<s>[INST] Area fractal pentagrams I When I saw this image I was a little curious.
How can I find the area of this fractal?

 [/INST] Each segment of the pentagram is the initiator of the fractal. Take its length to be 1. Now the generator consists of 2 line segments each of length $\frac{1}{3}$.
Hence on each iteration $n$ the area can be expressed as follows:
$$A_n=10\sum_{k=0}^{n}2^kS_k +S_{p}$$
Where $S_p$ is the area of the regular pentagon and:
$$S_k=\frac{1}{2}\frac{1}{3^{2k}}\sin\frac{\pi}{5}$$
is the area of the "k-th generation" petal.
$$10\sum_{k=0}^{n}2^kS_k=5\pi\sin\frac{\pi}{5}\sum_{k=0}^{\infty}\left(\frac{2}{9}\right)^k=\frac{45\pi}{7}\sin\frac{\pi}{5}$$
The area of the pentagon is:
$$S_p=\frac{t^2\sqrt{25+10\sqrt{5}}}{4}$$
where $t=2\sin\frac{\pi}{10}$
Finally,
$$A=\frac{45\pi}{7}\sin\frac{\pi}{5}+\sqrt{25+10\sqrt{5}}\left(\sin\frac{\pi}{10}\right)^2$$
Or equivalently
$$A=\frac{45\pi\sqrt{2(5-\sqrt{5})}}{28}+\frac{\sqrt{25+10\sqrt{5}}}{4\phi^2}$$
or any other way you wish to think of it.
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