Would a biplane be able to fly if the upper and lower wings were in opposite dihedral? Imagine a wing configuration similar to the fictional "X-wing" design from StarWars. What would be the theoretical pros and cons of using such an approach? Ignore the engines, fuselage, tail, etc
What happens if you take it to its logical extreme and make all wings at 45 degree angles?
There's a line of airplanes that have already done it, at least a bit.
The Rutan tandem wing Quickie designs are exactly that configuration, except that the wings are radically staggered because the forward one is the trimming/pitch control surface, and the dihedral/anhedral angles are shallower. But looked at from straight ahead, there it is.
The anhedral of the front wing is mostly there because that's where the wheels go. Versions of the 2 seat Q2 that have a conventional tricycle gear, have a nearly straight lower wing because the most of the anhedral is "in the way" (tips near the ground become a problem now without wheels there).
The problem with the configuration is other than being able to put the gear at the wing tips in Rutan's application, there's no point to it. That anhedral wings cancel out the dihedral wings, form a roll stability perspective, making the result more or less the same as making all the wings straight. If you do want to put the wheels there, you have to make the spars do double duty as really long leaf springs (bending loads from landing are much harder on the spars than flight loads), and there are ground handling issues with the wheels so far apart.
So might as well avoid the hassles of fabricating bent spar beams and make them straight across in the first place, or at least with more normal dihedral angles. It's more or less just something that looked cool from a science fiction perspective.
There's no particular need, from a physics perspective, to have dihedral in an airplane. It's there for stability. If you roll to one side, you naturally start slipping to that side. In an aircraft that has a positive dihedral, this slip means the lower wing will have greater lift than the upper wing. This creates a rolling moment back towards level.
An anhedral (negative dihedral) wing would have the opposite effect, increasing roll rate away from level. Note that this isn't necessarily a bad thing: it's possible for planes (such as the An-225) to be too stable, and thus, too difficult to roll when you actually want to. Having anhedral wings fixes that issue.
But there is another effect. Both dihedral and anhedral slightly reduce lift. Since the lift vectors of each wing aren't parallel, a small portion of the lift the wings might have generated is lost. This translates into slightly higher stall speeds, and slightly more drag.
In short, any "good" that the two opposite sets of wings might do would cancel each other*, while the "bad" would add together. You could get such an airplane to fly, it would just be less efficient than it otherwise would have been.
*Of course, if they're not at equal angles, or one is bigger than the other, or they have different airfoil shapes, etc., then they won't perfectly cancel, and you'll still get some effect based on which one is "stronger".
Back in the 1980s (when Star Wars was still fairly fresh) there were a couple different radio control model designs that had the "X-wing" layout -- a strutless biplane with approximately equal upper dihedral and lower anhedral. One was a canard, the other was tailless (in order to look even more like an X-wing from the movie).
They flew fine, with the very neutral "goes where you point it" stability of a (pre-3D, pre-Turnaround) Pattern aerobatic airplane. They couldn't fly knife edge, because they had little rudder authority and couldn't keep the nose up at 90 degree bank (and were sloppy in point rolls for the same reason), but in non-aerobatic flight, they handled about as well as a conventional mid-wing, zero-dihedral Pattern design of the day.
The main issue with biplanes is the separation between the planes, in relation to the wing chord. They need to be well separated to avoid interference and hence inefficiency. Anhedral lower plus dihedral upper increases the separation towards the tips, making them more efficient. Another way of looking at it is that the centre sections are bunched close together, making them inefficient. But the tips already suffer greater inefficiencies due to air spilling round the ends, with the centre sections providing the main lift. So what you are actually doing is making the whole wing inefficient. This is why many biplanes raise the upper centre section well above the fuselage on cabane struts.
As far as dihedral goes, the two angles balance each other out. If you want a stable plane you will need to increase the dihedral over the anhedral.
At extreme angles, say 45 deg, the wing offers significant side area. This configuration is in fact common on high-speed air-to-air missiles, in conjunction with a cruciform tail, as the control surfaces can now generate side forces and this can greatly improve manoeuvrability.
