Month: February 2019

# Pentagon Intrigue

I wrote this post recently about some idle maths I did while trying to doodle regular pentagons and pentagrams.

My parents (both mathematicians) were immediately interested, and started trying to create pentagons on paper via origami.

We found these videos on YouTube and had fun making the apparent regular pentagon and pentagram, but discovered the maths was slightly out, and they weren’t quite regular. As my dad said, “The cosine of the double angle (that’s the first one you make) is 1/square root of 10, but it should be 1/4(root 5 – 1). So the actual angle, which should be 72 degrees, is … (gets out old Casio calculator, dusts it, Oh it’s dead, my phone’s not charged, where’s the calculator on my computer?) … oh, you do it. It would be right if the square root of 5 was 2.200 instead of 2.236.”

I had a dim memory that I was taught how to fold a pentagon from an A4 sheet when I was doing my maths teacher training. I went hunting in some folders in a corner of my study, and found this:

Start with a sheet of A4 and fold along the diagonal:
Now fold that in half along a vertical axis:
Now open it out again:
You now have a new slanted fold across the middle:
Fold along this new fold:
You’re now going to create two new folds on either side:
You’ll do this by bringing the edges into the centre:
Turn it over to get a slightly better view of the pentagon:
Whether this is really a regular pentagon, I don’t know, I haven’t tried to do the maths yet.
It’s an exercise for the reader!
Edit: Look away now if you want to work it out for yourself…
According to my dad, “No, that’s not a regular pentagon. The angle at the top has cosine —1/3 but it ought to be — (sqrt{5} — 1)/4. Close, but …”
Also Colin Wright (@ColinTheMathmo) spotted that “It’s very close, but the angle where the first corners are folded to meet is not 108 degrees.”
So if you want to make something that looks like a pentagon, the above solution is pretty neat.
But if you want an accurate pentagon, we think this solution is probably right.

# Pentagons, Pentagrams, Doodles and Trigonometry

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. 🙂

# .Net Rocks podcast – “Teaching Experienced Developers”

While I was at NDC London last week I had a lot of fun chatting to ⁩ ⁦and ⁩ for the .Net Rocks podcast.

We talked about a lot of things related to teaching and learning: different learning models, standards for education in software and how to create a culture that allows people to not know things.

You can listen to the interview here.