Introduction
Consider the
hexagram applied to a right
angled hexagon.
If we apply
Desargues' theorem
in
two ways,
confirms the suspicion that
the intersection
points alternately lie on a pair of lines.
Let's apply the
octagram twice to a
right angled octagon. Here is the
first application
and here is the
second application.
The cool fact is that the final 8 points-4 of which are shown-alternately
lie on a pair of lines.
If we apply the
10-gram
thrice to a right angled 10-gon, then the
10 colored points alternately lie on a pair of lines.
The pattern continues
forever!
Expressed in slightly more
compact notation ,
this pattern takes a
simple form.
The hexagon case is a consequence of Desargues' theorem,
a result which is proved by projecting a certain
arrangement of planes
in 3-space
back into the plane. It turns out that the
octagon case
is the consequence of a theorem proved by projecting
a certain arrangement of hyperplanes in 4-space back
into the plane.
The 10-gon case involves the
projection of a 5-dimensional configuration, and so on.
This pattern generalizes.
Applying a variant of the
map above to a 9-gon whose sides are cyclically
parallel to three distinct directions, we get
9 points which are distributed on 3 lines.
(This case is again a
consequence of Desargues' theorem.)
If we apply the
same construction twice to a 12-gon whose
sides have the same property, then we get 12 points
cyclically distributed on 3 lines. The pattern
holds more generally (and more naturally) for
polygons
whose sides cyclically contain 3 points , and
continues forever.
If we generalize
the construction
to polygons which are
satellites
we arrive at a fairly general
theorem.
I discovered this theorem on the computer, while
trying to understand the dynamics of the constructions
above. It was unexpected.
I am slowly including the proof of this result in
this document. To see a precise statement of the
theorem, as well as a proof, check out my
preprint.