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Problem of the Day #25: Building an Infinite Crane April 13, 2011

Posted by Billy in : potd , trackback

Albert is constructing a crane. His basic building block is shown in Fig. 1.

Fig. 1

Fig. 1

Pentagon $ABCDE$ consists of isosceles triangle $ABC$ with $\overline{AB}=\overline{AC}$ and rectangle $ACDE$. Let $\angle ACB = \theta$, $\overline{CD}:\overline{AC} = k$, and $\overline{BC}=1$. Albert builds his crane by applying the following iteration ad infinitum starting with pentagon $ABCDE$:

  1. Call the most recently placed pentagon $ABCDE.$
  2. Construct a new pentagon $A’B'C’D'E’ \sim ABCDE$ with $\overline{B’C'}=\overline{DE}$.
  3. Move pentagon $A’B'C’D'E’$ so that $B’=E$ and $C’=D$.
  4. Repeat.

Fig. 2 shows Albert’s completed crane when $\theta=\frac{\pi}{6}$ and $k=\frac{3}{4}$.

Fig. 2

Fig. 2

It can be shown that as long as $0\leq\theta<\frac{\pi}{3}$ and $0\leq k$, the “tip” of the crane will converge on a single point. Let $P(\theta,k)$ represent the limiting point of a crane with angle $\theta$ and ratio $k$, and let $\mathcal{A}$ be the set of all points $P(\theta,k)$ for $0\leq\theta<\frac{\pi}{3}$ and $\frac{1}{2}\leq k\leq 2$. What is area of $\mathcal{A}$? Note that the pentagons are allowed to overlap.

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