The Trace-Determinant Plane

The essential behavior of a planar linear dynamical system, x' = Ax where A is a 2 by 2 matrix, can be characterized by the trace and determinant of the matrix A. Below is an animation of a collection of such systems when the matrix is rotated through all of the differing possibilities.

A happy halloween tridecahexaflexagon

I've been thinking a lot about hexaflexagons lately; here is a 13-faced hexaflexagon pattern: Double-sided pattern To construct it, you need to fold matching triangles together when they are adjacent. If you haven't made a more basic one before, I recommend making a hexahexaflexagon after watching some Vi Hart videos:

• https://www.youtube.com/watch?v=VIVIegSt81k
• https://www.youtube.com/watch?v=paQ10POrZh8
• https://www.youtube.com/watch?v=AmN0YyaTD60
• https://www.youtube.com/watch?v=GTwrVAbV56o