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

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 $x+y = \alpha/16$. Let us note, however, that the angle $y$ equals $DOB$ minus $EOB$. Thus $y = 3\alpha/8 - \beta$, whence $x = \beta - 5\alpha/16$.

In the triangle $OED$, the side $DE$ is twice the side $OD$ by the construction, therefore the law of sines gives $\sin y = 2 \sin x$. Consequently, $\sin(3\alpha/8 - \beta) = 2 \sin(\beta - 5\alpha/16)$. Solving this for $\beta$ we arrive at: \[ \beta = \frac{5}{16} \alpha + \arctan \frac{\sin(a/16)}{2+\cos(a/16)} = \frac{1}{3} \alpha - \frac{1}{2^{12}\cdot3^4} \alpha^3 + O(\alpha^5) = \frac{1}{3} \alpha - \frac{1}{331776} \alpha^3 + O(\alpha^5). \]

We see that the trisection error $e(\alpha) = \alpha/3 - \beta$ is given by: \[ e(\alpha) = \frac{1}{48}\alpha - \arctan \frac{\sin(a/16)}{2+\cos(a/16)}. \] (This formula is also given in Jamison's article.) The function $e(a)$ is monotonically increasing in $\alpha$. The worst error on the interval $0 \le \alpha \le \pi/2$ is $e(\pi/2)$ = 0.0000117 radians = 0.00067 degrees. The worst error on the interval $0 \le \alpha \le \pi$ is $e(\pi)$ = 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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