day I spent doing pro database work for a client in
Hillsboro, not far from OGI. But that's another story for another list.

Kirby
Instructor
Saturday Academy
Portland State University
Oregon

===============