Regular Star Polygons

4. Distances and Angles

The last three results would be very helpful for someone who wanted to make a star pattern in something like a quilt, a mosaic, or a stained glass window. However, it would be even more helpful for such a person if we could specify the lengths of the edges and the angles at each of the points. The convex regular n-gon in the middle is a well known figure. As for the quadrilaterals, we will first find the angles in the figures.

The quadrilaterals look like

The point labeled Pj is a point on the {n/j} star, the points labeled Pj-1 are points on the {n/j-1} star, and the point labeled Pj-2 is a point on the {n/j-2} star.

Theorem 4.1: The angle at Pj is 180-360j/n.

The angles at Pj-1 are 360(j-1)/n.

The angle at Pj-2 is 180-360(j-2)/n.

Remark: Note that if j = 2 the bottom angle is a straight angle, and triangles of the {n/2} star are actually a special case of our general result.

In addition to the angles, it would be useful for a quilter to know the lengths of the sides in the regions. The following result will be of central importance in finding all of the distances.

Theorem 4.2: (Coxeter [4]) The perpendicular distance from the center of the circle to the lines which make up the star is

ro = rcos(180k/n)

 

The distance from O to Po is very important, because Po is inside all of the inside stars. Since it is common to all the stars, we can use it as a reference for finding all of the distances in all of the stars. It is called the apothem of the star.

The Po points form a discrete star inside the n-gon. Moreover, Theorem 1.3 gives us a correct value of 180 for the angle at the Po points. The Po points are not counted in Theorem 3.3 because they are not formed by the intersection of two different lines.

We can now determine the polar coordinates of all the points in the star. We will assume that the star is oriented so that one of the points is pointing straight up.

Theorem 4.3: (Zeitler [14]) Let an {n/k} star be placed in a coordinatized plane in such a way that the center of the star is at the origin and the polar coordinates of one of the points is (r, 90). Let rj be the distance from the points Pj to the center of the circle. Then

rj = rcos(180k/n)sec(180j/n)

The angle at which the Pj points are found is

90 + 360j/n

if j is congruent to k mod 2

and

90 + 180(2j - 1)/n

if j is not congruent to k mod 2

i = 1, . . . , n, j = 1, . . . , k

The only other thing we would need to make quilts or mosaics or stained glass windows would be to know how long the sides were. This will follow from the following general result concerning the distance from the point to Pj the Pi point on a line which forms the star. Without loss of generality we can assume that i is less than or equal to j.

Theorem 4.4a: In an {n/k} star inscribed in a circle of radius r, if 0 < i < j < n/2 then the distance from the Pj points to the Pi points on a line of the star is

If the points are on the same side of Po use the minus sign. If they are on opposite sides, use the plus sign.

While this formulation is useful in some situations, there is a simplification which will prove more effective in the next section.

Theorem 4.4b: In an {n/k} star inscribed in a circle of radius r, if 1 < j < n/2 then the distance from the Pj points to the Pi points on a line of the star is

If the points are on the same side of Po use the minus sign. If they are on opposite sides, use the plus sign.

Remark: In the n-gon, i and j can only be 0 or 1. If i = 0 and j = 1, we get the familiar distance from the middle of an edge of an n-gon to the end of the edge. In the asterisk where k = n/2, we get radii and distances of 0 for all the interior points and indeterminate forms for radii and distances involving points on the asterisk where j = k = n/2. We will discuss methods of dealing with those indeterminate forms in the next section.

A quilter would probably find it easier and more effective to prepare an accurate drawing of the star on paper and cut out the regions with scissors than to use these formulas in order to make patterns. However, formulas do exist, which we can state as we have here, and which are useful for further observations.

Remark: Note that in the formulas j could be larger than k. If j is larger than k then we get points on stars outside the {n/k} star by extending the sides of the {n/k} star. j cannot increase indefinitely, because, once j = n/2, the formulas give us points at infinity or indeterminate forms.

Once j gets to be larger than n/2, the secants and tangents become negative telling us to look in the other direction for the points of intersection. If we can meaningfully interpret the indeterminate forms and negative distances, these formulas will work for any values of i, j, and k. If not then with the possible exception of the asterisk, we can find values for i, j, and k where 1 < k < n/2 and 0 < i < j < n/2 which will work. Any of the {n/j} stars could be used as a starting point, and all of the distances in all of the other n pointed stars, both inside and out, could be expressed in terms of the radius of that star.

If one were to start with a discrete star of radius r and draw n lines tangent to the circle through the n points extending a distance of rtan([(n-1)/2] /n) in both directions, one would construct a star which would contain all of the n pointed stars inside except for the asterisk if n were even.

While stars can be used to provide students at all levels from elementary school to upper division university students with interesting activities, they are not easy to draw freehand. The interior stars generally come out all wrong unless care is taken to produce accurate drawings. The results will be much better if the students are provided with materials such as straight edges and handouts containing n points arranged equidistantly on circles. There are several computer programs which will draw stars. While computers will theoretically draw accurate figures, the delicacy of the interior structure of stars with large n and k can easily exceed the tolerances of accuracy for standard computer graphics programs.

Section: 5. Areas and Perimeters