## Sequence of right triangles

Consider the right triangle $\triangle_1$ with vertices at $(0,0)$, $(0,1)$, and $(1,1).$ Note that the angle at the origin is $\frac{\pi}{4}.$ Use the hypotenuse of $\triangle_1$ as one of the legs of another right triangle, $\triangle_2,$ with an angle of $\frac{\pi}{8}$ at the origin, and whose second leg extends towards the $x$–axis. Continue constructing adjacent right triangles $\triangle_n$ in the same fashion, with one leg supported on the hypothenuse of the previous right triangle $\triangle_{n-1},$ with angle $\frac{\pi}{2^{n+1}}$ at the origin, and with a second leg extending towards the $x$–axis.

Note that, since the angles at the origin form the series $\sum_{n=2}^\infty \frac{\pi}{2^n},$ (that add up to $\frac{\pi}{2}$) the limit of the previous process is an actual segment on the $x$–axis.

1. What is the length of this segment?
2. Find an equation in polar coordinates $(\rho,\theta)$ of a curve passing through all the vertices of the resulting triangles, except possibly $(0,0).$
3. Find an equation for that curve in rectangular coordinates.

Let us denote $h_n$ the hypotenuse of triangle $\triangle_n.$ By construction we have $h_1=\sqrt{2}$ and, for each consecutive steps, the definition of cosine in our setup gives us that

$\cos \dfrac{\pi}{2^{n+1}} = \dfrac{h_{n-1}}{h_n}.$

We may solve the recurrence with the aid of the formula of sine of double angles: $\sin 2\alpha = 2 \sin \alpha \cos \alpha.$ In our case, setting $\alpha = \frac{\pi}{2^{n+1}}$, we have

$\dfrac{h_{n-1}}{h_n} = \cos \dfrac{\pi}{2^{n+1}} = \dfrac{\sin \frac{\pi}{2^n}}{2\sin \frac{\pi}{2^{n+1}}}$

or equivalently,

$h_n = \dfrac{2\sin \frac{\pi}{2^{n+1}}}{\sin \frac{\pi}{2^n}}\, h_{n-1} = \dfrac{2\sin \frac{\pi}{2^{n+1}}}{\sin \frac{\pi}{2^n}}\, \dfrac{2\sin \frac{\pi}{2^n}}{\sin \frac{\pi}{2^{n-1}}}\, h_{n-2} = \dotsb = 2^{n-1} \dfrac{\sin \frac{\pi}{2^{n+1}}}{\sin \frac{\pi}{2^2}} h_1$

which is the same as

$h_n = 2^n \sin \frac{\pi}{2^{n+1}} = \dfrac{\pi}{2}\, \dfrac{2^{n+1}}{\pi}\, \sin \frac{\pi}{2^{n+1}}.$

We use this identity to obtain the answer to the first two questions. For instance, an expression in polar coordinates of a function that goes through all the vertices of the triangles could be given by

$\rho(\theta) = \dfrac{\pi}{2}\, \dfrac{\sin \theta}{\theta}.$

And in that case, the length of the segment at the limit is given by

$\displaystyle{\lim_n h_n = \lim_{\theta\to 0} \rho(\theta) = \frac{\pi}{2}}.$

I leave the third part of the puzzle an an exercise to the reader.

## Miscellaneous

This is the $\LaTeX$ code used with tikz to produce the diagram above. Note the power of the plot command, that allows us to obtain a very accurate graph of any function.

\begin{tikzpicture}[scale=2]
\draw[very thick,-&amp;gt;] (-0.49,0) -- (1.99,0) node[right]{$x$};
\draw[fill=red] (0,0) -- (0,1) -- (1,1) -- cycle;
\draw[fill=red!60!white] (0,0) -- (22.5:1.5307) -- (1,1) -- cycle;
\draw[fill=red!30!white] (0,0) -- (11.25:1.5607) -- (22.5:1.5307) -- cycle;
\draw[fill=red!10!white] (0,0) -- (5.625:1.56827) -- (11.25:1.5607) -- cycle;
\draw[domain=0.001:3.141,smooth,variable=\t,very thick,blue] %
plot ({\t r}:1.570796*{sin(\t r)}/\t);
\draw (90:0.2) arc (90:45:0.2);
\draw (45:0.3) arc (45:22.5:0.3);
\draw (22.5:0.4) arc (22.5:11.25:0.4);
\draw (11.25:0.5) arc (11.25:5.625:0.5);
\draw[very thick,-&amp;gt;] (0,-0.49) -- (0,1.49) node[above]{$y$};
\draw[fill=white] (0,1) circle (0.5pt);
\draw[fill=white] (1,1) circle (0.5pt);
\draw[fill=white] (22.5:1.5307) circle (0.5pt);
\draw[fill=white] (11.25:1.5607) circle (0.5pt);
\draw[fill=white] (5.625:1.56827) circle (0.5pt);
\end{tikzpicture}

See also the post Interaction with sage for some other techniques to produce similar figures.

### References

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