hexagram applied to a right
If we apply
confirms the suspicion that
points alternately lie on a pair of lines.
Let's apply the
octagram twice to a
right angled octagon. Here is the
and here is the
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
thrice to a right angled 10-gon, then the
10 colored points alternately lie on a pair of lines.
The pattern continues
Expressed in slightly more
compact notation ,
this pattern takes a
The hexagon case is a consequence of Desargues' theorem,
a result which is proved by projecting a certain
arrangement of planes
back into the plane. It turns out that the
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
whose sides cyclically contain 3 points , and
If we generalize
to polygons which are
we arrive at a fairly general
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