Is there a theoretical (or practical) definition of $n$-gon, for $n < 0$? **Background** This is purely a "sate my curiosity" type question. I was thinking of building a piece of software for calculating missing properties of 2D geometric shapes given certain other properties, and I got to thinking of how to failsafe it in case a user wants to calculate the area of a $2$-gon, $1$-gon, $0$-gon, 'aslkfn'-gon or maybe even $-4$-gon. **Question** Are there any definitions for $n$-gons where $n < 0$? **Valid assumptions** Let's, for the sake of simplicity (if possible) say that $n \in \mathbb Z$, although I might come back later and ask what a $\pi$-gon is.

For regular $n$-gons with side-length $1$, the area is given as $$\frac{1}{4}n \cot \frac{\pi}{n}$$

Here are some values of the formula for negative values of $n$:

\begin{array}{c|c} n & \cot\frac{\pi}{n} \\\ \hline -1 & \text{complex } \infty \\\ -2 & 0\\\ -3 & -\frac{1}{\sqrt{3}} \\\ n \leq 3 & <0 \end{array}

How you want to interpret that is up to you, but the function is there.

This kind of thing (where you extend something beyond what's intuitive) is done many places within mathematics. Take for instance the $\Gamma$-function (see here), which is an extension of the factorial function. However, I don't know if it is useful in this particular case (with the $n$-gons), but why not try?