Sorry, no pictures. Explanation here.
The construction described in this page is due to Chris Alberts, who sent it to me in an email on March 15, 2011. I have paraphrased and rearranged his construction, but the differences from the original are cosmetic. The error analysis is mine.
Consider the circular arc $AB$ on the circle $C$ centered at $O$, shown in the diagram above. Assume that the angle $AOB$ is between 0 and 90 degrees. To trisect $AOB$, do:
The calculation of the coordinates of all the points that appear in the construction is elementary but the resulting expressions are massively large, therefore I will refrain from putting them on this web page. (I did the calculations in Maple).
Let $\alpha$ and $\beta(\alpha)$ be the sizes of the angles $AOB$ and $AOT$, respectively. Express the trisection error as $e(\alpha) = \frac{\alpha}{3} - \beta(\alpha)$. It turns out that $e(0) = e(\pi/2) = 0$. In the range $0$ to $\pi/2$ the error is the largest near 1.22175 radians = 70.0013 degrees. The maximum error is $2.32\times10^{-18}$ radians = $1.33\times10^{-16}$ degrees.
Expanding $\beta(\alpha)$ in power series we get: \[ \beta(\alpha) = \frac{1}{3} \alpha + \frac{5^9}{2^{13} \cdot 3^{40}} \alpha^{27} + O(\alpha^{29}) = \frac{1}{3} \alpha + \frac{1953125}{99595595440594360737792} \alpha^{27} + O(\alpha^{29}). \]
The extraordinary precision of Chris Alberts' trisection is a result of the application of a refinement technique which I will call Alberts' refinement. The 10 steps of his trisection procedure, described above, consist of three distinct stages:
The lines $L_1$ and $L_2$ intersect at the point $O$. Suppose that the line $OX$ is a crude trisector of the angle between $L_1$ and $L_2$. The rest of the diagram shows a straightedge and compass construction that produces a much finer trisection. Here are the details of the construction:
A quite straightforward calculation, involving an application of the law of sines in the triangle $APG$ leads to the equation: \[ \delta' = \delta - \arcsin\Big(\frac13 \sin3\delta\Big). \] Expanding this into power series in $\delta$, we obtain: \[ \delta' = \frac43 \delta^3 - \frac45 \delta^7 + O(\delta^9). \] This explains the notable efficiency of the refinement. For instance, if the value of $\beta$ has two significant digits after the decimal point, the value of $\beta'$ will have six significant digits after the decimal point.
Remark 1: Move the point $X$ in the diagram and note how insensitive the angle $LAG$ is to the choice of $X$. This indicates that even a crude initial approximation produces an excellent trisection.
Remark 2: If you examine closely Chris Alberts' trisection described earlier in this page, you will find buried in it two instances of Alberts' refinement.
Comparing the precision of the trisection described in this page to those of others presented on my website may not seem to be quite fair. After all, any approximate trisection may be applied iteratively to refine its own result. Nevertheless, I am making an exception in this case because in Chris Alberts' trisection, the iterative refinement is an inherent feature of the method.
This applet was created by Rouben Rostamian using David Joyce's Geometry Applet on March 23, 2011.
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