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Angle Trisection


TrisectionAngle

Angle trisection is the division of an arbitrary angle into three equal angles. It was one of the three geometric problems of antiquity for which solutions using only compass and straightedge were sought. The problem was algebraically proved impossible by Wantzel (1836).

Trisection

Although trisection is not possible for a general angle using a Greek construction, there are some specific angles, such as pi/2 and pi radians (90 degrees and 180 degrees, respectively), which can be trisected. Furthermore, some angles are geometrically trisectable, but cannot be constructed in the first place, such as 3pi/7 (Honsberger 1991). In addition, trisection of an arbitrary angle can be accomplished using a marked ruler (a Neusis construction) as illustrated above (Courant and Robbins 1996).

An angle can also be divided into three (or any whole number) of equal parts using the quadratrix of Hippias or trisectrix.

AngleTrisectionSteinhaus

An approximate trisection is described by Steinhaus (Wazewski 1945; Peterson 1983; Steinhaus 1999, p. 7). To construct this approximation of an angle A having measure alpha, first bisect A and then trisect chord BE (left figure above). The desired approximation is then angle DAB having measure t (right figure above). To connect t with alpha/3, use the law of sines on triangles DeltaDAB and DeltaEAD gives

 (sint)/(DB)=(sinx)/(AD)=(sinbeta)/(ED),
(1)

so sint=2sinbeta. Since we also have beta=(alpha/2)-t, this can be written

 sint=2[sin(1/2alpha)cost-sintcos(1/2alpha)].
(2)

Solving for t then gives

 t=tan^(-1)((2sin(1/2alpha))/(1+2cos(1/2alpha))).
(3)
AngleTrisectionError

This approximation is with 1 degrees of alpha/3 even for angles alpha as large as 120 degrees, as illustrated above and summarized in the following table (Petersen 1983), where angles are measured in degrees.

alpha ( degrees)alpha/3 ( degrees)t ( degrees)s ( degrees)
103.3333333.3338043.332393
206.6666666.6704376.659126
3010.00000010.0127659.974470
4013.33333313.36372713.272545
5016.66666716.72637416.547252
6020.00000020.10390919.792181
7023.33333323.49973723.000526
8026.66666726.91751126.164978
9030.00000030.36119329.277613
9933.00000033.48623432.027533

t has Maclaurin series

 t=1/3alpha+1/(648)alpha^3+1/(31104)alpha^5+...
(4)

(OEIS A158599 and A158600), which is readily seen to a very good approximation to alpha/3.


See also

Angle Bisector, Archimedes' Spiral, Circle Squaring, Conchoid of Nicomedes, Cube Duplication, Cycloid of Ceva, Maclaurin Trisectrix, Morley's Theorem, Neusis Construction, Origami, Pierpont Prime, Quadratrix of Hippias, Tomahawk, Trisectrix

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References

Bogomolny, A. "Angle Trisection." http://www.cut-the-knot.org/pythagoras/archi.shtml.Bogomolny, A. "Angle Trisection by Hippocrates." http://www.cut-the-knot.org/Curriculum/Geometry/Hippocrates.html.Bold, B. "The Problem of Trisecting an Angle." Ch. 5 in Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 33-37, 1982.Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 190-191, 1996.Courant, R. and Robbins, H. "Trisecting the Angle." §3.3.3 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 137-138, 1996.Coxeter, H. S. M. "Angle Trisection." §2.2 in Introduction to Geometry, 2nd ed. New York: Wiley, p. 28, 1969.Dixon, R. Mathographics. New York: Dover, pp. 50-51, 1991.Dörrie, H. "Trisection of an Angle." §36 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 172-177, 1965.Dudley, U. The Trisectors. Washington, DC: Math. Assoc. Amer., 1994.Geometry Center. "Angle Trisection." http://www.geom.umn.edu:80/docs/forum/angtri/.Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 25-26, 1991.Klein, F. "The Delian Problem and the Trisection of the Angle." Ch. 2 in "Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, pp. 13-15, 1980.Loy, J. "Trisection of an Angle." http://www.jimloy.com/geometry/trisect.htm.Ogilvy, C. S. "Solution to Problem E 1153." Amer. Math. Monthly 62, 584, 1955. Ogilvy, C. S. "Angle Trisection." Excursions in Geometry. New York: Dover, pp. 135-141, 1990.Peterson, G. "Approximation to an Angle Trisection." Two-Year Coll. Math. J. 14, 166-167, 1983.Scudder, H. T. "How to Trisect and Angle with a Carpenter's Square." Amer. Math. Monthly 35, 250-251, 1928.Sloane, N. J. A. Sequences A158599 and A158600 in "The On-Line Encyclopedia of Integer Sequences."Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.Wantzel, M. L. "Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas." J. Math. pures appliq. 1, 366-372, 1836.Wazewski, T. Ann. Soc. Polonaise Math. 18, 164, 1945.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 25, 1991.Yates, R. C. The Trisection Problem. Reston, VA: National Council of Teachers of Mathematics, 1971.

Cite this as:

Weisstein, Eric W. "Angle Trisection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AngleTrisection.html

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