## A circle inscribing a pentagon

This is also from The Irrationals – though I had to ask for assistance over at Stack Exchange to get the right answer (as is so often the case the solution is reasonably obvious when you are presented with it).

Anyway, the question is: given a regular pentagon (of sides with length of 10 units) which is inscribed by a circle, what is the diameter of the circle?

This figure helps illustrate the problem: We are trying to find $2R$ and we know that $x=10$.

If we knew $r$ then we could answer as $R^2 = r^2 + (\frac{x}{2})^2$. We do not know that but we do know that $\sin(\frac{\pi}{5})=\frac{x}{2R}$ and hence, from the $\cos^2 + \sin^2 = 1$ identity: $\sin\frac{\pi}{5} = \frac{10}{2R} = \sqrt{1-\cos^2(\frac{\pi}{5})}$.

From our knowledge of the pentagon with sides of unit length (you’ll have to trust me on this or look it up – it’s too much extra to fit in here) we also know that $\cos(\frac{\pi}{5}) = \frac{\phi}{2} = \frac{1}{4}(1+\sqrt{5})$, where $\phi$ is the golden ratio.

Hence $2R =$ … well, the rest is left as an exercise for the reader 🙂

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## More than a game: the Game of Life

Conway’s Game of Life has long fascinated me. Thirty years ago I wrote some Z80 machine code to run it on a Sinclair ZX80 and when I wrote BINSIC, my reimplentation of Sinclair ZX81 BASIC, Life was the obvious choice for a demonstration piece of BASIC (and I had to rewrite it from scratch when I discovered that the version in Basic Computer Games was banjaxed).
But Life is much more than a game – it continues to be the foundation of ongoing research into computability and geometry – as the linked article in the New Scientist reports.