There exist Star Polygons created (for some $n$) by tracing a line along the consecutive vertices $V_0, V_1, \ldots, V_{n - 1}$ of a regular $n$-gon using $m$ steps, e.g.

$$V_0 \rightarrow V_m \rightarrow V_{2m \pmod{n}} \rightarrow V_{3m \pmod{n}} \rightarrow \cdots$$

(This actually requires $\frac{n}{\gcd(n, m)}$ such cycles, obviously depending on the orbit of $m \pmod{n}$.)


Does there exist a term for the "star" formed by extending the lines through adjacent vertices until they meet?

For example, the lines $\overrightarrow{V_{n - 1} V_0}$ and $\overrightarrow{V_2 V_1}$ (using the notation from above) would meet at some external point $E_0$ after being extended past $V_0$ and $V_1$, respectively.

The triangle formed -- $\triangle V_0 V_1 E_0$ -- would be isosceles with the base $V_0 V_1$ a side of the regular $n$-gon and the equivalent angles $\angle E_0 V_1 V_0$ and $\angle E_0 V_0 V_1$ equal to $\frac{2 \pi}{n}$.

By similarly extending these throughout the circle to all the points $E_1, E_2, \ldots, E_{n - 1}$, the star would be formed by tracing

$$V_0 E_0 V_1 E_1 \cdots V_{n - 1} E_{n - 1} V_0$$

Anyhow, is there a name for this "star" formed?

  • 2
    $\begingroup$ A stellation, then? $\endgroup$ – J. M. is a poor mathematician Apr 3 '13 at 17:13
  • $\begingroup$ Indeed, but a very specific stellation. Since "A regular $n$-gon has $(n-4)/2$ stellations if $n$ is even, and $(n-3)/2$ stellations if $n$ is odd.", there are quite a few choices. $\endgroup$ – bossylobster Apr 3 '13 at 18:13
  • $\begingroup$ Seems this is always $\left\{n/2\right\}$. $\endgroup$ – bossylobster Apr 3 '13 at 18:16
  • $\begingroup$ @J.M. after thinking about it and chatting with some mathematicians around here it seems stellation is correct and my view of the "star" was too limited. Feel free to add it as an answer and I'll upvote/mark as correct. $\endgroup$ – bossylobster Apr 3 '13 at 23:09

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