Corkscrews in 4d

[anderscolingustafson]

In 3d an object that is moving forward has two degrees of freedom that could cause it to go off course. It could go vertical or sideways instead of moving forward. In 3d spinning around in circles that are perpendicular to the direction it is moving forward in helps to stabilize the object and helps it to move straight.

In 4d simply spinning an object moving forward would have one more degree of freedom from an object moving forward in 3d. In 4d if an object that was moving forward was to spin in circles perpendicular to the direction it was moving forward in then it would still have two degrees of freedom left over that could cause it to move off course. For instance if an object was to spin clockwise or counter clockwise the Ana-Kata directions would be left over to destabilize a corkscrewing object. In order for an object moving forward in 4d to use spinning to stabilize it would have to spin in spheres perpendicular to the direction the object is moving in.

In 3d it is impossible for an object to have two independent directions of spin so in 4d it is impossible for an object to corkscrew in two independent directions. It would take five dimensions for an object to corkscrew in two independent directions as it takes 4 dimensions for an object to have two independent directions of spin. It is still possible for an object to spin in multiple directions in 3d but the directions, just not in two independent directions.

In 3d if the polls of none rotation move slightly from their original location after each spinning cycle and if they keep moving in the same direction a second direction of spinning will be produced. So while it is impossible to have two independent directions of spin in 3d it is possible to have an object spin in two directions in which the second direction of spin is based on the first direction of spin. So in 4d it would also be possible for an object to corkscrew in spheres for stability.

[Keiji]

Very interesting. It's been known for a long time that rivers would corkscrew in 4D simply due to having freedom in two directions to traverse the landscape.

But for a bullet moving through the air, you raise a good point. I imagine in 4D the best you could have is moving forward, and spinning in two of the remaining three dimensions. Surely it's not possible to spin or corkscrew "in spheres", as the momentum would reduce itself to two dimensions of movement and one would be still?

[Secret]

Inspired from the toroquator of the 3-sphere

I think it is possible for something to corkscrew in 4D

Recall in 3D how you can bend the corkscrew motion into a circle, thus making a torus shaped corkscrew.

Now going to 4D, you can have a particle that does the torus shaped corkscrew in say the xyz hyperplane. Now there's still one direction w left for it to move forward

Then you effectively end up with a "double corkscrew" like motion

In 5D, it gets more interesting since you can have a swirl compounded with moving forward, thus generating a swirl corkscrew
E.g. first 4 dimensions=swirl (clifford rotations)
Remaining dimension=moving in straight line
E.g. First 3 dimension make corkscrew
The 4th one move in a circle and make a double corkscrew
The remaining one travel forward

[ICN5D]

Wow, so it does a helical corkscrew! Very complex!

[quickfur]

Wendy has said before that any rotating object in 4D will eventually settle into a Clifford double rotation, where it rotates in two orthogonal planes with equal rates of rotation. The reason for this is twofold: (1) since we're dealing with imperfect real-world scenarios, a pure 2D rotation is very unlikely; there's bound to be some perturbations that causes slight angular momentum in the orthogonal plane. So there will always be two orthogonal rotations, even if the rotation rate of one is pretty small compared to the other. (2) As soon as you have two orthogonal rotations going on, points on the object will trace out spiralling paths, which is unstable; the rotational energies will tend to equalize (precess) and settle in the equal double rotation, where the spiralling paths flatten into circles (basically the circles of the Hopf fibration of the 3-sphere) and energy is at an equillibrium.

This double rotation has an interesting feature that it does not fix a single stationary direction, as a rotating object in 3D would, but it does fix the orientation of the two orthogonal planes (it would probably resist change to that orientation, similar to a 3D corkscrewing bullet resisting change in lateral orientation). However, rotation within those planes would be allowed; so the 4D bullet would not be able to maintain pointing in a single direction! In the best-case scenario, the bullet's shape would need to have duocylinder symmetry, then it would be able to maintain its effective orientation in spite of any double rotation that may happen. Now, the duocylinder itself is ill-suited for this job, because it has two flat surfaces that increase air friction, so it will slow down quickly. But its dual, the bicircular tegum (pyramid product of two circles) seems better-suited: it has duocylinder symmetry, and has a sharp circular edge along both planes of rotation, so this edge should be able to reduce air friction.

In fact, you can have a bicircular tegum of a large circle and a small circle: the resulting shape would be very sharp around the large circle, and quite blunt in the orthogonal circle. When firing the bullet, you would shoot it in one of the directions of its large circle, and introduce a spin in the small circle. The spin in the small circle would spontaneously equalize with the large circle into a double-rotation, but since the large circle is symmetric with respect to its spin, the resulting double-rotation will maintain the bullet's sharp edge in the direction of travel at all times. Since it is also symmetric in its small circle, the orthogonal rotation does not change the orientation of the facing sharp-edge either, neither does it introduce uneven air friction. The double rotation will serve as an angular momentum that resists change in the orientation of the bullet's planes of symmetry, so its sharp edge will always be directed parallel to its direction of travel.

The closest 3D analogue of this curious 4D phenomenon would be a spinning disk travelling parallel to its plane of rotation (perpendicular to its rotational axis), like a frisbee or a shuriken. (But in 3D, there is no orthogonal rotation, whereas in 4D, the small_circle-large_circle tegum would be rotating in both planes, with the large circle's plane being parallel to the direction of travel.)

Such a shape would be much more stable than any tube-shaped bullet with spherical symmetry along its axis, since in the latter case, the double rotation introduced by uneven air friction will quickly cause the bullet to lose its ideal orientation and slow down rapidly.

[anderscolingustafson]

Also I was just thinking about how flagella are used for locomotion in 3d life and flagella move in a corkscrew motion. Would cells in 4d also use flagella and if so what would a 4d cell use instead of a flagella?

[ICN5D]

It would probably use a thin noodle of biomass with a spherical cross-section. Like a spherindric tentacle flagella. But there's probably more types. If a 3D flagella has a cylindrical shape, then a 4D flagella could be several kinds of cylindrical 4D ones. I suppose there's only 5 of these kinds in 4D, the spherinder, duocylinder, cubinder, cyltrianglinder, and torinder. Each one has a unique ability and weakness. Maybe a 4D being could be bristling with an array of specialized flagellum, be it for defense, transportation, communication, reproduction, or what have you. Like a difference between legs and arms, but using geometrically distinct, analogous limbs.

[wendy]

Here is my 4d worth.

I suspect that corkscrews in 4d would be the good old helix you see in 3d. The hairy sphere does not exist, so you will have to deal with 'end-problems' of the screw-prism.

Flagella would not be latrous (line-like), but hedrous. The way these work, is the motion is intended to 'cut space', eg by creating a mass of pressure behind the sweep of the flagellum. A line won't do this. You need a hedrous thing to do it.