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6.BU. CONSTRUCTION OF N-GONS
New Section. This is really a proper geometric topic, but there is some recreational interest in two aspects.

A. Attempts to construct regular n-gons for impossible values of n, e.g. n = 7, either by ruler and compass or by origami or by introducing new instruments -- see 6.BV.

B. Attempts to construct possible cases, e.g. n = 5, by approximate methods.

Aspect A is closely related to the classic impossible problems of trisecting an angle and duplicating a cube and hence some of the material occurs in books on mathematical cranks -- see Dudley. Further, there were serious attempts on both aspects from classic times onward.

Abu’l-Jūd. 11C. He "devised a geometrical method to divide the circle into nine equal parts." [Seyyed Hossein Nasr; Islamic Science -- an Illustrated Study; World of Islam Festival Publishing Co., London?, 1976, p. 82. Q. Mushtaq & A. L. Tan; Mathematics: The Islamic Legacy; Noor Publishing House, Farashkhan, Delhi, 1993, p. 70.]

Pacioli. De Viribus. c1500. These problems are discussed by Mackinnon, op. cit. in 6.AT.3, pp. 167, 169, citing Agostini, p. 5. Let l_n be the side of a regular n-gon inscribed in a unit circle.

Ff. 146r-147r, XXIII afare la 7^a fi^a dicta nonangolo. cioe de .9. lati difficile (XXIII to make the 7th figure called nonagon, that is of 9 sides, difficult) = Peirani 198 199. Asserts l₉ = (l₃ + l₆)/4. Mackinnon computes this gives .6830 instead of the correct .6840.

Ff. 148r-148v, XXV. Documento della 9 fi^a rectⁱ detta undecagono (XXV. on the 9th rectilinear figure called undecagon) = Peirani 200. Asserts l₁₁ = φ (l₃ + l₆)/3, where φ is the golden mean: (1 - 5)/2. Mackinnon computes this gives .5628 instead of the correct .5635.

F. 148v, XXVI. Do. de' .13. (XXVI. on the 13th) = Peirani 200. Asserts l₁₃ = (1 φ)·5/4. Mackinnon computes this gives .4775 instead of the correct .4786.

Ff. 149r-149v, XXVIII. Documento del .17. angolo cioe fi^a de .17. lati (XXVIII on the 17-angle, that is the figure of 17 sides) = (Peirani 201-202). Peirani says some words are missing in the second sentence of the problem and Agostini says the text is too corrupt to be reconstructed. MacKinnon suggests l₁₇ = (l₃ - l₆)/2 which gives .3660 instead of the correct .3675.

Barbaro, Daniele. La Practica della Perspectiva. Camillo & Rutilio Borgominieri, Venice, (1569); facsimile by Arnaldo Forni, 1980, HB. [The facsimile's TP doesn't have the publication details, but they are given in the colophon. Various catalogues say there are several versions with dates on the TP and colophon varying independently between 1568 and 1569. A version has both dates being 1568, so this is presumed to be the first appearance. Another version has an undated title in an elaborate border and this facsimile must be from that version.] Pp. 26-27 includes discussion of constructing a regular heptagon, but it just seems to say to divide the circumference of a circle into seven equal parts -- ??

Christian Huygens. Oeuvres Complètes. Vol. 14, 1920, pp. 498-500: problem dated 1662, "To inscribe a regular heptagon in a circle." ??NYS -- discussed by Archibald.

R. C. Archibald. Notes (to Problems and Solutions section) 24: Problems discussed by Huygens. AMM 28 (1921) 468 479 (+??). The third of the problems discussed is the construction of the heptagon quoted above. Archibald gives an extensive survey of the topic on pp. 470-479. A relevant cubic equation was already found by an unknown Arab writer, c980, and occurs in Vieta and in Kepler's Harmonices Mundi, book I, Prop. 45, where Kepler doubts that the heptagon can be constructed with ruler and compass.

An approximate construction was already given by Heron of Alexandria and may be due to Archimedes -- this says the side of the regular heptagon is approximately half the side of the equilateral triangle inscribed in the same circle. Jordanus Nemorarius (c1230) called this the Indian method. Leonardo da Vinci claimed it was exact. For the central angle, this approximation gives a result that is about 6.5' too small.

Archibald then goes on to consider constructions which claim to work or be good approximations for all n-gons. The earliest seems to be due to Antoine de Ville (1628), revised by A. Bosse (1665). In 1891, A. A. Robb noted that a linkage could be made to construct the heptagon and J. D. Everett (1894) gave a linkage for n-gons.

Italo Ghersi; Matematica Dilettevoli e Curiosa; 2nd ed., Hoepli, 1921; pp. 425-430: Costruzioni approssimate.

He says the following construction is given by Housel; Nouvelles Annales de Mathématiques 12 (1853) 77-?? with no indication of its source. Ghersi says it also occurs in Catalan's Trattato di Geometria, p. 277, where it is attributed to Bion. However, Ghersi says it is due to Rinaldini (probably Carlo Renaldini (1615-1698)). Let AOB be a horizontal diameter of a circle of radius 1 and form the equilateral triangle ABC with C below the diameter. Divide AB into n equal parts and draw the line through C and the point 4/n in from B. Where this line hits the circle, say P, is claimed to be 1/n of the way around the circumference from B. Ghersi obtains the coordinates of P and the angle BOP and computes a table of these values compared to the real values. The method works for n = 2, 3, 4, 6. For n = 17, the error is 36'37".

On pp. 428-430, he discusses a method due to Bardin. Take AOB as above and draw the perpendicular diameter COD. Divide the diameter into n equal parts and extend both diameters at one end by this amount to points M, N. Draw the line MN and let it meet the circle near B at a point P. Now the line joining P to the third division point in from B is claimed to be an edge of the regular n-gon inscribed in the circle. Ghersi computes this length, finding the method only works for n  5, and gives a table of values compared to the real values. This is exact for n = 6 and is substantially more accurate than Renaldini's method. For n = 17, the error is 1'10.32".

The "New" School of Art Geometry, Thoroughly Remodelled so as to Satisfy all the Requirements of the Science and Art Department for Science Subject I. Sections I. and II, Practical Plane and Solid Geometry, (Cover says: Gill's New School of Art Geometry Science Subject I.) George Gill and Sons, London, 1890.

Pp. 26-27, prob. 66 -- To describe any regular Polygon on a given straight line, AB. He constructs the centre of a regular n-gon with AB as one edge. Taking the side AB as 1, the height h_n of the centre is given by h_n  =  (n 4) 3/4     (n-6)/4, while the correct answer is ½ cot π/n. For large n, the relative error approaches 14.99%. He gives no indication that the method is only approximate and doesn't even work for n = 5.

Pp. 74-75, prob. 188 -- To inscribe any Regular Polygon in a given circle. He gives three methods. The first is to do it by trial! The second requires being able to construct the regular 2n-gon! The third construction is Renaldini's, which he does indicate is approximate.

R. C. Archibald, proposer; H. S. Uhler, solver. Problem 2932. AMM 28 (1921) 467 (??NX) & 30 (1923) 146-147. Archibald gives De Ville's construction and asks for the error. Uhler gives values of the error for n = 5, 6, ..., 20, and the central angles are about 1^o too large, even for n = 6, though the error seems to be slowly decreasing.

T. R. Running. An approximate construction of the side of a regular inscribed heptagon. AMM 30 (1923) 195-197. His central angle is .000061" too small.

W. R. Ransom, proposer; E. P. Starke, solver. Problem E6. AMM 39 (1932) 547 (??NX) & 40 (1933) 175-176. Gives Dürer's method for the pentagon and asks if it is correct. Starke shows the central angle is about 22' too large.

C. A. Murray, proposer; J. H. Cross, E. D. Schell, Elmer Latshaw, solvers. Problem E697 -- Approximate construction of regular pentagon. AMM 52 (1945) 578 (??NX) & 53 (1946) 336-337. Describes a method similar to that of de Ville - Bosse and asks if it works for a pentagon. Latshaw considers the general case. The formula is exact for n = 3, 4, 6. For n = 5, the central angle is 2.82' too small. For n > 6, the central angle is too large and the error is increasing with n.

J. C. Oldroyd. Approximate constructions for 7, 9, 11, 13-sided polygons. Eureka 18 (Oct 1955) 20. Gives fairly simple constructions which are accurate to a few seconds.

Marius Cleyet-Michaud. Le Nombre d'Or. Presses Universitaires de France, Paris, 1973. Méthode dite d'Albert Dürer, pp. 45-47. Describes Dürer's approximate method for the pentagon and says it fails by 22'.

Underwood Dudley. Mathematical Cranks. MAA Spectrum, 1992. This book discusses many related problems, e.g. duplication of the cube, trisection of the angle. The chapter: Nonagons, Regular, pp. 231 234 notes that there seem to be few crank constructors of the heptagon but that a nonagoner exists -- Dudley does not identify him. Actually he constructs 10^o with an error of about .0001', so he is an excellent approximater, but he claims his construction is exact.

Robert Geretschläger. Euclidean constructions and the geometry of origami. MM 68:5 (Dec 1995) 357-371. ??NYS -- cited in next article, where he states that this shows that all cubic equations can be solved by origami methods.

Robert Geretschläger. Folding the regular heptagon. CM 23:2 (Mar 1997) 81-88. Shows how to do it exactly, using the result of his previous paper.

Dirk Bouwens, proposer; Alan Slomson & Mick Bromilow, independent solvers. An early protractor. M500 171 (Dec 1999) 18 & 173 (Apr 2000) 16-17. Draw a semicircle on diameter AOB. Draw the perpendicular through O and extend it to C so that BC = BA (the diameter) [this makes c = OC = 3] . If P divides AO in the ratio λ : 1-λ, then draw CP to meet the semicircle at D and OD divides the arc ADE in approximately the same ratio. He finds the exact value of the angle AOD and finds that the maximum error in the process is only .637^o (when λ  .18). Second author provides a graph of the error and says the maximum error is at about 18^o. [I get .637375^o at .181625^o.]

Ken Greatrix. A better protractor. M500 175 (Aug 2000) 14-15. Taking c = 1.67721 in the previous construction gives a more accurate construction, with maximum error about .32^o at about 13^o. [I get .324020^o at .130164^o.]