This is tangentially related. Do you know why we have hair in armpits but not on back of the knees or on inner side of the elbow (between biceps and forearm)? Both knee and elbow can move in a way that skin touches itself (just like armpit), but the knee and elbow have only 1 degree of freedom, no matter how you move them, the skin cannot rub on itself. Skin can only rub on itself in armpit (2 degrees of freedom), groin (2 degrees of freedom), and butt crack (technically 1 degree but they are very close together and fulcrum are hips which are far apart not like single-jointed elbow so it can rub on itself).

The information content in hair distribution is substantial. The genetic program for facial hair must have dozens of bits. Like eyebrows are a pretty specific shape in a specific place.

Eyebrows may have a practical function (avoiding sweat dripping into the eyes) but I suspect the main evolutionary pressure is through mate selection. People with eyebrows halfway up their forehead or below their eyes were sufficiently unattractive that those combinations of genes didn't propagate.

Since hair for the whole body is presumably encoded by some very non-modular genes, hair distribution on parts that aren't important for function or appearance may just be side-effects of changes that affected parts that are important.

Our species lost most of its body hair fairly recently, so the evolution of hair patterns is relatively incomplete.

The main theory I'm aware of is that eyebrows serve to better communicate facial expressions, on top of being a sweat guard.

Our eyebrow region is exceedingly expressive in terms of movement, similar to our mouth when not speaking (think of all the variations of smiles, frowns, smirks, etc.). The bands of hair make it much easier to identify that movement, especially from a few meters away. Eyebrows are for direct emotional signaling.

> People with eyebrows halfway up their forehead or below their eyes were sufficiently unattractive that those combinations of genes didn't propagate.

What we find attractive is also genetically determined, though. If very fit (in an evolutionary sense) people started to develop eyebrows half way up their face, that would quickly lead to a population of people who found that hot.

Because we have sweat glands for pheremones in our armpits. The main theory is that armpit hair serves to waft that.

We don't have those types of glands on the backs of our knees or inner side of elbow, etc. So no extra hair there, and no need for antiperspirant there either.

And pubic and anal hair is meant to waft the pheromones from those regions too? :)

Unironically yes: https://en.wikipedia.org/wiki/Skatole

However, it strikes me as more likely that pheromones are emitted in places already adapted for evaporative cooling, which in turn would be hairy places, which in turn could be lubricative.

OK but why are the pheromones in your armpit instead of your elbow?

You might be reversing cause and effect.

Evolution, though directed by natural selection, is a random process. So sometimes the answer's just because that's how it happened.

That's not an answer to why selection made the decisions it did.

Just because it's not satisfying doesn't mean it's not an answer. Evolution is driven by random mutations in DNA. Sometimes they're beneficial, sometimes they're harmful. In those cases, natural selection determines whether the individuals with a given mutation will survive and pass on their genes. But sometimes, mutations are neither beneficial nor harmful, and which one beats out the other (if either does) is simply random chance. So many times, questions like yours are unanswerable.

Local maxima that can not be left anymore with the given mutation rate.

Better alternatives for so many functions that were never explored/reached by evolution.

Not sure that holds up to strict scrutiny. It's a fine hypothesis for the armpit but makes no sense with the groin. There the hair is heaviest on skin that touches nothing else and almost indistinguishable from background on the actual sliding surfaces. Same for butt crack.

Hair act as mechanical lubricant.

Exactly, so why isn't there more hair on places that need mechanical lubrication other than the armpits?

What places?

The creases between the tops of the thighs and the abdomen, the crease between the butt cheeks.

The angle would not be right, you want hair growing slightly off an roll parallel where it should lubricate. Armpit is bit of problem because it normally goes 180 deg so they cannot choose preferential direction. There the optimum is probably more radial then linear.

Counting equations is dubious. Is dx^2/dt^2 = c one or two equations?

The fact that this paper is written in Word doesn't help it.

I have never heard of applying Pearson-Coefficients for system identification like that.

This paper smells like someone applying insufficient math to a problem. The world is not linear. In fact the human body also requires Algebraic constraints.

This is just a bad paper.

> With animate objects, though, and with humans, in particular, we do not know the cardinality of the set of equations that define their trajectory.

Does it really matter? You can imitate human motion by multiplying a bunch of matrices to the point that the difference with "reality" is negligible. Learning Blender is the hardest part.

I'm pretty sure both Galileo and Isaac Newton already worked out the equations for a non-slipping spherical human on an inclined plane. The answer is 2: one for the linear acceleration and one for the rolling condition.

Omg! You cant derive science from first principles and give up all the physics/maths techniques used in the last 200 years. They didn't even take into account the error would increase is the you added more time derivatives.