4D Helicopter

[PatrickPowers]

A helicopter with its blades rotating in a plane would be unstable in 4D. If, say, the blades were rotating in the sideways plane then the copter would be unstable in the complementary up-forward plane. A solution would be to have two sets of blades rotating in two planes that intersect in a point. Both sets can rotate off of a single hub, at the expense of a yet more complex set of gears. The question remains, what are the ideal planes? We want the two planes to be as level as is practical while otherwise being as perpendicular as is possible.

Let's say our coordinates are [ana,right,up,forward]. Then the first plane is the set [w,x,C(w-x),0] and the second [w,0,C(w-z), z]. Set C so that it has the smallest magnitude that is practical. The two planes intersect only at the origin, in the sideways plane they are perpendicular.

How about the tail rotors. To prevent the body of the copter from counterrotating each has one dimension that is perpendicular to the corresponding main rotor. There's some freedom here because there is no need for the rotors to be level. A possible choice could be [w,0,C(w-z), z] and [w,x,C(w-x),0].

[quickfur]

I think there's more to this than just having rotating blades in some 4D configuration. The purpose of a helicopter's blades are to provide upward thrust, i.e., they direct air downwards. In 3D, rotating blades do the job. In 4D, I'm not so sure. Since the complement-space of the 2D plane of rotation is another 2D plane, the air wouldn't be forced in a particular direction, but would have 2 dimensions in which to scatter. Would that provide enough thrust to lift a helicopter? I can't say for sure. But it would seem that such a scheme would require far more power expenditure than it may be worth, for the amount of thrust it would provide (or wouldn't provide). I'm not even sure any meaningful thrust would be provided in this case.

[PatrickPowers]

I think there's more to this than just having rotating blades in some 4D configuration. The purpose of a helicopter's blades are to provide upward thrust, i.e., they direct air downwards. In 3D, rotating blades do the job. In 4D, I'm not so sure. Since the complement-space of the 2D plane of rotation is another 2D plane, the air wouldn't be forced in a particular direction, but would have 2 dimensions in which to scatter. Would that provide enough thrust to lift a helicopter? I can't say for sure. But it would seem that such a scheme would require far more power expenditure than it may be worth, for the amount of thrust it would provide (or wouldn't provide). I'm not even sure any meaningful thrust would be provided in this case.

I thought the same way then realized the airfoil could be curved downward only. No curvature in the undesired direction,.

[quickfur]

Hmm, airfoil. If we could come up with a workable 4D airfoil, then this would work: you'd just attach several elongated airfoils around the rotor and you're good to go. You'd then need two perpendicular propellors in order to maintain thrust in the right orientation. Can this work? Maybe. But it depends on whether it's possible to design a 4D airfoil that does the job.

Making the (big) assumption that we could just extrude a 3D airfoil into 4D, we'd have a sort of cube-like blade extending in 2 directions and having an airfoil cross-section. WLOG, we could imagine starting with two cubes, displaced in the 4th direction from each other, and consider one cube as the "top" and the other the "bottom". Assume the flight direction is along the +Z axis. Then the +Z half of the top cube would have larger curvature into 4D, being the leading edge of the airfoil, and would ease back towards the bottom cube as you go along the -Z direction. If we use perspective projection such that +W is closer to the 4D viewpoint (and appears larger) and -W is farther (appears smaller), then the top cube would look like it's bulging outwards in its upper half, and narrows down as you go down to the bottom face.

In order to maintain lift, this airfoil needs to extend in 2 directions. So presumably we could attach say 3 of these cube pairs around some point, oriented such that their forward directions lie along some circle around the origin. This would leave 30° gaps between each cube pair. Suppose this circle lies along the XZ plane. So the first blade would point in +Z, the second blade in a combination of +X and -Z, and the third a combination of -X and -Z. As they rotate around the rotor at the origin, they would generate lift in the +W direction. There would be angular momentum in the XZ plane. Since the Y direction is not constrained in any way by this setup, there's a possibility that the aircraft might rotate in a plane involving the Y axis. Presumably we'd need a second rotor, oriented in the YZ plane. That rotation would also provide lift along +W, but would have angular momentum in the YZ plane. Together, the respective angular momentums should prevent the aircraft from rotating as it flies.

(Though interestingly, now that I think of it, even if the helicopter were to "roll" in this way it'd still face forward, the rotation would be completely in the lateral dimensions and wouldn't affect the direction of flight or the direction of lift. It'd be extremely confusing for the pilot to maneuver, though. )

In any case, the main potential issue here would be the size of the blades, which must span a 2D area in order to provide sufficient lift. This would make them very bulky, and potentially add too much weight to the aircraft. Also, we may potentially need four propellors: two pairs that cancel out the other's angular momentum in their respective planes of rotation. Could be either four main propellers, or two large mains and two on the tail fins, like in 3D.

[PatrickPowers]

Here's what i have in mind.

Start with a familiar blade from our world. It has length, width, thickness\height, and is cured downward in the height direction to provide lift. Take it to the 4D world and it has a surface area but no volume. Add copies of the this surface in the 4th dimension until it has a second thickness that is strong enough to be sturdy. Now you have a blade that has far too much in mass, surface area, and lift. But better too much than too little. Shrink the width and two thickness and decrease the curvature until you possibly get something workable. What you get is a blade of the same length -- you evidently need that length to get sufficient airspeed -- but radically reduced in the other three dimensions and less curvature.

You can go further and argue that in 4D helicopters of the same mass as ours would be an inch or so long. But I'm not going to go there.

[PatrickPowers]

Hmm, airfoil. If we could come up with a workable 4D airfoil, then this would work: you'd just attach several elongated airfoils around the rotor and you're good to go. You'd then need two perpendicular propellors in order to maintain thrust in the right orientation. Can this work? Maybe. But it depends on whether it's possible to design a 4D airfoil that does the job.

Making the (big) assumption that we could just extrude a 3D airfoil into 4D, we'd have a sort of cube-like blade extending in 2 directions and having an airfoil cross-section. WLOG, we could imagine starting with two cubes, displaced in the 4th direction from each other, and consider one cube as the "top" and the other the "bottom". Assume the flight direction is along the +Z axis. Then the +Z half of the top cube would have larger curvature into 4D, being the leading edge of the airfoil, and would ease back towards the bottom cube as you go along the -Z direction. If we use perspective projection such that +W is closer to the 4D viewpoint (and appears larger) and -W is farther (appears smaller), then the top cube would look like it's bulging outwards in its upper half, and narrows down as you go down to the bottom face.

In order to maintain lift, this airfoil needs to extend in 2 directions. So presumably we could attach say 3 of these cube pairs around some point, oriented such that their forward directions lie along some circle around the origin. This would leave 30° gaps between each cube pair. Suppose this circle lies along the XZ plane. So the first blade would point in +Z, the second blade in a combination of +X and -Z, and the third a combination of -X and -Z. As they rotate around the rotor at the origin, they would generate lift in the +W direction. There would be angular momentum in the XZ plane. Since the Y direction is not constrained in any way by this setup, there's a possibility that the aircraft might rotate in a plane involving the Y axis. Presumably we'd need a second rotor, oriented in the YZ plane. That rotation would also provide lift along +W, but would have angular momentum in the YZ plane. Together, the respective angular momentums should prevent the aircraft from rotating as it flies.

(Though interestingly, now that I think of it, even if the helicopter were to "roll" in this way it'd still face forward, the rotation would be completely in the lateral dimensions and wouldn't affect the direction of flight or the direction of lift. It'd be extremely confusing for the pilot to maneuver, though. )

In any case, the main potential issue here would be the size of the blades, which must span a 2D area in order to provide sufficient lift. This would make them very bulky, and potentially add too much weight to the aircraft. Also, we may potentially need four propellors: two pairs that cancel out the other's angular momentum in their respective planes of rotation. Could be either four main propellers, or two large mains and two on the tail fins, like in 3D.

Aha so you have three blades in the XZ plane and another set in the YZ plane with W being up-down. But the XZ and YZ planes intersect in a line. This is just like the situation in 3D, where having two perpendicular sets of rotating blades is practically impossible. You have to tilt the two planes in the W dimension in such a way that the intersection of the two planes is a point, then tilt more to provide clearance between the two rotation collars.

The angular momentum of the XZ rotor is in that plane. There has to be a counterrotating force in that plane. By golly, you could get that force out of the YZ set of rotors. Just curve the airfoils so that some of the net thrust goes in the needed sense in the X direction. No tail rotors are needed. Great!

Note that in 4D there are two angles between the planes. In this case we want one of the angles to be 90 degrees, while the other is something like 10 degrees. I need some elegant way to quantify this.

[DonSoreno]

In 4d an airfoil would be 3-dimensional, not 2-dimensional. It needs 3d surcell area, otherwise the amount of air that is deflected downwards would be insignificant (like a 1d spear thrusted forward in 3d). Then Just tilt this cube-shaped air foil in the combined upward-forward plane and the thing will produce lift.

[PatrickPowers]

In 4d an airfoil would be 3-dimensional, not 2-dimensional. It needs 3d surcell area, otherwise the amount of air that is deflected downwards would be insignificant (like a 1d spear thrusted forward in 3d). Then Just tilt this cube-shaped air foil in the combined upward-forward plane and the thing will produce lift.

Yes. Also if it were 2D it wouldn't have enough strength.

[jerkjerry47]

interesting information