What is the Lute of Pythagoras?

  • Draw a regular pentagon (lightly).
  • Draw its diagonals (darkly)

The outline of the dark shape is a concave, equilateral decagon. Pick one of the concave ((“Inny”)) corners and its two neighbours; the edges linking these can be two edges of a regular pentagon.

  • Draw this regular pentagon (lightly)
  • Draw its diagonals (darkly)

Repeat these two steps for as long as you want to. You’ll end up (in the limit) with a lightly-drawn kite with pretty diagonals drawn.

Why is it interesting?


Also, it’s just fun to mess around with it. What if you go off in random directions? What if you start with an irregular pentagon, can you make that work? Can you make it tessellate? Why does it fit together nicely? There’s all sorts you can do.

What has it to do with Pythagoras?

Bog all, so far as anyone knows. The earliest known mention of it is from 1990, which is generally classified as “post-Pythagorean.”

Who was Pythagoras of Samos?

He’s the one mathematician almost everyone has heard of. If you stop a random person on the street and ask them to name a theorem, they’ll probably avoid eye contact and walk on. But failing that, they’ll probably say Pythagoras.

He was born is Samos, in the eastern Aegean Sea, about 570BCE; he died around 495BCE. Little is known for sure of his life, and many of the stories surrounding him are almost certainly legendary (his approach to Hippasus’s proof that $\sqrt{2}$ is irrational may have inspired generations of Reviewer 2s, but almost certainly didn’t happen). He’s reputed to have done good work on triangles, astronomy and music (among other things).

Oh, and Pythagoras’s Theorem? He didn’t discover that, the Babylonians did, likely a millennium before the bearded wonder made his entrance.