This was originally intended for the toss-it contest, but it didn't quite fit, I got hung up on the video, and they the local virus swept through the family and I never got around to finishing it. Now that the toss-it contest is over, I can just publish without the video, and maybe it will still be interesting...
This isn't so much plans for a particular paper airplane, as a concept and technique that could be used for a whole new style of airplane, not TOO much more complex or expensive than the typical all-paper airplane.
Step 1: Weighing Paper
Here, we have a regular 8.5x11 inch piece of paper. It weighs about 4.6 grams (Isn't it fun to mix those metric and american units?) If we fold this into a paper airplane, it will still weigh the same (of course), but only PART of the weight will be used in "functional" parts of the plane like the wing surfaces. In fact, in a model like the classic Dart, only about half of the paper will end up in the wings (and a good part of that is folded in multiple layers, so it's not particularly efficiently applied to the wings.
Step 2: Weighing Other Things.
So what else is shaped like the fuselage (body) of an airplane? How about one of those long, skinny balloons that are typically twisted into animal shapes?
One of these surprisingly rigid when inflated, at least compared to a piece of paper. You can attach wings and a tail with tape, weight the nose, and create a number of interesting plane designs. And it's still cheap and still easy to experiment with!
Here we have some balloons, weight in at 1.6 to 2.1 grams. Slightly more when inflated, of course (thereby demonstrating that air has mass!) That may not seem like a lot less than the paper when you're looking at the un-inflated balloon, but that 2.1g balloon is about five feet long when inflated. That's either a very large plane, or ... we can use PART of the balloon!
Step 3: Cutting a Balloon in Pieces
There are probably several ways to cut a balloon into pieces, and have each of the pieces remain usable. Here, I demonstrate partially inflating the balloon first, which has the advantage that you can see the "finished" size. Inflate as much as you want, tie it off, and then cut. Tie the remaining piece at the end so it can be inflated too.
I wind up with a fuselage about 20 inches long and 1.2g. That's more reasonable!
Step 4: Attach Wings and Things
Now that you have your near-massless fuselage, you can attach paper wings, tail, canard, and/or whatever, using tape. If you use Post-It like, or painter's tape, it should be relatively easy to remove and reposition the pieces. You can use more tape in the nose as balancing weight, or resort to heavier items like paper clips or cardboard (also taped on.)
One thing to watch out for is that this sort of fuselage has its own curvature, and using typically sized pieces of tape will tend to impart that curvature to the paper as well, where it will usually be in the "wrong" direction.
The design pictured here uses a 1x11 inch strip of paper to form a forward canard "loop", and the remainder the single sheet as a large wing slightly behind the midpoint. The large sheet has the forward edge folded over to provide some stiffness and (hopefully) to produce a slightly airfoiled shape. It works "ok" (see the video.)
Step 5: Advanced Topics
I suppose you can get even lower mass (and negative weight!) by filling the balloon with helium instead of air... It won't last long; helium diffuses through balloon-weight latex quite rapidly, and even filled with air these fuselages tend to shrink over a period of several days, changing the aerodynamics of your creation...
You can add more rigidity to a balloon fuselage by putting strips of tape along the sides where you don't want it to bend. There's a piece of composite materials science that says the rigidity of a composite structure depends on the elasticity of the outside layers and the distance between them. That's why you see things like carbon fiber combined with relatively weak 'core' materials like balsa - the core provides the separation without much mass, and the carbon fiber provides very in-elastic outer layers, so you get a very rigid, very light core. The balloon fuselage has a pretty big distance, but the outer layers aren't at all inelastic (they're RUBBER!) Tape stretches less, so adding it makes the whole structure more rigid. Of course, the tape also adds mass...
Another variable to play with is the balloon itself. The common "twisting" balloon is actually a Qualtex 260Q (2 inch diameter, 60 inches long) balloon. If you hunt around for A store that caters to Balloon Twisting Artists, you'll discover that there are several other sizes of long balloons. There's a 160Q that is only 1 inch in diameter, and there are varieties that are 3x50 and 6x46 inches. And of course the balloons are available in a myriad of colors (including clear!) that you don't usually see at your local variety store. Or you can color on any balloon using a permanent marker...
Experiment! And have fun...