Sorry, no pictures. Explanation here.
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:
The line OP′ is an approximate trisector of the angle AOB.
Wayne as provided this javascrip application which provides an interactive interface to his construction. Thanks, Wayne!
Let α and β=τ(α) be the sizes of the angles AOB and A′OP′, respectively. It is straightforward to show that β=2arcsin(43sinα8)=α3+727⋅34α3+O(α5)=α3+710368α3+O(α5).
The error e(α)=τ(α)−α3 is monotonically increasing in α. The worst error on the interval 0≤α≤π/2 is e(π/2) = 0.002695 radians = 0.154 degrees. The worst error on the interval 0≤α≤π is e(π) = 0.0237 radians = 1.360 degrees.
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 A′P′, P′P″, 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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