Polyominoes, Polygons, and Polyhedra Discovering Geometry through Exploration

When asked to draw a star, most people will draw a five point star polygon without lifting their pencil even though they are unfamiliar with the terminology. We will generalize this to create star polygons with different number of points. We will also see that our procedure will sometimes give us a regular polygon or no polygon at all.

To ensure that the star polygons we create are regular, we begin with equally spaced points around a circle as shown in Figure 6.1.1.(a). We draw segments from point 1 to point 3, from point 3 to point 5, from point 5 to point 2, from point 2 to point 4, and then returning to point 1 to get the star polygon in Figure 6.1.1.(b). Note that we are connecting every second point and continuing until we return to our starting point. There was no need to lift our pencil in this construction.

The notation \(\{_{2}^{5}\}\) signifies that the star polygon has 5 points where every second point is connected. To create \(\{_{1}^{5}\}\text{,}\) we sketch a continuous path that connects each of the five points in Figure 6.1.1.(a) to the first point from it (its neighbor). The resulting figure does not look like a star but rather the convex regular pentagon shown in Figure 6.1.2.

Similarly, we can draw star polygons on six vertices. The \(\{_{1}^{6}\}\)-star polygon is simply a regular hexagon.

As we create \(\{_{2}^{6}\}\) skipping every other vertex, we return to the starting point after creating a triangle. The three even-numbered vertices have not been visited.

We now repeat this process starting at an unvisited vertex and joining every second vertex until we return to this new starting vertex. As shown in Figure 6.1.6, this creates a second triangle congruent to the first but rotated 60 degrees about the center of the circle containing the six vertices. To aid in our exploration, a new color was used to draw the second triangle. You are also encouraged to use a new color each time you need to lift your pencil.