Category Archives: geometry

Orthogonal polygons

It’s time to say more about PWW #21, in which I exhibited things like this: Quite a few commenters figured out what was going on, and mentioned several nice (equivalent) ways to think about it. Primarily, the idea is to … Continue reading

Posted in combinatorics, geometry | Tagged , , | 5 Comments

Post without words #21

Posted in combinatorics, geometry, posts without words | Tagged , , | 17 Comments

Paper torus with Villarceau circles

I made another thing! This is a torus, made from 24 crescent-shaped pieces of paper with slots cut into them so they interlock with each other. I followed these instructions on There is also a template with some ideas … Continue reading

Posted in geometry | Tagged , , , , ,

Computing sums of primitive roots

Remember this picture? It, and other pictures like it, express the fact that for a given , if we take the primitive roots for each of the divisors of , together they make up exactly the set of all th … Continue reading

Posted in geometry, pictures | Tagged , , , , , , | 9 Comments

Sums of primitive roots

In my previous post, we saw that adding up all the complex th roots of unity always yields zero (unless , in which case the sum is ). Intuitively, this is because the roots are symmetrically distributed around the unit … Continue reading

Posted in geometry, pictures | Tagged , , , , , , | 4 Comments

Sums and symmetry

Let’s continue our exploration of roots of unity. Recall that for any positive integer , there are complex numbers, evenly spaced around the unit circle, whose th power is equal to . These are called the th roots of unity. … Continue reading

Posted in geometry, pictures | Tagged , , , , , , | 2 Comments


Everyone should go check out this Kickstarter project for CURVAHEDRA, a sort of construction kit that lets you build beautiful stuff like this: So cool! You can read more about the math behind them here. These are designed by Edmund … Continue reading

Posted in geometry, links | Tagged , , ,