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260 MATHEMATICS MAGAZINE A New Angle on an Old Construction: Approximating Inscribed n-gons R O B E R T S. M I L N I K E L Kenyon College Gambier, OH 43022 To the 21 st -century mathematics student compass-and-straightedge constructions— such as constructions of regular polygons—might constitute a fun puzzle, an en- tryway into the rich ideas of axiomatic geometry, or simply a relic of ages past. Before computer-assisted drafting technology was widely available, the compass and straightedge were vital tools for the engineer, the architect, and anyone else who needed precise drawings. There are innumerable sources for more background on compass-and-straightedge constructions, e.g., [1], [10], and [6]. Compass-and-straightedge constructions of some regular n-gons, such as for n = 5, 6, 8, and 10, were known in Euclid's time and likely well before. For others, such as n = 7 and 9, no exact construction was known, and for good reason. In the early 19 t h century, Gauss conjectured in [5] and Pierre Wantzel proved in [11] that exact compass-and-straightedge constructions of regular n-gons are possible only if n is a power of 2 or a product of one or more Fermat primes times some power of 2. The only known Fermat primes are 3, 5, 17, 257, and 65537; see [3] for much more on these numbers. The fact that no exact construction was known did not, however, keep draftsmen from having need of a method for drawing regular n-gons for values such as 7 and others. In this case, necessity was the mother of approximation, and engineers and draftsmen developed a single construction that would produce a reasonably accurate regular n-gon inscribed in a circle for any value of n between 4 and about 15. The accuracy degrades quickly as n increases toward 25, at which point the construction fails completely. The draftsman's construction. We use n = 7 in the illustrations, but the same con- struction works for any n ≥ 4. The first two steps are classical and can be found in Euclid's Elements ([4]); the details of how they are accomplished are not crucial to the rest of the construction. The procedure is: 1. Find a diameter of the given circle, with endpoints A and B ([4], Book III, prop. 1). 2. Divide the diameter into n equal segments A D 1 , D 1 D 2 , . . . , D n−2 D n−1 , D n−1 B ([4], Book VI, prop. 9). 3. Use the compass to make two circular arcs with radius A B, one with center A, the other with center B. These arcs will meet at a point which we will call C directly below the center of the circle. 4. Draw a line segment connecting point C to D n−2 , the second mark from the right on the diameter, excluding the right endpoint. Let E 1 be the intersection of the circle and the line C D n−2 that is on the opposite side of A B from C . We now have two points of the n-gon, B and E 1 . See Figure 1. Math. Mag. 88 (2015) 260–269. doi:10.4169/math.mag.88.4.260. c Mathematical Association of America MSC: Primary 51M15.

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