Figure 1

A polyhedron is any closed region of 3space
cut out by a finite set of planes.
Take any polyhedron and do the following:
on each face, place a vector perpendicular to that face
with length proportional to the area of that face.
No matter what polyhedron you started with,
the sum of all those face vectors will be zero!
Presentation Suggestions:
First draw several polyhedra, and make sure some of them
are very irregular. They do not even have to be convex...
this fact is most surprising for the irregular polygons.
Draw the face vectors so that students can see what you
mean.
The Math Behind the Fact:
This Fun Fact can be proved by first showing that is
is true for any tetrahedron. Use vector geometry to
do this; express the face vectors in terms of the
cross product of coincident sides... when you sum
them, everything will cancel. Since any polyhedron
can be built up by tetrahedra, and since the sums
of the face vectors of two coincident faces cancel,
the theorem can be proved for arbitrary polyhedra!
A more sophisticated proof uses the divergence theorem from
multivariable calculus:
the component of each vector in the i direction is the
flux of the constant vector field i across the closed surface,
which by the divergence theorem is zero. The same is true
in the j and k directions as well.
How to Cite this Page:
Su, Francis E., et al. "Polyhedral Face Vectors."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
