In our 3D world, it is basically impossible to evolve living wheels, or helicopter rotors, or anything similar because free rotation of an axle dictates that there is no fixed connection to the body of the creature or machine--thus, there is no way to route nerves, blood vessels, etc.

If you just want to rotate a compact object, though (not an axle or similar non-compact object that has significantly extent in one dimension beyond its attachment point), there are arrangements like this that allow you to do so. Adding an additional spatial dimension would seem to provide a way to extend an axle out from the cube in the center of that assembly to transfer continuous rotation to a wheel or other continuously-rotating structure. However, as with my previous question on 4D biomechanics, my capacity for higher-dimensional visualization kind of fails here, which makes it difficult for me to be sure that this would actually work, with an extended axle simultaneously being capable of transmitting the mechanical motion, and avoiding intersecting any of the support structures during a full 720-degree rotation.

So: Does having an extra spatial dimension to work with actually make it possible for creatures to grow freely-rotating wheels with fixed (although flexible) connections to the rest of the body, or does even a 4D wheel end up requiring a mechanically-isolated axle after all?


You asked for this, this is a lengthy subject and if you were to really go into detail you could write hundreds of pages about it. I've tried to keep it a bit short but that's impossible.

Bio-wheels can be done in 3D! There is just no evolutionary path there because of the simple question "how does it move while the wheel is still evolving".

I've actually designed a living wheel once when a biomechanics teacher said it couldnt happen. After I told him about the idea he amended it with the "no evolutionary path". The wheel worked like this:

Bloodflow does not have to be constant to keep an appandage alive. So when turning the bloodflow will be severely hindered. During arm surgery for example they often take the blood out and can operate 7 hours without permanent damage to the arm. Since this wheel appandage will be in use that time will be significantly shorter before damage is sustained, and without a blood reservoir in the wheel itself and a local heart-like pump any muscle-movement while the wheelnis in motion will last a maximum of 2 minutes, probably less. This problem can be circumvented by making the axle that the wheel spins on contain the bloodvessles, and having the blood flow down into a small open area surrounding the axle where fresh blood can be pumped through the wheel while de-oxynegated blood is collected in a funnel at the top and flows away. The turning of the wheel and constant muscle contractions can pump the blood around, thus is actually how de-oxynegated blood in your legs is pumped upwards during excercise.

Nerves is trickier. While there is no need for muscles on the wheel end (more later) it would be necessary to have nerves to feel injury to the wheel and changes in the ground it moves on. This can be solved through a ring that forms one giant synaptic gap. There would be thousands of these rings in a row, one for each nerve in the wheel. To reduce the amount of input needed it is likely the wheel will use the same self-thinking capabilities as octopus arms. The nerves travel through the axle of the wheel and will all connect to the synaptic gap. This is the hardest part as a synaptic gap is incredibly small and movement shouldnt interfere with this gap, but since we have functional synaptic gaps right now it should work. If the neurotransmitters cross the gap they are received at whatever spot was at that time turning passed the nerve, and then the signal is created and travels around the ring until it reaches the axons and dendrites of the actual nerve cell(s) it belongs to.

Then there's muscle movement, and that may actually be much simpler than it seems. Take a look at this picture https://images.app.goo.gl/QuokdPB9jnZsatds7

You can see a schematic of how a muscle is build up. Each sarcomere is a round tube with an actin filament on the outside and a myosin filament on the inside that is suspended by a titin filament. You have one end facing left anf the other facing right, with the actin filaments apart from each other. When activated the myosin filament has tons of "arms" that grab the actin filament above it and pull it inwards, which causes a contraction as the left and right actin filaments are pushed closer to each other. The myosin "arms" also release when they cant push any further, move backwards to grab the actin filament and then push again. The muscle can only conteact until the left and right actin filaments touch each other, at which point the muscle is maximally contracted.

The thing is, what if you only had an actin filament attached to the wheel, and only half of the myosin filament attached to the body around the axle of the bio-wheel? Now instead of making many muscles with many filaments in tandum, you make it one continuous filament around the axle? Now the myosin will never be able to maximally contract the actin filament, and keep pulling it over itself as the wheel spins and spins. Another advantage is that since the muscle is on the body-end and not the wheel-end you can supply it with blood and nerve input (which would kind of make the octopus-like control redundant). Now add as many of these circular muscle rows as necessary around the axle to push it. Additionally by making the axle thicker and the wheel part a thinner edge around the axle you can increase the amount of muscle fiber per circle. It would also mean that the wheel part only has to be as thick as it's bone support needs to be limiting the amount of blood and nerves you have to connect to it.

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    $\begingroup$ This is interesting, but not really an answer to the question.... $\endgroup$ – Logan R. Kearsley Jul 3 at 19:10
  • $\begingroup$ You can use parts of it to reduce the headache of 4D thinking where necessary. $\endgroup$ – Demigan Jul 3 at 20:06
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    $\begingroup$ Um... Why isn't it an answer? The question was "Can you do it in 4-D?" The answer is "yes, you can do it in using only 3 of the 4." $\endgroup$ – puppetsock Jul 3 at 21:22
  • $\begingroup$ @puppetsock because I dont mention the 4rth dimension he asks for. Despite it not actually not being necessary for a bio-wheel to have a 4rth dimension he asks for "is it possible with a 4rth dimension to have a bio-wheel", which most people on this site take to mean "there needs to be mention of a 4rth dimension". $\endgroup$ – Demigan Jul 3 at 21:43
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    $\begingroup$ @puppetsock Because the question wasn't "can you make a living wheel". The question was "Does having an extra spatial dimension to work with actually make it possible for creatures to grow freely-rotating wheels with fixed (although flexible) connections to the rest of the body". The wheel described here explicitly does not have fixed connections, relying on sliding synaptic connections and a disconnected circulatory system served by an open blood ventricle and being supported by a sliding-bearing axle. $\endgroup$ – Logan R. Kearsley Jul 3 at 23:01

Maybe. But I'm having a lot of difficulty doing the visualization. Consider this video. https://vimeo.com/62228139

It shows a thing that has lots of names, in this case "Dirac's Belt Trick." The idea is that by twisting and looping, you can rotate something that is connected to a surrounding solid structure. The only thing is, you need to loop the connecting things all the way around the rotated thing. But nothing gets disconnected.

There's another version of this where you turn a cup around 720 degrees by passing it first under your arm then over. If you are careful you never lose any water.

The looping-round part is tough because, for example, if it was a wheel, you would wind up running over the connection every time you looped around. Not as convenient as one might hope.

Now the part I'm having difficulty visualizing is doing the rotation in 4-D. Maybe in 4-D you can loop it around so that it does not get run over every time.

So if you allow this looping thing, it's possible in 3-D, at the cost of flinging the connection around or under. Though as several others have pointed out, the evolutionary path is tough to imagine. In 4-D it might be possible to avoid that.

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    $\begingroup$ I think the idea is not that the belts (or arms) behave any differently, but that the video of the rotating cube is treated as one 3D cross-section of a larger 4D system, and that there is an axle extending in the 4th dimension (i.e. orthogonal to that 3D cross-section) to the center of the cube to a different object (a 'wheel' of some kind) in a parallel 3D "hyperplane". The 3D analogue would be a square rotating in one 2D plane (though you couldn't use the Dirac's belt trick to do it), and an axle running from the center of the square to the center of a circle in a parallel plane. $\endgroup$ – Hypnosifl Jul 3 at 23:27

I find it easiest to consider 4d options for a 3d world via the Flatland method: consider 3d options for a 2d world.

Fortunately a wheel can easily be considered in 2d. It is a circle, rotating in the tabletop-like plane of Flatland. You could supply your wheel with a cylindrical 3d axle coming up out of the plane of Flatland.

The 3d cylindrical axle can be considered a stack of infinitely many 2d circles. The issue is the final 2d circle which is the interface with the wheel and (necessarily!) is in the same Flatland plane as the wheel. The fact that the rest of the axle is extraplanar to the wheel does not help with the interface in the same plane.

If 2 things have a normal mechanical interacton, they must at least in part occupy the same dimensional plane. The problems encountered with that interaction are not made easier by the fact that parts of the objects occupy different dimensional planes.

  • $\begingroup$ But you can approximate a real 3D pair of wheels on an axle as a pair of 2D circles in parallel 2D planes, connected by a rod whose axis is orthogonal to both planes, and which connects to each circle in their center. The fact that the interface is in the same plane as the wheel in the approximation doesn't prevent you from creating something basically similar in 3D where the wheel is not actually perfectly flat and so the axle/wheel connecting point is not in exactly the same plane as the cross section of the wheel at its exact center. $\endgroup$ – Hypnosifl Jul 3 at 18:04
  • $\begingroup$ @Hypnosifl: if the connecting point of the axle is not in the same plane as the wheel, how does it exert a vector force on the wheel? It has no option for a vector in plane with the wheel and the wheel is immune to vector forces from outside of its plane (which would push the wheel outside of its plane). $\endgroup$ – Willk Jul 3 at 18:11
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    $\begingroup$ Are you asking how this would work assuming some kind of 2D physics in flatland, or are you asking how this works with the real physics of 3D rigid bodies? Of course in the second case there is no problem with imagining a 2D plane passing through a 3D body and then imagining a force applied to the body at a point outside the plane causing parts of the body to rotate within that plane, just think of applying force with your feet to bicycle pedals and causing the chain wheel to rotate within its own plane. $\endgroup$ – Hypnosifl Jul 3 at 18:34

Nice video Logan, I've never thought something like that was possible.

Now, please follow and check my thoughts as I am not very sure myself.

First of all, I think it is really not possible to use this construction in 3D, because any of the six bands sweeps around the whole cube, not leaving any space for shafts. Even the two bands ending at the top/bottom of the cube would end up wound around shafts. So this is probably not just some accidental ignorance of 3D evolution.

Rotation in any number of dimensions greater than one looks like swapping two of the dimensions cyclically. Things pointing to X will turn to Y, then -X, then -Y, then back to X. Remaining dimensions, if any, stay in the same relative position. So, in 2D, the whole thing changes directions; In 3D, there is an axis of rotation, whose direction stays fixed; In 4D, things rotate around a fixed plane.

We could think of the cube as an endpoint (or middle point) of a 4D "shaft" extending away from the 3D hyperplane shown in the video. This "shaft" rotates in XY plane and stays fixed in ZW. The construction with cube and bands works in 3D, so it can all happen in a single XYZ slice of the 4D space, oblivious to anything far away in the W direction (even though the cube and bands need some non-zero W thickness for sure). So, the "shaft" extending in W can have bearings, wheels, propellers, anything attached at other W positions, and these all will still spin in the XY plane!

Please keep in mind that this is no mathematical proof, there is a ton of things I could have missed. I can't really imagine things rotating around planes. I'm not quite sure about construction of 4D bearings, or how wheels would work on a 3D surface. On the other hand, this is just a demonstration example, a real 4D evolution would probably come up with more elaborate solutions using the same principle.


Having mulled this over for a while longer in my own head, and building on the reasoning in Willk's answer and Nicol Wollaston's answer, I think I can now definitively say "Yes! A joint being spun by a 3D arrangement of fixed muscle attachments can indeed drive a continuously-rotating shaft extending in a 4th dimension."

We start by considering rotation in a plane. There is a single stationary point at the center of rotation. Adding a third dimension, we can extend this point into a stationary line, or axis. By itself, that line cannot transmit rotational motion; in order to generate torque, you must have an attachment point with some finite offset in a direction parallel to the plane of rotation--or, in other words, a driveshaft centered on the axis must have non-zero thickness.

Now, here's the key part: consider a different third dimension. If we are working in a 4D space, either of two dimensions perpendicular to the plane will work for extending a shaft. The two dimensions are indistinguishable, in that either of them can form a 3D subspace of the 4D world in combination with the plane of rotation, and either one can have lines extended from it parallel to that plane defining the radius of a driveshaft. In fact, in general, we could have an entire drivesheet, or an arbitrary number of driveshafts all intersecting and rigidly connected to the plane of rotation at the same place, but rotated at arbitrary angles between the orthogonal cardinal directions of whatever coordinate system we choose.

Now, we just have to determine whether or not such a driveshaft must necessarily intersect the belts/muscles/whatever that attach to the joint at some point during a full double-rotation. And in fact, we can show that it need not do so. First, note that the motion of the belts can be confined entirely to a 3D hyperplane--we know this because the belt mechanism works in 3D! All of the components will occupy some 4D depth, but only a finite 4D depth. The driveshaft must also have finite radius in all 3 dimensions perpendicular to its axis--this means that it will have some finite extent in the dimension used to defined the 3D hyperplane within which the belt arrangement operates. However, we can also arrange that, as each belt passes over alternating faces of the central cube (or hypersquare prism, hypercylinder, or whatever), it can be made to pass arbitrarily close to the surface of that joint. Now, we imagine the driveshaft attached to the center of the 3D hypersurface of the joint; supposing we have belts of thickness $t$, we can arrange that no part of a belt extends further from the surface of the joint than distance $t$ by building the entire mechanism compactly, so that the belts are always sliding in contact with the surface of the joint. If the driveshaft has a radius of $r$, then as long as the joint itself has a minimum radius of $t+r$, there is no way that the shaft can ever come into contact contact with the belts, let alone intersect their paths.

I strongly suspect that the reality of the situation is actually much looser than that, but that much I feel I can prove. In fact, it should be possible to attach a nearly-complete 2D drivesheet to the joint, with mere holes cut out to permit the passage of muscle belts. When viewing only the 3D hyperplane containing the muscle belts, the intersection of the drivesheet with that subspace would appear as a set of disconnected axle sections floating unsupported, with the belts passing through the gaps in between.


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