How can a piece of paper be shaped to guarantee it falls slower than a plain sheet of paper?

My son and I are playing "MAKE" (think "HORSE") and we challenged ourselves to build the slowest falling object using only a sheet of paper. We cannot build anything that falls slower than a plain sheet of paper. Any ideas?

• to make it more interesting, increase the height, add some wind.
– Abel
Commented Aug 25, 2021 at 0:34
• Unrelated to your getting an answer, but I'm intrigued by your game. Do you just think of things to ....make? Then whoever does "worse" gets the letter? A few more examples would be fun! Commented Aug 25, 2021 at 16:33
• “HORSE”...? Even thinking it, I'm not sure what you're talking about. (Different countries, different continents, different cultures...)
– DaG
Commented Aug 25, 2021 at 20:17
• @DaG HORSE is a game where players take turns making shots with a basketball. When someone makes it, the other person has to make the same shot or they earn a letter. Whoever gets all the letters first loses and is a HORSE. I assume MAKE is similar to that, but I'm intrigued where OP gets their challenges from Commented Aug 25, 2021 at 22:56
• One of the winners in the the Great International Paper Airplane Contest (you can see it on the book cover) was a simple folded gyrocopter which worked quite well. Commented Aug 26, 2021 at 14:03

For regular paper weight and size A6, roughly a card postal size one can tear from the middle vertically the top half and bend the two flaps 90 degrees in opposite directions like a helicopter's blades and fold the bottom half on itself like a narrow rod to make it act like balast.

This will be a primitive gyro- copter and will descend in a slow controled manner as opposed to a paper swinging laterally and then diving in a possible sharp angle of attack.

After a few prototypes one will find the good cut size.

Larger papers could have trouble holding their geometry unless they are thicker.

• We are trying this and reinforcing the "wings" to prevent them from collapsing (we are using legal paper) but it still descends faster than the sheet of paper from 8 feet. We will continue pursuing this approach because it at least gives a more predictable descent. I'm thinking we want to maximize our wing size without compromising stability or adding too much weight but that is easier said than done! Commented Aug 25, 2021 at 15:03
• @CoreyAlix - Am I right in assuming you're dropping a piece of paper, which then kind of wavers right/left as it descends to the ground? That's going to be tough to beat IMO, so great question. Commented Aug 25, 2021 at 16:35
• I joined this site just to point that Nasa created a student project (project Make a Paper Mars Helicopter) which, IMHO, is exactly what you're looking for. Instructions on how to build a paper helicopter and clues as how to change the design to make it take even longer to hit the ground. Commented Aug 25, 2021 at 18:13
• @BruceWayne, there are falling modes that are even slower than side-to-side motion. For example, a strongly-rectangular piece of paper (eg. a dollar bill) spinning along its long axis, with the axis parallel to the ground.
– Mark
Commented Aug 25, 2021 at 22:19
• Note, these are called whiligigs or whirlybirds: scientificamerican.com/article/make-a-whirlybird-from-paper (at least in the U.S.) Commented Aug 26, 2021 at 15:08

I really love this game idea! I think a flat piece of paper falling face-down is probably the least aerodynamic thing you could possibly drop if it maintained its shape and orientation, but naturally the paper bends and changes angle (edge-down) as it descends, tending towards the path of least resistance.

Therefore, my thought is that your goal should be to keep the paper as close to its original shape as possible while providing rigidity to prevent bending and stability to prevent tipping its edges down.

The solution I came up with was to fold the edges of the paper as shown (roughly) in the image below. The creases in the paper help to prevent it from bending into a more aerodynamic shape, and the flaps act like rudders that cause the paper to spin rather than flip into an edge-down position. I only tested it out a couple of times, but it seemed to work for me.

What a fun challenge! We had a good time trying out various designs :)

I did eventually come up with a helicopter-style design that falls about 50% slower quite consistently.

Build instructions:

1. Cut diagonally along the red line.
2. Fold one half in the opposite direction.
3. Fold up the edges along green lines to add rigidity.
4. Put a single staple through the sheets at the blue line. This both adds a bit of weight for balance and strengthens the connection.

Here is a set of videos of testing this against a plain A4 paper. To make the launches more consistent, I'm using two sheets of transparent plastic. A typical drop is significantly slower. But sometimes the rigidity of the center strut fails and it falls faster.

• "using only a sheet of paper", though. Commented Aug 26, 2021 at 16:57
• @EricDuminil Hmm yeah, I guess one would have to try and bend the paper to reinforce the joint instead of the staple.
– jpa
Commented Aug 26, 2021 at 17:52

Extrapolating from the Gyrocopter idea for A4 paper size, I arrived at following conclusions:

• Bending (i.e. not creasing a fold) part of the helicopter blade along its length provides ample structural stability at the cost of some air resistance. The result however has a nice wing profile.
• Structural stability is most important at the blades' root, so you should bend diagonally. You can leave about 1/3 of the length of an A4 paper sheet without such structural reinforcement.
• Tape these stability bends to the hub (where both blades meet), so the blades don't fold up under load. With the hub straight and the bends diagonal, a short stretch of tape will be left hanging in the air.
• You can crease the outmost point of the stability bend to reduce air resistance. The wing profile will still extend to the blade tips.
• Angular momentum (in direction of the height axis) increases drastically with the diameter of the rotor disc (aka blade length). This causes the Gyrocopter to spin very slowly, which decreases its effectiveness at slowing down the fall.
• Angular momentum (in direction along the blades) is very low and decreases with increasing blade length. This leads to poor stability.

However the poor stability led to a lucky discovery: If the Gyrocopter flips over, it can assume a stable state of continually flying loopings. If optimized for narrow loopings (henceforth Gyrocopter being renamed to Looper) with

• both stability bends on the same edge
• a hub of small height (long blades/wings and the rest folded into a triangle about as high as the stability bends)

the Looper seems to achieve additional lift. This may be caused by Magnus effect (circular movement of wing surface around an axis along the wing's length affecting the airflow). The effect can be increased by releasing the Looper in downwards position with a little downwards flick to start the first looping.

My Looper build performs up to 8 loopings, taking about 5 seconds for 2m of height.

PS: I build this yesterday evening after seeing the question, before reading Mark's and PcMan's comments about the fluttering dollar bill.