I was in a workshop the other day and I started a doodle which I often do (or some version of it) when I’m in meetings and such:

I Iove drawing pentagrams because they’re so satisfying – five quick lines is all you need, without lifting your pen from the paper.

But whenever I do this doodle, it always bothers me that these are not really pentagrams, and the contained shapes are not pentagons. I often wonder what would happen to the tesselation if I was drawing real pentagons. I also wonder how I could draw proper pentagrams without a protractor.

(For clarity, a pentagram is a five-pointed star. A pentagon is a five-sided shape. And when I say “real” I mean “regular” – ie shapes with rotational symmetry, where every point, every angle, every side is equal. And “tesselation” is a word that describes the way different shapes slot together side by side, with no gaps (I learnt that word in primary school, in relation to Roman mozaics):

).

So during this workshop I did some trigonometry to work it all out. For those of you who were there with me, this is what I was scribbling in the breaks when I was being so antisocial:

The conclusion I came to was that on lined paper, I could get a reasonable approximation of a pentagram using the following proportions:

If you’re wondering what w represents, I drew it on a different diagram when I realised that G and w were not the same distance – they only appeared to be because I was drawing non-regular pentagrams:

Based on these proportions and the dots that each horizontal line was made of, I came up with the following not-bad pentagrams (they’d probably be better if I had a ruler available instead of drawing freehand):

…and now the pentagons are all regular pentagons, and the pentagrams are regular too, but they’re forced to collide with each other as a result.

For the sake of aesthetics I think I prefer the non-regular versions at the top of this post, but the mathematician in me is now happy. 🙂