Here are some thoughts about Magnetix (tm). It's a kit you can buy at
the toy store.
-- It contains some _struts_ about 1" long. They are permanent magnets.
-- It also contains some _balls_ about 1 cm in diameter. They exhibit
soft (i.e. non-permanent) ferromagnetism.
The point is that you can build stuff.
-- When two struts stick together, they will be colinear
-- Multiple struts can stick to a ball, at any angle, provided the
angle is not too small.
The kit has the remarkable property of being fun for 3-year-olds, adults,
and everybody in between.
However, for today I will concentrate on physics-related stuff that you can
teach to high-school-age kids.
The first thing to do is build a tetrahedron, an octahedron, and an icosahedron.
--> Point out the S4 symmetry of the tetrahedron, i.e. improper rotation,
i.e. rotation of 90 degrees followed by reflection. Kids will generally
not notice this unless you point it out.
--> Also point out the more-obvious symmetries.
--> Build an Egyptian pyramid (square base plus three triangular sides) and
point out that it is an _irregular_ polyhedron ... in contrast to the
tetrahedron that is completely regular. The tetrahedron should be called
a tetrahedron, not a pyramid.
--> Also point out that the octahedron is in some sense much more symmetric
than two Egyptian pyramids stuck together, because it is completely regular.
Yes, inside it you can find a square in the XY plane, but you can also find
a square in the YZ plane and a square in the ZX plane. The latter two are
absent from the Egyptian pyramid.
--> Mention that tetra-, octa-, and icosa- come from ancient Greek roots for
four, eight, and twenty. Get them to count faces on each of the models ...
which is harder than it sounds, because it is easy to lose count.
--> Ask them to _calculate_ the number of edges and the number of vertices
on each solid. Get them to calculate it first, then verify it by counting.
Hint: number of edges per face * number of faces / sharing factor.
--> Ask what's special about the series: tetrahedron, octahedron, icosahedron.
Answer: At each vertex, we have three triangles meeting, four triangles meeting,
and five triangles meeting (respectively). These are three consecutive members
of a series ... and there are no other members. Kids are unlikely to notice
this if you don't point it out ... whereupon for some kids, their eyes will bug out
of their heads. They begin to see (possibly for the first time) "the unreasonable
power of mathematics" and they readily believe it when you say smart people
have been fascinated by these things for 2400 years or perhaps longer.
----> In particular, ask why you can't have just two triangles meeting at a
vertex. (Answer: it would be flat, not three-dimensional. It would be a
double-thickness flat thing. This is easier to demonstrate with paper cut-outs
than with Magnetix.)
----> Then ask what happens if you have six triangles meeting at a point.
(Answer: It would be flat, not three-dimensional. It would be a single-
thickness flat thing.)
----> They will ask what happens with seven or more. Build such a thing
and show that it has negative curvature (too little radius per unit
circumference) in contrast to spheres and such, which have positive
curvature (too much radius per unit circumference). [This has connections
to general relativity that the kids will find helpful in a few years, but
you don't need to say much about that now.]
----> Consider having three squares meeting at a point, or three pentagons.
Convince everybody that no higher polygons can be used as faces.
Partially relevant reference: Feynman volume II section 30-5.
--> Go ahead and build the cube and the dodecahedron. Point out that you now
have a complete set: all the regular polyhedra.
--> Point out the cube is friends with the octahedron. Faces map to vertices
and vice versa. Similarly point out that the dodecahedron is friends with
the icosahedron. You can inscribe dodecahedrons inside icosahedrons and
vice versa, ad infinitum.
Of course, for each pair, their relationship is strictly Platonic.
The tetrahedron is friends with itself.
--> It will be obvious that the cube and dodecahedron (as built out of
Magnetix) are floppy, in contrast to the other three, which are rigid.
Point out the importance of triangulation as a sufficient (albeit not
strictly necessary) way to create rigidity.
Actually, a plain dodecahedron is so heavy and so floppy that it is virtually
impossible to build out of Magnetix; you may want to build one using
toothpicks and a hot-glue gun, just so you'll have one to use as a
hands-on example.
Now it's time to start building stellated polyhedra.
--> The stellated tetrahedron is remarkably ugly. Point out that it
still has the S4 symmetry characteristic of tetrahedra.
--> I don't have much to say about the stellated octahedron.
--> The stellated cube looks nice. The remarkable thing is that it is
not floppy ... very unlike the plain cube.
--> The stellated dodecahedron is beautiful. It is also large. It is
not floppy; indeed it is strong enough that you can play catch with
it if you are careful and gentle.
Point out that some of the nodes on the stellated dodecahedron are
six-coordinated, which may be unexpected. (This is in addition to
the five-coordinated nodes, which are more easily expected.) Point
out that the six-coordinated nodes do not actually have sixfold
symmetry, since the bonds are alternately above and below the plane
of the "hexagon". These points had threefold symmetry in the original
dodecahedron, i.e. the points where three pentagons met.
--> The stellated icosahedron is remarkably pointy. It has pointy
things sticking out everywhere.
===========
--> Buid a big flat triangular lattice. It lies in 2D like a mat.
Then choose one of the nodes in the middle somewhere and reduce it
from six-coordinated to five-coordinated. This introduces some
curvature. Near the chosen point, the mat will curl up into 3D.
Explain that by introducing a little bit of curvature here and
there, you could build almost any arbitrary shape ... even a coat
of chain mail (to borrow an example featured in Misner, Thorne,
Wheeler).
===========================================
I am quite aware that this is highly teacher-intensive. It works fine
with two or three students, but I have no idea how to make it work in
a large classroom.
You can buy Magnetix in sets with 150 pieces. That gives you enough
parts to build any of the stellated regular polyhedra (whereas the
smaller sets don't).
There are of course astronomically many things that can be built
with such a kit; the point of this note was to call attention to
the ones that teach something about math and physics.
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