# Counting Self-intersecting n-gons

David Wilson davidwwilson at comcast.net
Fri Apr 22 13:20:16 CEST 2005

```Define a self-intersecting polygon as a polygon in which any two
edges may intersect at a single nonvertex point, but no three edges
may intersect.

Let P be the space of self-intersecting polygons, and define two
polygons in P as equivalent if there is a continuous tranformation
between them in P.

Assuming this definition means anything, I am pretty sure that
equivalent polygons will dissect the plane into the same number
of regions and corresponding regions will have the same number
of edges.

The question is, according to this definition, how many distinct
self-intersecting n-gons are there?

Clearly, a(3) = 1, since all triangles are equivalent.

I believe a(4) = 2, all self-intersecting quadrilaterals begin equivalent
to either a square or a bowtie.

I have found 6 inequivalent pentagons, so a(5) >= 6.

- David W. Wilson

"Truth is just truth -- You can't have opinions about the truth."
- Peter Schickele, from P.D.Q. Bach's oratorio "The Seasonings"

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