2013년 10월 24일 목요일

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)
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

t has Maclaurin series

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

(Sloane's A158599 and A158600), which is readily seen to a very good approximation to alpha/3.
Wolfram

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