Let (n \geq 2) be an integer. The regular (n)-gon inscribed in the complex unit circle and having (1) as a vertex, a.k.a., the convex hull of the (n)th roots of unity, is closed under complex multiplication.
The set of (n)th roots of unity is closed under multiplication, and
The product of two convex linear combinations of the vertices is itself a convex linear combination.
It's crucial to take the "standard" (n)-gon whose vertices are the (n)th roots of unity, i.e., not to take an arbitrary regular (n)-gon inscribed in the unit circle. The animations show the situation for (n = 7), with roots of unity ("standard") on the left, and the polygon rotated by one-tenth of a radian ("non-standard") on the right.
If you’ve been following me for a while, you know I used to have a mild obsession with drawing black holes. I drew this one the day my friend brought over an inexpensive set of alcohol markers. The ruler on the right explains how big everything is. It’s worth noting that it’s in logarithmic base four because: aliens. #doodle#markers#BlackHole#MathArt
Happy birthday to one of greatest #mathematicians of all time Emmy Noether (1882-1935), here with her eponymous theorem, the backbone of modern physics. Noether's theorem links any symmetry of a system with a conservation law. In my portrait, I chose to depict a young Emmy in front of a blackboard with a more simple formulation of her theorem & 3 specific applications of it, 🧵1/n