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I know that the vertices of a regular $n$-gon determines the total of $n$ different slopes. We nest the total of $m \in \mathbb{N}$ polygons by drawing a $(1/2)n$-gon inscribed inside the original polygon (for example, an equilateral triangle inside a regular hexagon, a square inside of a regular octagon and so on). We repeat this for every nested polygon.

Now, if we are given $m$ nested polygons, is there a formula for determining the total number of slopes in this configuration?

Slopes in the regular hexagon are defined by drawing a line from each point in the hexagon to every other point in the hexagon. By adding another hexagon, we draw a line from each point in the new hexagon to every other point in the configuration, including the pre-existing points.

Below is an example of $2$ nested hexagons.

enter image description here

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    $\begingroup$ What is the point of the nested hexagon? Doesn't it have exactly the same slopes as the outer hexagon, hence has no effect on the total number of different slopes? Also, in what sense does a regular hexagon have 6 different slopes rather than the same 3 slopes repeated twice? $\endgroup$
    – Erick Wong
    Commented Mar 12, 2021 at 1:07
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    $\begingroup$ Yes, it does. I am asking about the new slopes created by pre-existing points in the old hexagon and the new hexagon. $\endgroup$
    – user830143
    Commented Mar 12, 2021 at 1:08
  • $\begingroup$ And the slopes in a regular hexagon come from joining every point to every other point and counting the total number of slopes or directions that way. $\endgroup$
    – user830143
    Commented Mar 12, 2021 at 1:15
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    $\begingroup$ That makes more sense. Please update your question to reflect this detail. For instance, the first sentence would be clearer if it read “the vertices of a regular $n$-gon”. The sides of the polygons have almost no role to play in this question except that they are integral to what it means to “inscribe”. $\endgroup$
    – Erick Wong
    Commented Mar 12, 2021 at 1:20
  • $\begingroup$ I added the edit. Thanks. $\endgroup$
    – user830143
    Commented Mar 12, 2021 at 1:20

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