Rolling a Duocylinder & 4D Rugby

[PatrickPowers]

So how would a duocylinder roll in "real" 4D life. I believe it would usually follow a complex waddling path that is somewhat unpredictable and chaotic. How it is rotating as it rolls makes a big difference.

It so happens there is a very similar shape in our world : the rugby ball. The minimum aspect ratio of an official rugby ball is very close to that of a duocylinder. If you spin a rugby ball (or many things longer than wide) rapidly enough it will spontaneously stand on end like a spinning top. A duocylinder can continue to roll while spinning like that. If the duocylinder is solid and heavy then there is the complexity of precession of a double rotation due to gravity. I suspect a solid duocylinder would nutate.

https://en.wikipedia.org/wiki/Nutation.

So, would 4D rugby teams use a duocylinder as a ball? Sure, why not. How about USA football teams? The USA football has an aspect ration of 5/3 -- that makes it 20% slimmer -- so it can be gripped with one hand for throwing. The football also has that pointed end for improved flight. Nothing stopping you from accentuating/sharpening the ridge of the duocylinder and making it more squat/slim for the same reasons.

For such a sporting good is there any particular advantage over a tapered spherindrical shape? I'm guessing the duocylinder would be easier to make.

[PatrickPowers]

A main difference is that instead of the maximum extent of the rugby ball being two points, with the duocylinder the maximum extent is what I call a 2D torus. The ball can and will roll on this torus like a disc rolling on its edge, but the surroundings of this circle are rounded like the rugby ball instead of sharp like a disc. What direction will the ball roll while rolling on this torus? It can be any direction in a 2D plane that is a subset of the 3D plane the duo is rolling on.

[PatrickPowers]

A main difference is that instead of the maximum extent of the rugby ball being two points, with the duocylinder the maximum extent is what I call a 2D torus. The ball can and will roll on this torus like a disc rolling on its edge, but the surroundings of this circle are rounded like the rugby ball instead of sharp like a disc. What direction will the ball roll while rolling on this torus? It can be any direction in a 2D plane that is a subset of the 3D plane the duo is rolling on.

My next question is, would a duocylinder exhibit the Dzhanibekov Effect while rolling? https://www.youtube.com/watch?v=vklY1bHIi1I I belief that if the vertical 2D plane in which it is rolling contains points approximately in both the maximum and minimum extents then it would. For the effect it's necessary to have three dimensions of freedom of motion, which the duo has while rolling on a surface. Here in 3D rolling objects have only two dimensions of freedom so objects can exhibit the Dzhanibekov Effect only when rotating unconstrained.

The duo rolls in a simple way only at low energies. At higher speeds and rotations the rolling duosphere could show a mind-bogglingly complex motion, combining all the weirdness of a spinning top, a rolling rugby ball, and the Dzhanibekov Effect to top it all off. To get an idea I'd have to write a computer program to simulate it. Not going to happen.

[PatrickPowers]

Instead of being a prolate spheroid like a rugby ball, a duocylinder is a oblate spheroid. It is oblate in two perpendicular planes.

To see this, compare with a sphere. With a sphere we build the sphere out of latitude tori. These tori have a minor radius r0 and a major radius r1 such that r0^2 + r1^2 = C with 0<=r0<=C.

The duocylinder is the same, built out of latitude tori, but the major radius is a constant.

r0^2 + C^2 with 0<=r0^2<=C^2
and
r1^2 + C^2 with 0<=r1^2<=C^2.

A maximal circle on a duocylinder has radius C*sqrt(2). There is much more surface area is close to a maximal circle than there is surface area close to a minimal circle.

I can have a glass map made of the surface of a duocylinder. It will be constructed via the same algorithm as the map of the 4D sphere but will be more oblate. In the map of the 4D sphere the diagonals of the rectangles are all of the same length. In the map of the 4D duocylinder the length of the diagonals increases gradually, the maximum being larger than the minimum by a factor of sqrt(2). Looked at from the sides the duocylinder map will appear about the same as the map of the sphere. The difference comes when looking from above. From above the map of the sphere appears round while the map of the duocylinder appears square.

Compare a sphere with radius R with a duocylinder with minor radius R. The duocylinder has twice as much surface area that close to the equator.

So a duocylinder wouldn't be used as a rugby ball, the only use I could think of for prolate spheroids. Here in our mundane 3D world the oblate spheroid is the shape of an old school cauldron. Such a cauldron is usually twice as oblate as a duocylinder. Though 4D people would not use an oblate spheroid as a cauldron. The planes of maximum oblateness must have a vertical component, rendering the cauldron is less stable -- bad! An oblate spherinder seems like the way to go. But for 5D people the two planes of oblateness can both be horizontal, which would be better for stability. So oblate spheroids are used as old school cauldrons only in odd dimensional spaces, where a vertical axis is possible.

So, what would 4D people use as a rugby ball? A prolate spheroid, just like here on Earth. Take the map of the sphere, make that less oblate, then you've got a map of the surface of a 4D rugby ball. Work back from the map to make the actual thing.

[PatrickPowers]

So how do a duocylinder roll? It shares character with 3D prolate and oblate spheroids both. A 3D prolate spheroid has a minimal circle it can roll on but not a maximal circle. An oblate spheroid has a maximal circle but not a minimal one. The duocylinder has two minimal circles and the good old infinity of maximal circles so it can roll like a rugby ball, like a coin on its end, or anything in between such as the rugby ball waddle.

Another way to look at it is that the 3D rugby ball prolate spheroid is stretched in one dimension, the 3D cauldron oblate spheroid is stretched in two dimensions, and the duocylinder is also expanded in two dimensions.

In 3D spheres on a level surface roll in the forward-up plane. A 4D sphere usually doesn't roll in the plane in which it is rotating. It has two planes of rotation so the direction in which it rolls is the vector sum of the velocity of the two planes. For a long time I've looked for some weird effect due to this but no luck so far. Indeed it seems it would remove a weird effect -- would 4D things precess? I think usually not, such forces would simply add to the rotation in a straightforward way, but it's hard to be sure.

At any rate it seems to me that a duocylinder is prolate and oblate BOTH.