How Many Regular Polygons Can Be Drawn Through the Points of Intersection of the Sides of Regular Star Polygons?

In this Instructables, we show that if regular star polygons have n sides and n vertices with each vertex connected to the mth vertex further along from each vertex, (such polygons are notated by the Schläfli symbol {n/m} as described in an earlier Instructables), then m regular n-sided polygons can be drawn through the points of intersection of their sides. This statement will be illustrated in Steps 1 and 2 for:

• all unicursive regular star polygons (polygons that can be drawn without lifting the drawing instrument from the page) having eleven sides known as hendecagons (using the Greek prefix) or endecagons (using the Latin prefix);
• {9/3}, the only non-cursive regular star polygon (the polygon cannot be drawn without lifting the drawing instrument from the page; {9/3} is a tri-cursive polygon) having nine sides known as an enneagon (using the Greek prefix) or a nonagon (using the Latin prefix).

When defining a regular star polygon in terms of its Schläfli symbol {n/m}, m is referred to as the density of the star polygon. In Step 3, we present a brief introduction to the concept of the density of a regular star polygon, and in Step 4 conclude that it is related to the number ofregular polygons can be drawn through the points of intersection of the regular star polygon’s sides.

Step 1: Cursive Regular Star Polygons With Eleven Sides

The black dashed lines in the diagrams in the Introduction show all four cursive star polygons with eleven sides, namely, {11/2}, {11/3}, {11/4} and {11/5}. An enlarged version of the central part of the {11/5} star polygon is shown above (below the heading for this Step). The solid red lines show the 11-sided regular polygons drawn through the points of intersection of the sides of these star polygons. It can be seen in each of the four cases of these cursive regular star polygons, {11/m}, that the number of 11-sided regular polygons that can be drawn through the points of intersection of the sides of these star polygons equals m.

Step 2: Non-cursive Regular Star Polygon With Nine Sides

There is one non-cursive regular star polygon with nine sides, namely {9/3} as shown in the above tri-cursive nonagram. As in the previous step, the black dashed lines show this star polygon and the red lines show the three 9-sided regular polygons drawn through the points of intersection of the nine sides of this star polygon. The statement that that only three 9-sided regular polygons can be drawn through the points of intersection of the sides of this tri-cursive regular star polygon only holds if {9/3} is regarded as a non-unicursal nonagram and does not hold if {9/3} is regarded as a triply-wound triangle (for a discussion of the terms used used in this sentence see the earlier Instructables mentioned in the Introduction).

Step 3: Brief Introduction to Density of Polygons

The density of a polygon tells us the number of times that, starting from a point on the polygon, this point can be rotated around the center of the polygon until it returns to the starting point on the polygon. For a regular convex polygon the density is 1. For a regular star polygon, it turns out that its density can be readily determined by counting the minimum number of edges that are crossed by a line drawn from the center of the polygon to any point outside the polygon.

Step 4: Conclusion

Assuming the above results hold in general, we can say that for cursive and non-cursive n-sided regular star polygons, we have shown that there is another way to describe the density of such star polygons, namely, counting the number of n-sided regular polygons that can be drawn through the points of intersection of their sides.