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An angle trisection

Construction by Wayne Baker

Sorry, no pictures. Explanation here.

The Basic Construction

Here is a very simple straightedge and compass construction of an approximate angle trisector due to Wayne Baker.

Let us represent the angle by the circular arc AB centered at O; see the diagram above. The angle's size may be anything from 0 to 180 degrees. To trisect, do:

  1. Quadrisect the angle AOB, that is, divide it into four equal parts. The arc AP in the diagram above represents one quarter of the original arc AB. Let L be the length of the chord AP (shown in green).
  2. Draw a circular arc (shown in orange) centered at O and radius 3/4 of OA. Mark A and B its intersections with the rays OA and OB, respectively.
  3. Swing an arc (not shown) of radius L centered at A and mark P its intersection with the arc AB, as shown.

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

Note added October 2024

Wayne as provided this javascrip application which provides an interactive interface to his construction. Thanks, Wayne!

Error Analysis

Let α and β=τ(α) be the sizes of the angles AOB and AOP, respectively. It is straightforward to show that β=2arcsin(43sinα8)=α3+72734α3+O(α5)=α3+710368α3+O(α5).

The error e(α)=τ(α)α3 is monotonically increasing in α. The worst error on the interval 0απ/2 is e(π/2) = 0.002695 radians = 0.154 degrees. The worst error on the interval 0απ is e(π) = 0.0237 radians = 1.360 degrees.

Iterative Improvement

As we see in the asymptotic expansion shown above, the angle τ(α) is slightly larger than the target value of α/3. Making three copies of the constructed angle, and putting them end-to-end as in arcs AP, PP, and P''P''' shown in the diagram below, we arrive at the endpoint P''' which is very slightly off the point B', and just outside the arc A'B'. The constructible angle B'OP''' is exactly three times the error e(\alpha). If we were able to trisect B'OP''' exactly, then we would know the error, and consequently will have achieved the exact trisection of the original angle. Of course the exact trisection of B'OP''' is impossible in general, but we may repeat the method outlined in the Basic Construction above to obtain an approximate trisection of B'OP''', which will yield \tau\big(3\tau(\alpha) - \alpha\big) , and consequently an improved trisection \tau_{\mathrm{improved}}(\alpha) of the original angle: \tau_{\mathrm{improved}}(\alpha) = \tau(\alpha) - \tau\big(3\tau(\alpha) - \alpha\big) = \frac{\alpha}{3} - \frac{7^4}{2^{28}\cdot3^{13}} \alpha^9 + O(\alpha^{11}). The error \ds e_{\mathrm{improved}}(\alpha) = \frac{\alpha}{3} - \tau_{\mathrm{improved}}(\alpha) is monotonically increasing in \alpha. In particular, e_{\mathrm{improved}}(\pi/2) = 1.5\times 10^{-9} radians = 8.6\times10^{-8} degrees.


This applet was created by Rouben Rostamian using David Joyce's Geometry Applet on May 31, 2010.

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