An angle trisection

William R. Raiford, An approximate trisection, American Mathematical Monthly, vol. 68, no. 9, Nov 1961, p. 917.

Sorry, no pictures. Explanation here.

Construction

The construction described in the article cited at the top of the page, is quite straightforward. Consider the angle $AOB$ represented by the circular arc $AB$ centered at $O$, as shown in the diagram above. To trisect $AOB$ do:

  1. Erect a perpendicular to $OA$ at $A$ (shown in green).
  2. Construct the bisector $OC$ of the angle $AOB$.
  3. Connect $B$ to $C$ and extend to intersect the green line at a point $T$.
The line $OT$ is an approximate trisector of the angle $AOB$.

Error Analysis

Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $AOT$, respectively. One may verify that \[ \beta = \arctan \Big( \sin\alpha - (1 - \cos\alpha) \cot \big( \frac{3}{4}\alpha \big) \Big) = \frac{1}{3}\alpha + \frac{1}{2^3\cdot3^4} \alpha^3 + O(\alpha^5) = \frac{1}{3}\alpha + \frac{1}{648} \alpha^3 + O(\alpha^5). \]

The error $ \ds e(\alpha) = \beta - \frac{\alpha}{3} $ is monotonically increasing in $\alpha$. The worst error on the interval $0 \le \alpha \le \pi/2$ is $e(\pi/2)$ = 0.0063 radians = 0.361 degrees. The worst error on the interval $0 \le \alpha \le \pi$ is $e(\pi)$ = 0.06 radians = 3.435 degrees.

Raiford, whose affiliation is given as IBM, states that he has calculated the error in increments of one degree in an IBM 709. Computers were novelties when that article was published.

This applet was created by Rouben Rostamian using David Joyce's Geometry Applet on June 14, 2010.

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