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An angle trisection

Free Jamison, Trisection Approximation, American Mathematical Monthly, vol. 61, no. 5, May 1954, pp. 334–336.

Sorry, no pictures. Explanation here.

The construction

This construction, due to Free Jamison (see the reference at the top of this page) is a more accurate variant of the construction described in a simpler construction.

Consider the circular arc AB centered at O, shown in the diagram above. Assume the angle AOB is between 0 and 180 degrees. To trisect AOB, do:

  1. Pick the points F and D on the arc BA such that arc BF = 2/8 of the arc BA and arc BD = 3/8 of the arc BA.
  2. Extend FO to intersect the circle at a point C.
  3. Draw the line CD and extend it to a point E such that DE equals the circle's diameter.

The line OE is an approximate trisector of the angle AOB.

Error Analysis

Let α and β be the sizes of the angles AOB and EOB, respectively. The angle FOD equals α/8 by the construction, therefore the angle FCD, which is half the central angle FOD, is equal to α/16. The triangle DOC is isosceles, therefore the angle ODC also equals α/16.

In the triangle OED, let x and y be the sizes of the angles OED and EOD, respectively. Since the sum x+y of the triangle's internal angles equals the triangle's external angle ODC, we have x+y=α/16. Let us note, however, that the angle y equals DOB minus EOB. Thus y=3α/8β, whence x=β5α/16.

In the triangle OED, the side DE is twice the side OD by the construction, therefore the law of sines gives siny=2sinx. Consequently, sin(3α/8β)=2sin(β5α/16). Solving this for β we arrive at: β=516α+arctansin(a/16)2+cos(a/16)=13α121234α3+O(α5)=13α1331776α3+O(α5).

We see that the trisection error e(α)=α/3β is given by: e(α)=148αarctansin(a/16)2+cos(a/16). (This formula is also given in Jamison's article.) The function e(a) is monotonically increasing in α. The worst error on the interval 0απ/2 is e(π/2) = 0.0000117 radians = 0.00067 degrees. The worst error on the interval 0απ is e(π) = 0.000093756 radians = 0.00537 degrees. Quite impressive!


This applet was created by Rouben Rostamian using David Joyce's Geometry Applet on July 22, 2002.
Cosmetic revisions on June 7, 2010.

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