Stars

Mathematical stars, unlike celestial ones, are pointy, usually symmetric, shapes that burst outward from a central “core.”  In the plane, regular star-shaped polygons are traced by line segments that connect equally-spaced dots on a circle.  If the dots are connected in cyclic order, a regular convex polygon results.  But if the dots are connected in order, skipping over one dot each time, a “star polygon” will result.  If the number of dots is odd, the traced path will return to the starting point, completing the star.  But if the number of dots is even, the traced path will close after visiting only half of the points, and a second path must connect the remaining points to produce the star.  The final star consists of two  identical convex polygons, one turned to overlap the other, like two triangles composing a familiar 6-pointed star.  Other star polygons can be traced in a similar way, by repeatedly skipping over more than one point as dots on a circle are connected.

In space, intricate polyline paths can trace out circuits with star-shaped projections.  Here, with 3-dimensional freedom of movement, a path can even trace out a figure having all 90° corners, traveling over and under itself, at times in one plane, then in a plane perpendicular to that one, repeating the same travel instructions.   The completed circuit might be an intricate knot, where in each of several projections, we see a symmetric star.