Can humans survive by eating only nails and hairs of their own?
As they grow they consume their own hair and nails. Hair includes even body hair. What is the maximum number days a human can survive on this?
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Your question asks how much longer they'd live, not if they could live forever
I think people are misreading your question. Can someone survive forever? No, and the laws of thermodynamics are the reason. But, how long would it extend life? Obviously recycling buys you something... or does it?
The problem is that nutrition isn't just a function of mass
Hair and nails are made of keratin, a difficult to digest protein, and pretty much nothing else. Because it's difficult to digest, simply cutting off your hair and ripping off your nails and eating them will provide so few calories that you likely burned more trying to eat them, speeding your demise.
However, researchers have found if you properly powder and prepare keratin,1 it becomes a mediocre source of protein.
Does this buy you much? Most calories come from carbohydrates, not proteins. Therefore, it's not surprising to discover commercially available keratin is listed as having no calories per 250mg serving. [1, 2, 3] Granted, a full head of hair may weigh a couple of kilograms2 (the world record holder for length is 5+ meters, but it doesn't look very full...), but even if we're generous and suggest 1 calorie per gram, 2Kg of hair is only worth 2,000 calories, your single day allowance.
And there are other problems
Your hair and nails are, well, dirty. We don't even want to think about what bacteria and nasty germs are living in your hair and under your nails, much less the hair spray and conditioner you use and the finger- and toenail polish you might be wearing. Perhaps you took the time to scape off the polish, wash your hair thoroughly, and wash your hands and feet before yanking those nails off. Total calories used to avoid illness: gazillions.
You pulled all 20 nails off. Ouch. There's a better than average chance of infection.
If you don't powder the hair, there's a better than average chance it can ball up in your intestinal tract and block it. Maybe you don't die any faster, but you'll wish you did.
If you took the time to cut all your hair and rip off all 20 nails, then grind them into a very fine powder (preferably without causing heating, per the linked article), you would maximize the value of the keratin to your body. But you probably expended far more nutritional calories than you could possibly receive from them.
Net effect: you die one day faster, but you won't feel as hungry
1 Have you ever had to do anything after breaking a big chunk of nail off? It hurts. It hurts a lot. Hitting it with a feather duster hurts even more. And our earstwhile survivalist just ripped all 20 off to the root. And the next thing you're going to do is pick up a rock and hit it against another rock to powder everything. And you're not going to slip and hit a finger or toe even once. Not even once.
2 Unless you're like Max Klinger from M*A*S*H, the odds are your body hair won't add substantially to your supply of Keratin. It's worth noting that there's a huge variation in hair weight. I've seen women with so much hair I wondered how they could hold their hair up. I've also seen Patrick Stewart. It's a good thing hair's worthless or we'd have to figure out the average weight of hair on a person's head.
(If they're their hair and nails, this is forbidden by the Third Law of Thermodynamics - you can't get the energy to live and grow hair from the energy you put earlier in those same hair).
This is a plot point in Serge Brussolo's Sommeil de sang, 1982 (Blood Sleep, but I don't know if it was ever translated into English), where the Autonomous people survive on their own hair, and baldness is a wasting disease (I must confess I almost put the book down then and there).
What about surviving on someone else's hair and nails? It can't work (also: eeew), because hair is not digestible by humans; to be digestible we would need specialized bacteria similar to those that help in digesting cellulose, which has some similarities to keratin (both are polymers, the former made of D-glucose units, the latter of protein units). While such bacteria do exist (they're called dermatophytes and infect humans, feeding on hair and skin squamae), they're not optimized for survival in the gut. Dust mites also are able to digest the softer squamous keratin by drenching it in peptidase enzymes, but you can't do that in a normal human gut because at the required concentrations the enzymes would digest the gut itself first. Also, digestive protease production in humans isn't all that great (one of the reasons humans aren't full carnivores, but rather omnivores; and even obligate carnivores like cats will rather regurgitate hairballs than digest them).
The caloric value of keratin is also way below that of cellulose, because the digestion process requires the undoing of much stabler disulphide bridges. Even setting up the digestion process is expensive as it requires dedicated enzymes and/or strong acids (cellulose can be hydrolyzed much more easily).
(Hair and nails supply different kinds of keratin; the latter is more easily digested, at least in rats).
By the same token, perversely, keratin synthesis is more expensive than cellulose synthesis. Crudely, L-cystein supplies around 70% of the caloric output of D-glucose, and that without the waste management costs (D-glucose burns to easily disposed of CO2 and water, while L-cysteine produces, among other things, ureic acid and nasty sulfur compounds. You really don't want to stay downwind of a keratin-digesting, human-sized organism).
I haven't been able to come across hard numbers, but I feel that a still very optimistic estimate is a ratio of perhaps 1 to 20 - that is: to be able to produce one gram of keratin, you need to efficiently consume around 20 grams of keratin (we knew from thermodynamics that we could never reach a 1:1 ratio anyway).
The number of "enough people" is likely to be high, though, because standard caloric intake is around 2000 kCal per diem, and this translates to a wildly optimistic 500 grams of hair (that is the absolute minimum, without expenses and interests. I'm treating hair as if it was prime steak). On eBay - you don't want to know - they say that a full head of 50-cm hair weighs 250 grams. So you need one meter of hair per day.
Various sources on the Internet claim both that
According to the U.S Center for Disease Control, hair grows at an average of 0.50 inches (1.25cm) per month which means the average person grows 6 inches (0.15 m) per year.
and that you can grow your hair 1 cm per week with several magical remedies.
Even allowing 0.5 m per annum instead of 0.15, every day you would need two years' worth of hair growth, that is 730 man-days.
Riding the wave of optimism, we thus end up with a people ratio of 1:730 - you need 730 people, surviving on something else, to allow one gene-tailored person to survive on their hair.
(Until oligoelements and vitamin deficiencies do him in, that is).
We can turn up the gross factor to eleven by noticing that oily hair contain more easily digested and caloric substances (Wikipedia: "triglycerides (~41%), wax esters (~26%), squalene (~12%), and free fatty acids (~16%)"), with a final calorific power about five times that of hair.
No, they cannot survive on such a diet. Nails and hair are made of keratin. Keratin is very much not digestible:
Keratin is highly resistant to digestive acids if it is ingested (Trichophagia). Because of this, cats (which groom themselves with their tongues) regularly ingest hair which will eventually result in the gradual formation of a hairball that is occasionally vomited when it becomes too big. Rapunzel syndrome is an extremely rare but potentially fatal intestinal condition in humans that is caused by Tricophagia.
Even if it was digestible, it is very low in usable energy, which is why almost nothing eats it except a few fungi, such as the fungus behind Athlete's Foot.
How long would they survive eating only their own nail and hair? Well, nail growth and hair growth account for an insignificant fraction of the energy expenditure of a human being. Accordingly, you should see negligible increase in survival times over not eating anything at all. How long you can survive as such depends on a multitude of factors, but somewhere between 3 weeks and 8 weeks seems to be the limit. However, if you are unusually obese, you can last much longer, such as one man who weighed 456 pounds before he went 382 days without eating any solid or liquid food (he did get supplements to ensure he had sufficient nutrients, just not any calories).
But the OP does not mandate it must be on the fly. Suppose I stored a large quantity of my hair and nails over time, while eating meals at Arbys. When it came time to live on hair and nails I would have a large supply, enough to meet my nutritional needs.
One would of course suffer from nutritional deficiencies (e.g. scurvy, pellagra) in the intermediate and certainly long term.
In addition to CortAmmon's answer on the merits of keratin, there is the simple problem of energy conversion.
That is to say, how do they grow hair and nails in the first place? When you are a baby, you don't have much of either. You need some external source of energy to even begin growing extra hair and nails.
As pointed out by everyone, of course not indefinitely (because thermodynamics), but this is not what is being currently asked, rather "the maximum number of days a human can survive by eating hairs and finernails".
If you don't have a decent storage, from the low nutritional value of hair+nails, a very good approximation to the answer would be "the same maximum number of days as without eating hair nor fingernails". There is also a consensus here.
What if you have a unusually large storage, as suggested by Willk? Eww, sure, but a good suggestion nonetheless.
I'd be not making the eww-factor any better with my answer, but let me point out that neither hair nor nails do have to be clean. I'm going for an abundant reservoir of dirty hair.
One can get some nutitional value from eating dirt (even more so if it contains compost!). In the same line, but for conventionally-dirty hair: dandruff, sebaceous secretions (yay fat!), different kinds of fungi (yay proteins!), non-microscopic bugs and their eggs (lice, ticks). The more complex the dirtiness of the hair, the better its potential nutritional value. Nails of course can also be a good source of fungi and various dirt.
Still, how much fat+muscle does one have on oneself? Typically, a lot, compared with all but the most extreme reservoirs of dirty, oily hair. It's really hard to compete with that. One's own typical methabolic mechanisms, where you slowly&internally eat yourself, will typically overwhelm the kinds of eww-tricks we can come up with. So my best attempt at a numeric answer is the same maximum number of days as without eating hair nor fingernails, plus maybe 1.