FR: Kirby Urner, teacher trainer, 4D Solutions, Oregon
TO: HS and JHS math/geometry teachers (in Oregon especially)
RE: implementing a 'Beyond Flatland' curriculum
As advertised at my "shingle" page, I'm into moving my students
"Beyond Flatland" into spatial geometry early, then coming back
to the more traditional Euclidian demonstrations as "scribed
in the sand on a beach" with the planetary context already
firmly in view.[1]
Towards this end (of moving beyond flatland) I use the still
little-known "concentric hierarchy" of nested polyhedra.[2]
Although nested polyhedra have a venerable tradition, comprise
an important thread in many cultures east and west, their use
today might be disturbing to some parents worried about
"occult" influences.[3] So a first step in some classrooms may
be to hook in an appropriate, non-polarizing aesthetics, such
as Bill Nye the Science Guy's (which some object to for other
reasons -- jumps around too much, too "Pee Wee's Playhouse").
If parents approach you with concerns about "the occult", point
out that the flatlander curriculum currently in vogue is just
as overrun by pentacles (flat) and other Crowley-type insignia,
that five-fold symmetry in a volumetric context is not
necessarily the root of the problem, and that kids with spatial
fluency are far better equipped for all the exciting design-
related jobs in our Silicon Forest (CAD-based, a lot of them).
In some communities it might work to point out that you're getting
a lot of this material from an American inventor awarded the
Medal of Freedom by president Ronald Reagan -- it all depends
on the demographics in your region.[4]
Presuming you can get free of sticky issues (cultural ectoplasm
has a way of keeping teachers mired in the past), you'll next
want to get some actual polyhedra showing up in your
classroom. The time-honored approach (and a good one) is to
have kids make these themselves, either alone or in groups or
both. What you will need to start are:
Tetrahedron (aka tetra or tet for short)
Cube <-- face diagonal = tetra edge
Octahedron <-- same edge as tetra
Icosahedron <-- same edge as tetra
Cuboctahedron <-- same edge as tetra
Important to make the cube have a face *diagonal* equal to the
tetra's edge, such that two interpenetrating tets (points
sticking out one another's faces) define this cube's eight
corners, with two sets of 6 edges per tet criss-crossing at the
cube's face centers.[5] This is old news to nested polyhedra
people (e.g. Kepler) which is no excuse to bleep over it.
Next you'll want some ping pong balls and maybe some old
(cut-upable) texts re crystallography -- doesn't matter if the
math in these latter is way above your students' level, as
we're just looking for interesting pictures.
Glue 6 ping pong balls in a "wheel" around a central one of a
different color (you can usually find day-glow green or orange
ping pong balls), then glue together 2 sets of 3 balls to make
triangles. Show how you can "sandwich" the 7-ball "hexagon"
between the two triangular sets (by nesting balls in valleys),
such that the top triangle balls line up over the lower ones
(hexagonal close-packing or hcp) or don't (face-centered cubic
or fcc) -- find corresponding pictures in your crystal book,
cut out and scan for web page (students learning HTML) if
scanner available (I appreciate getting all such URLs).
Show how you can make a tetrahedron by nesting one ball atop
one of your triangular groups -- identify this with the
tetrahedron constructed of paper, straws, clay, wood or
whatever.[6]
Show how the 12 balls around 1 (day-glow) in the fcc
arrangement define a cuboctahedron -- identify this with the
cuboctahedron constructed above as well.
You are now set to do sections on space-filling (cubes alone,
octahedra with tetrahedra, octahedra with cuboctahedra -- but
not icosahedra, nor tetrahedra by themselves), duals
(tetrahedron dual of itself, cube of octahedron), and
rotational symmetry (turning shapes around axes and noting
self-congruence at certain "click-stop" points).
As for the nested polys, you've got the 2 tets interpenetrating
to make a cube, an octahedron with the cube touching its
mid-edges, and a cuboctahedron with octahedron tips at the
centers of its square faces.[7] The respective volumes of
these shapes is:
Tetrahedron: 1
Cube: 3
Octahedron: 4
Cuboctahedron: 20
As for the icosahedron, you'll want to invest in one of those
"vector flexor" devices (or build one yourself, or with
students -- somewhat labor intensive but I've done it with 6th
graders). This device shows how a cuboctahedron may be
rotationally contracted into an icosahedron of same edge
lengths. Its volume is incommensurable vis-a-vis the above,
weighing in at about 18.51...
In a more advanced class, you'll want to invest in a rhombic
dodecahedron (dual of cuboctahedron) with long face diagonals =
tetra edge (volume 6). These fill space as "cells" with fcc
spheres inside, just "kissing" (technical term) through the
face centers.[8]
If your kids respond to this material and seem to wish deeper
explorations, by all means go for the A & B modules (4-fold
symmetry) and T mods (5-fold symmetry).[9] The difficulty here
is primarily the shortage of relevant curriculum supplies, and
few classrooms with the necessary computer power.
If you have further questions about moving "beyond flatland" in
your high school or junior high school, feel free to email or
post questions. If you're in Oregon, check out my availability
and email the bookkeeper if you'd like me to drop by. Again,
my aesthetics are more NASA's than occult, futuristic, not dark
ages. As a kid, OMSI pretty much defined my religion (Oregon
Museum of Science and Industry -- where I'm headed in a few
minutes, once my 4-year-old has finished her lunch).
For a better overview of the concentric hierarchy and its place
in the curriculum, see my "Beyond Flatland: Geometry for the
21st Century" at http://www.inetarena.com/~pdx4d/ocn/urner.html
=========================
Notes:
[1] http://members.xoom.com/Urner/working6.html -- note, I
support USA VP Al Gore's proposal to put a hidef TV webcam at
L1, aimed at the sunny side of Earth and transmitting in
real time.
[2] http://www.teleport.com/~pdx4d/volumes.html
[3] justifiably in many cases, as teenagers often conjure up
whatever cultural aesthetics appear most disturbing to parents,
as a way of asserting independence from what to young eyes may
appear a corrupt and arbitrary authoritarian regime -- again,
varies greatly from one household to the next, and the rage may
not even be at parents per se, but at some nebuluos "ruling
caste" of community elders (I can only speak to some situations
from personal experience -- don't proclaim blanket authority)
[4] http://www.teleport.com/~pdx4d/bio.html
[5] http://www.teleport.com/~pdx4d/volumes.html#cube
[6] A good way to make polys is with party straws and fuzzy
pipeclearners with folded ends wadded into the openings (thanks
to Karl Erickson for this suggestion)
[7] http://www.teleport.com/~pdx4d/quadshapes.html -- towards
middle of paper (includes rhombic dodeca in blue)
[8] http://www.teleport.com/~pdx4d/ivm.html
[9] http://www.teleport.com/~pdx4d/modules.html
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