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.