Angle Trisection
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).
Although trisection is not possible for a general angle using a Greek construction, there are some specific angles, such as 
and 
radians (
and 
, respectively),
which can be trisected. Furthermore, some angles
are geometrically trisectable, but cannot be constructed in the first place, such
as 
(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.
An approximate trisection is described by Steinhaus (Wazewski 1945; Peterson 1983; Steinhaus 1999, p. 7). To construct this approximation of an angle 
having measure
, first bisect 
and then trisect
chord 
(left figure above). The desired approximation
is then angle 
having measure 
(right figure above).
To connect 
with 
, use the
law of sines on triangles 
and 
gives
 |
(1)
|
so 
. Since we also have 
, this can be written
![sint=2[sin(1/2alpha)cost-sintcos(1/2alpha)].](/images/equations/AngleTrisection/NumberedEquation2.gif) |
(2)
|
Solving for 
then gives
 |
(3)
|
This approximation is with 
of 
even for angles 
as large as
, as illustrated above and summarized
in the following table (Petersen 1983), where angles are measured in degrees.
 ( ) |  ( ) |  ( ) |  ( ) |
| 10 | 3.333333 | 3.333804 | 3.332393 |
| 20 | 6.666666 | 6.670437 | 6.659126 |
| 30 | 10.000000 | 10.012765 | 9.974470 |
| 40 | 13.333333 | 13.363727 | 13.272545 |
| 50 | 16.666667 | 16.726374 | 16.547252 |
| 60 | 20.000000 | 20.103909 | 19.792181 |
| 70 | 23.333333 | 23.499737 | 23.000526 |
| 80 | 26.666667 | 26.917511 | 26.164978 |
| 90 | 30.000000 | 30.361193 | 29.277613 |
| 99 | 33.000000 | 33.486234 | 32.027533 |
has Maclaurin series
 |
(4)
|
(OEIS A158599 and A158600), which is readily seen to a very good approximation to 
.
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
REFERENCES:
Bogomolny, A. "Angle Trisection." https://www.cut-the-knot.org/pythagoras/archi.shtml.
Bogomolny, A. "Angle Trisection by Hippocrates." https://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." https://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." https://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."
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