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:

pentagon

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 🙂

 

More than a game: the Game of Life


English: Diagram from the Game of Life
English: Diagram from the Game of Life (Photo credit: Wikipedia)

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.

For me, it’s just fun though. When I wrote my first version of it back in 1981 I merely used the rubric in Basic Computer Games – there was no description of gliders or any of the other fascinating patterns that the game throws up – so in a sense I “discovered” them independently, with all the excitement that implies: it is certainly possible to spend hours typing in patterns to see what results they produce and to keep coming back for more.

  • “Life.bas” should run on any system that will support the Java SDK – for instance it will run on a Raspberry Pi – follow the instructions on the BINSIC page. A more up to date version may be available in the Github repository at any given time (for instance, at the time of writing, the version in Git supports graphics plotting, the version in the JAR file on the server only supports text plotting). On the other hand, at any given time the version in Git may not work at all: thems the breaks. If you need assistance then just comment here or email me adrianmcmenamin at gmail.