I'm planning to make a short video about hexaflexagons in which I'll demonstrate a kind of move I haven't seen described elsewhere: sliding the band "along" itself. This is not to be confused with flexing, which causes the band to slide "through" itself. "Along", "through" --- ordinary English prepositions aren't much help here. Is there any existing terminology for describing these kinds of motions of a band? To be clearer (or at least more techincal) about what I mean: Imagine we coordinatize the band with coordinates theta and y, where theta goes from 0 to 2pi and y goes from -1 to +1. These coordinates hide the details of the imbedding (and the three half-twists), but for present purposes that's desirable. If you imagine making each point (theta,y) travel in a circle of radius |y| lying in a plane locally perpendicular to the theta-axis, then what you're doing (more or less) is flexing the hexaflexagon. In contrast, what I'm talking about is moving the point (theta,y) along the band parallel to the theta-axis. I've gone through all Martin Gardner's writings, and watched a few YouTube videos about hexaflexagons, but haven't found anything along these lines. Jim Propp