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[Phys-L] just for fun : dissecting polygons and polyhedra



Teaser: Can you dissect a regular tetrahedron into
eight little tetrahedra, each with the same volume?

=========

Longer version: As previously mentioned, one of the rules for
learning, problem-solving, and reasoning says that whenever
you solve a problem, see if you can find a generalization.

Applying that rule to myself, the recent discussion of scaling
got me thinking.

*) Known result: A square can be dissected into four little
squares, all identical, indeed all similar to the original.
*) Ditto for any rectangle. All similar.
*) Interesting result: Less obvious, but no less true: Any
triangle can be dissected into four little triangles, all
similar to the original.

Here "similar" refers to strict geometric similarity,
i.e. same shape, differing only by a uniform scale factor.

*) Can we generalize that to more complicated polygons? That
depends on details.
-- Preliminary remark: Note that any polygon can be dissected
into triangles /without/ adding any new vertices, just by
cutting triangles defined by the existing vertices. Can you
prove this? This result does not generalize to 3D; see below.
-- If we are forbidden to Frankenstein the little polygons,
then the answer is no, we cannot dissect a general polygon
into little polygons of the same shape. We cannot tile the
interior of a pentagon with little pentagons.
-- OTOH if we are allowed to Frankenstein the little polygons,
trimming pieces here and there and rearranging, then we can
do anything we want, because an arbitrary polygon can be
dissected into triangles. Then it reduces to the problem
already solved: we dissect the triangles and rearrange the
pieces.



Now things get interesting. Rather than generalizing the number
of vertices, let's generalize the number of dimensions.

*) Known result: A cube can be dissected into eight little
cubes, each with 1/8th of the volume, each similar to the
original.
*) Ditto for any parallelepiped.
*) Ditto for a triangular prism.
*) However, the same is not true for a tetrahedron! Again
the devil is in the details.
-- Preliminary remark: It's fairly obvious that any
polyhedron can be dissected into nice simple tetrahedra.
What's less obvious is that in some cases this cannot
be done without adding vertices, which stands in contrast
to the 2D case. Proof by counterexample:
http://en.wikipedia.org/wiki/Sch%C3%B6nhardt_polyhedron
-- You cannot stack eight identical tetrahedra to make a
bigger tetrahedron. By the same token you cannot dissect
a big tetrahedron into eight little ones that are similar
to the original. So this is another way in which 3D
differs from 2D.
-- Now we get to a more refined version of the teaser question:
Can you dissect a regular tetrahedron into eight smaller ones,
all with the same volume, even if some of them are not regular?

I know the answer to this one. It took me a while to figure
out.

This is mostly just for fun. It's something you can do if you are
bottled up in a waiting lounge and haven't got anything better to
do. Dissections have no direct application to physics so far as
I know. However, the process involves skills such as generalization,
symmetry, and visualization in 3D. These skills get used again
and again.

Some people have trouble with spatial relationships in two dimensions.
Almost everybody struggles with three dimensions ... not to mention
four-dimensional spacetime.

The tetrahedron problem is sufficiently hard that the usual rule
of "draw the diagram" doesn't entirely solve the problem. I
considered using computer graphics, but I decided it would be
easier to build some tangible 3D manipulables using straightedge
and compass and paper and scissors. On the other hand, I would
have had no idea what to build until I had more-or-less entirely
solved the problem in my head. On the third hand, my confidence
in the solution and my /understanding/ of the solution went up
quite a bit after playing with the manipulables.

In particular, there was a symmetry that I found by visualizing
things in my mind's eye, which was key to answering the question.
On the other hand, there was a second symmetry, a higher symmetry,
that I didn't see until I made the models. Then, duhhhhh.

I mention this because on 12/20/2014 11:37 AM, Richard Heckathorn
suggested using 3D models. You can expect pushback from students
who think such things are beneath their dignity, i.e. too much
like pre-kindergarten. We should resist the pushback. Computer
graphic visualization has gotten a lot better recently, but even
so, sometimes a tangible 3D model is a big win. (Nowadays you
can split the difference using a 3D printer, but that's a whole
story unto itself.)

Remotely related: Here's a classic 3D brain-teaser puzzle that
makes a nice desk ornament or coffee-table ornament:
http://www.creativecrafthouse.com/index.php?main_page=product_info&products_id=765
It might take you a while to find the solution the first time.
I recommend the 5x5x5 version, not the 4x4x4 version.