~/imallett (Ian Mallett)

Approximate Construction of Regular Polygons

Ian Mallett, Brenda Mallett, Russell Mallett, 2006-05-22
Updated Ian Mallett 2012-01-29, 2017-06-16

Abstract:

Some regular polygons with $n$ sides are classically "constructible" using only a compass and a straightedge, whereas others have been proven by Gauss to be "unconstructable", at least exactly. We develop a practical general procedure for the approximate construction of regular polygons which works well for large $n$ and which can be adapted for small $n$.

Introduction:

Ian, then 14, was interested in finding approximate and practical ways to construct regular polygons that are impossible to construct exactly if using only a compass and straightedge.

The geometry computer program "Geometer's Sketchpad" was used to develop a construction for an approximate regular heptagon (for comparison, other procedures for doing this have been described by Dixon: Mathographics. New York: Dover, 1991, reprinted http://mathworld.wolfram.com/Heptagon.html).

The algorithm was then extended for the construction of a nonagon (comparing favorably with traditional constructions, as shown by Dixon: http://mathworld.wolfram.com/Nonagon.html). With assistance, the algorithm was generalized to construct polygons for any $n$. We term this algorithm, and its improvements, the "fraction of a radius" technique.

Discussion:

 Figure 1: Approximate construction of a regular heptagon via the "fraction of the radius" technique. To replicate Ian's initial efforts, using standard geometric techniques, one may draw a circle of radius $a$, then divide the radius of the circle into $15$ segments of equal length. Then, using a compass set to $c := \frac{13}{15}$ of the now-divided radius, one may mark off arcs around the circumference of the circle and connect the intersections to inscribe an almost-regular heptagon. The error in this construction is minimal, with central angles only about $0.07^{\circ}$ too small.

Ian then set out to come up with a method for nonagons. Using the above method of dividing the radius, a nonogon may be constructed by choosing $\frac{13}{19}$ of the radius. In this case, the error is even less; the central angles are a little over $0.01^{\circ}$ too large, each.

The pattern was generalized for any $n$ as follows:

To approximately construct a regular polygon of $n$ sides, mark off side lengths around a circle with a fraction $F$ of its radius equal to:$$F = \frac{13}{2 n + 1}$$

Harold Jacobs, author of Geometry: Seeing, Doing, Understanding, suggested we evaluate the algorithm using the Law of Cosines.

The Law of Cosines states that $c^2 = a^2 + b^2 - 2 a b \cos(C)$, where $a$, $b$, and $c$ are the sides of a general triangle and $C$ is the angle opposite side $c$.

In the case of the fraction of the radius procedure, both $a$ and $b$ are the radius of the circle, and $c$ represents the length of one side of the polygon. $C$ is the central angle opposite $c$, and for a regular polygon of $n$ sides is equal to $\frac{360^{\circ}}{n}$ (or $\frac{2 \pi}{n}$).

Since the number of interest for construction by the fraction of the radius procedure is the polygon side length $c$ divided by the radius of the circle $a$, one may solve the Law of Cosines for the fraction $\frac{c}{a} = \sqrt{2 (1 - \cos(C))}$.

A quick numeric comparison of this desired $\frac{c}{a}$ with that given by $\frac{13}{2 n + 1}$ for increasing $n$ shows that the approximation is a remarkably good fit for such a simple formula (see Figure 2).

Note that Equation 1 works exactly for a hexagon and extremely well for nonagons. The individual central angles constructed become less accurate up to $n = 23$, although they are still accurate to within a quarter of a degree, then very slowly improve for remaining $n$; however, the improvement is overcome by the multiplicative effect of increasing $n$, so that for the polygon as a whole the cumulative error slowly increases with $n$.

Russ, using the trig identity $1 - \cos(C) = 2 sin^2(C)$ rearranged the Law of Cosines ($C$ in radians):$\frac{c}{a} = 2 \sin\left( \frac{\pi}{n} \right)$For small central angle $C$ (i.e., large $n$), the first-order Taylor approximation $\sin(\theta) \approx \theta$ holds, giving:$\sin\left( \frac{\pi}{n} \right) \approx \frac{\pi}{n}\\ \frac{c}{a} \approx \frac{2 \pi}{n}$Equating $\frac{13}{2 n + 1}$ to this and solving for $\pi$, one obtains:$\pi \approx \lim_{n \rightarrow \infty} \frac{n}{2} \frac{13}{2 n + 1} = \frac{13}{4}$

Using a better fractional approximation of $\pi \approx \frac{22}{7}$, one obtains an improved "fraction of the radius" formula:$$\frac{c}{a} = \frac{44}{7 n}$$

One may continue refining the algorithm by using another approximation for $\pi$, such as $\frac{355}{113}$, but the physical compass and straightedge procedure would quickly become unrealistic, with merely an asymptotic gain in accuracy. The improvement from $\frac{13}{4}$ to $\frac{22}{7}$ is justified, however; Equation 2 is much better than Equation 1 for $n > 14$ (see Figure 3). For example, for $n = 25$, Equation 1 results in central angles more than $0.24^{\circ}$ too large, for a cumulative error over the $25$ angles of over $6^{\circ}$, while Equation 2 results in central angles just over $0.04^{\circ}$ too large, for a cumulative error of only a litle over $1^{\circ}$.

In contrast to Equation 1, Equation 2 only improves with increasing $n$, for both individual central angles and for polygons as a whole.

 Figure 2: Generating formulae for the target fraction $\frac{c}{a}$. Figure 3: Accuracy of fraction of the radius procedure. Values $n < 4$ are not included; both formulas are terrible for triangles, and exact constructions exist for $n < 7$ anyway.

After using common exact constructions for $n = 3, 4, 5,$ and $6$, one might reasonably continue with the Equation 1 approximation for $n = 7, 9, 11, 13,$ and $14$, with Equation 2 for higher $n$ where inexact constructions are impossible. However, Russ suggests that, for a highly accurate polygon of small $n$, one could merely construct a polygon with a convenient integer multiple of $n$ sides, but only connect a corresponding fraction of the vertices (for example, construct a regular $21$-gon, and connect every third vertex to form a heptagon. We call this process "superconstruction/division". It is worth noting that regular polygons of any degree may be constructed to arbitrary precision using this method.

Conclusion:

We have presented a successful general rule for the approximate construction of regular polygons. The rule may be extended in simple manners—either by better approximating $\pi$, or by superconstruction/division. Using these extensions, this simple and straightforward method can be used to construct polygons of any $n$ to any precision desired using only the classic geometric instruments.