In physics, "almost everything is already discovered, and all that remains is to fill a few unimportant holes." (See Jolly.) Therefore, on Physics SE, people are veering off into different directions: biology, for example.
Thus, it happens that a question about bicycles generates some discussion about evolution in biology and animals with wheels.
Three explanations are offered for the apparent lack of wheely animals (also on Wikipedia, where, by the way, most Physics SE questions are answered perfectly).
Evolutionary constraints: "[A] complex structure or system will not evolve if its incomplete form provides no benefit to an organism."
Developmental and anatomical constraints.
Wheels have significant disadvantages (e.g., when not on roads).
Now, I suggest that all three can be "solved".
With time.
With a symbiotic relationship between a wheel-like animal and a "driver"-like animal, although this gets awfully close to a "driver"-animal to jump onto an actual (man-made) wheel. (So, perhaps, you can suggest a better loophole around this constraint.)
Roads are presumably not the only ecological niche where animals with wheels could thrive. I'm thinking of frozen lakes, although there skates would be better than wheels.
What, therefore, is the explanation for there not being any wheeled animals? Please consider, in your answer, the counterfactual: What assumption of yours would be falsified once a wheely animal is discovered?
Answer
Wheels are possible on the molecular level — bacterial flagella are rotating cores inside a molecular motor, but wheels larger than the flagellum have not really been found.
A single animal with a wheel is an improbable* development that would require a single animal have two separable parts (axle/wheel and body).
[*read as: pretty much impossible]
It's hard to imagine how such a thing could evolve. A wheel and axle would need to be made of living tissue, otherwise it would be vulnerable to wear and tear. Wheels also have problems going over uneven terrain, which is really all terrain animals live in. It's difficult to imagine what sort of selection conditions would be strong enough to push animals away from legs.
If you include driver-vehicle symbionts where the 'car' and 'wheel' are actually two animals, then they have evolved. Parasites can have all sorts of symbiotic control over their victims including as means of transport. The Jewel Wasp is one which is the most suggestive of what you may be thinking. The wasp stings its victim (a cockroach) in the thorax to immobilize the animal and then again just behind its head. After this, the wasp can ride the beetle, steering it by holding its antennae back to its nest where the roach is immobilized to feed the wasp larvae there.
(see section "Pet cockaroaches" in this reference.)
As to the three schools of thought you added to the question, I would probably rather say there were two strong arguments against. The first is whether there is an evolutionary path to wheels (argument 1 in your question), which I doubt. Given even a large amount of evolutionary time you will not see a naked human being able to fly on their own power. Too many structural characteristics of the body plan have been made to all be reversed so that wings or other means of aerial conveyance will show up. The same can be said for wheels when the body plans have fins/legs/and wings already.
Argument 3, which I also tend to agree with, is perhaps more convincing. By the time a pair of animals makes a symbiotic relationship to do this, or a single macroscopic animal evolves wheels, they will literally develop legs and walk away. When life came onto the land this happened, and since then it's happened several times. It's sort of like saying that the random movement of water molecules might line up to run a stream uphill. There's just such a strong path downwards, that the statistical chances of you seeing it happen are nil.
This is a hypothetical case, but arguing this in a convincing way I think you would need to lay out: a) an environment whose conditions created enough of a selective advantage for wheels to evolve over legs or other similar adaptations we already see. Perhaps based on the energy efficiency of wheels; b) some sort of physiological model for the wheels that convey a reasonable lifestyle for the wheel.
There are lots of questions that would need to be satisfied in our thought experiment. Here are some: "the symbiotic wheel would be spinning constantly; if it died the driver creature would be completely defenseless"; "if the ground were bumpy, all these wheeled animals would get eaten"; "the wheel symbiont — how would it eat while its spinning all the time? Only fed by the driver? Even symbionts such as barnacles or lampreys on the flanks of sharks still have their own ability to feed."
For many similar questions the same sort of discussion ensues where there are many disadvantages which outweigh advantages for animals. e.g. "why are all the flying animals and fish and plants even more similar to airplanes than helicopters?"
Sorry if I seem negative, but way back in grad school I actually did go over some of these angles.
UPDATE: First Gear found in a Living Creature. A european plant hopper insect with one of the largest accelerations known in biology has been found to have gears! (There's a movie on the article page. )
The little bug has gears in its exoskeleton that synchronize its two jumping legs. Once again selection suprises.
The gears themselves are an oddity. With gear teeth shaped like cresting waves, they look nothing like what you'd find in your car or in a fancy watch. There could be two reasons for this. Through a mathematical oddity, there is a limitless number of ways to design intermeshing gears. So, either nature evolved one solution at random, or, as Gregory Sutton, coauthor of the paper and insect researcher at the University of Bristol, suspects, the shape of the issus's gear is particularly apt for the job it does. It's built for "high precision and speed in one direction," he says.
The gears do not rotate 360 degrees, but appear on the surface of two joints to synchronize them as they wind up like a circular spring. The gear itself is not living tissue, so the big solves the problem of regenerating the gear by growing a new set when it molts (i.e. gears that continually regenerate and heal are still unknown). It also does not keep its gears throughout its lifecycle. So the arguments here still stand; the exception still supports the rule.
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