Why do flat sheets of paper twist more than paper cones as they fall? If I cut a 10cm circle out of a piece of paper and drop it, it will twist as it falls, even if it's rigid. If I then cut out a one-quarter wedge of the circle and tape the remaining three-quarters into a cone, the cone will drop straight down stably.
What's the difference? If I make a series of cones with ever-smaller wedges cut out, I approach a flat circle, so presumably at some angle an instability develops that results in significant twisting. What happens at that instability? Can I predict roughly the angle where it occurs without a full-on numerical simulation? Can I make a "flutter number" out fluid and cone properties that tells me how the relevant parameters can scale with each other to produce similar behavior?
 A: So I think there's a couple things going on here, all related to fluid dynamics. I'm not entirely sure, but I'll start with the fluttering.
There are two types of fluttering you could be seeing. The first is if it could flutter like a flag in the wind as the paper drops sideways or moves diagonally through the air. This is a result you can derive in fluid mechanics, using some pretty heavy math. It's not an intuitive result. If there's a flag in the wind, the windspeed is the same on both sides of the flag, so it's symmetric. Why then does the flag decide to wave? It turns out this configuration is unstable, so any tiny motion in the flag gets amplified until you get full on fluttering.
The second type of fluttering could be what's called vortex shedding. Think of a car driving along the road. As the car drives, it moves the air in front of it out of the way and creates a vacuum behind the car as it moves forward. As air tries to rush into this vacuum it twists and creates what are called vortices. This is what causes leaves or debris in the road behind cars to swirl around. These vortices build up over time until they get too big and are shed from the object that formed them. Because the paper is flexible it bends while building up a vortex. Then as it sheds the vortex it springs back to it's original position.
Or the fluttering could be due to none of these effects and there could be some other type of instability at work. I would bet it's one or both of the effects I described, but fluid mechanics is an extremely complex subject that I'm still not wholly comfortable with.
The last effect that's worth noting is the fact that a cone falls with stability while a paper will twist. There's a theorem in fluid mechanics that says the most favorable way for an object to fall is the way that maximizes the drag force on that object. A piece of paper wants to fall completely flat, slowly towards the floor. The problem is that there are other forces that come into play. The paper bends and changes geometry and it creates turbulence around itself which makes it go every which way. When you make the paper into a cone you some additional structure and this makes it much easier to resist those other forces and fall flat. You'll notice though that if you make your cone too pointed that even though it's more aerodynamic, the cone will start to wobble on the way down. This is because there comes a point where a cone falling on it's side will create more drag than a cone falling point downwards.
