• The exciting new world of AI prompt injection via). Promote your business by running a bot that uses other people’s social media post text to prompt a text-writing AI that generates customized responses to those posts. What could go wrong?
• ## Counting paths in convex polygons

Let’s count non-crossing paths through the all points of a convex polygon. There is a very simple formula for this, $$n2^{n-3}$$ undirected paths through an $$n$$-gon, but why? Here’s a simple coloring-based argument that immediately gives this formula.

• ## Comparing distances along lines

I’ve written here several times about Gilbert tessellations, most recently in last year’s post about cellular automata that naturally generate them. These are polygonal subdivisions of the plane, generated from the tracks of particles moving at the same speed, where the particles start as oppositely-moving pairs with random locations and directions, and continue moving until they crash into the track of another particle. Here’s Wikipedia’s illustration of these things, generated in 2012 by Claudio Rocchini:

• ## Permuted points of interest

Suppose you have a map, with certain points of interest marked. To avoid cluttering the map with the labels of these points, you want to list the labels in a line down the side of the map, with each point of interest connected to its label by a line segment. Preferably, the line segments should not cross each other, as they did when I tried to match the sidebar to the points in this example from Google Maps: