A TRISECTRIX FROM A CARPENTER S SQUARE

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1 A TRISETRIX FRM A ARPENTER S SQUARE AVI RIHESN arxiv: v1 [math.h] 11 Mar 2016 Abstract. In 1928 Henry Scudder described how to use a carpenter s square to trisect an ange. We use the ideas behind Scudder s technique to define a trisectrix a curve that can be used to trisect an ange. We aso describe a compass that coud be used to draw the curve. In 1837 Pierre Wantze famousy proved that it is impossibe to trisect an arbitrary ange using ony a compass and straightedge [13]. However, it is possibe to trisect an ange if we are aowed to add additiona items to our tookit. We can trisect an ange if we have a marked straightedge [7, pp ], a Mira (a vertica mirror used to teach transformationa geometry) [4], a tomahawk-shaped drawing too [11], origami paper [5], or a cock [9]. We can aso trisect an ange if we are abe to use curves other than straight ines and circes: a hyperboa [15, pp ], a paraboa [3, pp ], a quadratrix [7, pp ], an Archimedean spira [2, p. 126], a conchoid [15, pp ], a trisectrix of Macaurin [10], a imçon [15, pp ], and so on; such a curve is caed a trisectrix. In many cases, we can use speciaydesign compasses to draw these or other trisectrices. For instance, escartes designed such a compass [1, pp ]. The iterature on different construction toos and techniques, new compasses, and their reationships to ange trisection and the other probems of antiquity is vast. A reader interested in earning more may begin with [1, 6, 7, 8, 15]. In this note we describe a trisection technique discovered by Henry Scudder in 1928 that uses a carpenter s square [12]. Then we use the ideas behind this construction to produce a new trisectrix, and we describe a compass that can draw the curve. 1. Ange Trisection Using a arpenter s Square A carpenter s square a common drawing too found at every home improvement store consists of two straightedges joined in a right ange. To carry out Scudder s construction we need a mark on one eg such that the distance from the corner is twice the width of the other eg. For instance, we wi assume that the onger eg is one inch wide and that there is a mark two inches from the corner on the shorter eg. Let s say we wish to trisect the ange A in figure 1. First, we draw a ine parae to and one inch away from A; this can be accompished using 1

2 2 AVI RIHESN a compass and straightedge, but a simper method is to use the ong eg of the carpenter s square as a doube-edged straightedge. We now perform the step that is impossibe using Eucidean toos: Pace the carpenter s square so that the inside edge passes through, the two inch mark ies on the ine, and the corner sits on the ine (at the point, say). Then the inside edge of the carpenter s square and the ine trisect the ange. This procedure works for any ange up to 270, athough the arger the ange, the narrower the short eg of the carpenter s square must be. 2 E A F A Figure 1. A carpenter s square or a T-shaped too can be used to trisect an ange. In fact, we do not need a carpenter s square to carry out this construction. A we need is a T-shaped device (shown on the right in figure 1) in which the top of the T is two inches ong. It is not difficut to see that this technique trisects the ange: the right trianges F, E, and E in figure 1 are congruent. 2. A New ompass We now use the carpenter s square as inspiration to create a compass to draw a trisectrix (see figure 2). The device has a straightedge that is one inch wide and a T-shaped too with pencis at both ends of the two-inch top of the T. The ong eg of the T passes through a ring at one corner of the straightedge. The T can side back and forth in the ring, and the ring can rotate. ne penci draws a ine aong the straightedge. The other penci draws the curve we ca the carpenter s square curve. We use the compass as foows. Suppose we woud ike to trisect A in figure 3. Pace the bottom of the straightedge aong A with the ring ocated

3 A TRISETRIX FRM A ARPENTER S SQUARE 3 2 Figure 2. A compass to draw the carpenter s square curve. at. Use the compass to draw the straight ine and the carpenter s square curve. Say that intersects the curve at. Use an ordinary compass to draw a circe with center and a two-inch radius. It wi intersect at two points. Labe the right-most point (viewed from the perspective of figure 3). Then trisects the ange. Use an ordinary compass and straightedge to bisect to obtain the other trisecting ray. A Figure 3. We can use the carpenter s square curve to trisect an ange.

4 4 AVI RIHESN 3. The arpenter s Square urve What is this carpenter s square curve? oes it have a cosed form? Is it agebraic or transcendenta? (In [14], Yates used a carpenter s square in a different way to generate a different curve a cissiod. Yates gives an agebraic expression for his curve and shows that it can be used to compute cube roots.) First we introduce x- and y-axes. Let be the origin and A be the positive x-axis (see figure 4). Let = (x, y) and = (x + a, 1). ecause = 2, a 2 + (1 y) 2 = 4, and hence a = (3 y)(y + 1). (Notice that a 0 throughout the construction.) Aso, E is the midpoint of, so E = (x + a/2, (y + 1)/2). ecause and E are perpendicuar, (y + 1)/2 x + a/2 = a 1 y. Substituting our expression for a and simpifying, we obtain x 2 = (y 2)2 (y + 1). 3 y This agebraic curve has a sef-intersection at (0, 2), and y = 3 is a horizonta asymptote. However, as we see in figure 4, our compass does not trace this entire curve. 3 y x 1 Figure 4. The carpenter s square curve. To see an interactive appet of this trisection, visit ggbtu.be/mjpanpat. References [1] Henk J. M. os. Redefining geometrica exactness: escartes transformation of the eary modern concept of construction. Springer-Verag, New York, 2001.

5 A TRISETRIX FRM A ARPENTER S SQUARE 5 [2] ar. oyer and Uta. Merzbach. A History of Mathematics. John Wiey & Sons, New York, 2 edition, [3] René escartes. The Geometry of René escarte: Transated from French and Latin by avid Eugene Smith and Marcia L. Latham. over Pubications Inc., New York, [4] John W. Emert, Kay I. Meeks, and Roger. Neson. Refections on a Mira. The American Mathematica Monthy, 101(6): , [5] Koji Fusimi. Trisection of ange by Abe. Saiensu (suppement), page 8, ctober [6] Thomas L. Heath. A history of Greek mathematics. Vo. I: From Thaes to Eucid. arendon Press, xford, [7] Wibur Richard Knorr. The ancient tradition of geometric probems. over Pubications Inc., New York, [8] George E. Martin. Geometric constructions. Springer-Verag, New York, [9] Leo Moser. The watch as ange trisector. Scripta Math., 13:57, [10] Hardy. Reyerson. Anyone can trisect an ange. Mathematics Teacher, 70: , Apri [11] ertram S. Sackman. The tomahawk. Mathematics Teacher, 49: , Apri [12] Henry T. Scudder. iscussions: How to trisect an ange with a carpenter s square. The American Mathematica Monthy, 35(5): , [13] P. L. Wantze. Recherches sur es moyens de reconnaître si un Probème de Géométrie peut se résoudre avec a rège et e compas. Journa de Mathématiques Pures et Appiquées, 2(1): , [14] Robert. Yates. The ange ruer, the marked ruer and the carpenter s square. Nationa Mathematics Magazine, 15(2):61 73, [15] Robert. Yates. The Trisection Probem. The Frankin Press, aton Rouge, LA, ickinson oege, arise, PA E-mai address: richesod@dickinson.edu

Section V.1.Appendix. Ruler and Compass Constructions

V.1.Appendix. Ruler and Compass Constructions 1 Section V.1.Appendix. Ruler and Compass Constructions Note. In this section, we explore straight edge and compass constructions. Hungerford s expression