# 3/4 and Kleiber’s Law – Numberphile

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THOMAS WOOLLEY: OK, so today

we’re going to be talking about the number 3/4, or 0.75. Unusual, I know, because it’s

not a whole number, like 0, 1, 2, 3, 4. But this one is very important

in biology. So Max Kleiber in the 1930s

posited a graph of mass of animal against metabolic rate. Down here we have the mouse,

he’ll have his little legs there and little ears. MALE SPEAKER: Is that a mouse? THOMAS WOOLLEY: That’s

a mouse. MALE SPEAKER: That’s the worst

mouse I’ve ever seen. THOMAS WOOLLEY: Wait until

you see my elephant. So somewhere up here,

1,000 times heavier, you’ll have an elephant. So let’s try this one. MALE SPEAKER: That’s not bad. THOMAS WOOLLEY: That’s

not bad. Well, you know, it’s

kind of like a big mouse, to be honest. And then all the way up here,

you have your blue whale. OK, so he plotted the mass

against the metabolic rate of all the animals. So mouse, elephant, blue

whale, there was also mayflies in there. And as you might expect, with

the bigger animals, they need more energy to survive. And that’s what metabolic rate

essentially measures. How much energy do you

need to survive? So small animals need a small

amount of energy, bigger animals need more energy. So one way it could happen is

that since the elephant is 1,000 times heavier than the

mouse, it would need 1,000 times much more energy. And if you had that sort of

scaling, it would be linear. And so it would have just a

straight line between the metabolic rate and the mass. So the whale, that’s a million

heavier than the mouse, so it would need a million

times more energy. Or it could be super linear. So the elephant is 1,000 times

heavier than the mouse, so it needs, say, a million

times more. It actually needs more than

the line originally, the linear line. What Max Kleiber found was that

it’s neither of those. The curve actually tapers

off, and so it’s below the straight line. What that means is that the

bigger you get, although you need more energy, you can spend

it more efficiently. So down here, with your mayflies

and your spiders and all your insects, they need very

little energy, but they don’t use it very efficiently. And this scale crosses all sizes

of animal, right down to cells, all the way up

to the blue whale. It’s amazing how this curve

describes all the animals. And how does this curve

link in the 3/4? Well, what Max found was that

this curve, is that the metabolic rate is proportional

to the mass to power 3/4. Now, statistically, it was

around 3/4, so it’s something like 0.74 or 0.76. But on average, that’s what

biologists now take. So this is Kleiber’s rule,

or the 3/4 rule. Well, when they found 3/4, one

thing we want to understand is where does that number

come from? Is it just a coincidence that

it’s 3/4, or is this some reasoning behind it that

we can get at? So originally when they were

looking for this 3/4, they were trying to link metabolic

rate and the mass. And so your metabolic rate is

how much you use energy. And you use energy, they

thought, mostly through heating yourself. And so what affects the heat? Well, your surface area. The bigger your surface area,

the more you lose heat. And the bigger you are in terms

of your mass, again, the more heat you will have. So they looked at spherical

animals. So just for the sake of

mathematics, we’re going to treat these animals as spheres,

because your main body is sort of spherical. And you can at least get an

approximation of what you’d expect to get at the end. We draw our sphere, and we

give it a radius, r. So that’s the body of

the animal, OK? So even if you’re a mayfly,

you’d be a tiny sphere. If you’re a blue whale, you’d

be an enormous sphere. So the area, the surface area

of this sphere, that’s proportional to r squared. As you increase r, if you double

r, your surface area increases by 4. And the volume of the sphere

is proportional to r cubed. So if you double the

radius, your volume increases by eight. So using this idea that the

metabolic rate is proportional to these two guys, you’d expect

it to be metabolic rate is proportional to r 2/3. So you’d get a 2/3 power out. So although 2/3 is quite close

to 3/4, we wanted to understand where that difference

is coming from, because they’re close. So it’s a good first

approximation, but there has to be some reason that we’re

not getting 3/4. And so what mathematicians have

looked at is using the fractal structure inside us

to deliver that energy. So the circulation system that

we have, the veins, the arteries, and the

capillaries, is self-similar all the way down. As you zoom into the

veins, they look like the whole system. And that’s what a fractal is. And using that fractal

structure, you can link the mass to the elephant

to the blue whale. And that does spits out

the 3/4 power law at the end of it. So as I said, yeah, the bigger

the animal, the more efficient you’re using your energy. But using this idea of the

circulation system being a fractal, you can then start

using it to look at cities, because what are cities? Well, they’ve got their own

circulation system, because they have roads that deliver

cars to places. They have water pipes, they

have electricity cables. And you can plot those certain

things against population, and you get very similar graphs that

follow this power rule. It may not be 3/4, but you’ll

get that it’s proportional to the population, and it’s

less than linear. And they usually are following

a quarter structure. So whatever this number may be

on top, you’ll usually find a 4 on the bottom.

Lawrence SarsamPost authorSpherical Chicken joke! 😀

IMortagePost authorUnfortunately, some strange cities aside, cities – both big and small – tend to be circles (2dimensional objects) and not spheres. Unless you build the city up and down just as much as sideways… this law no longer applies.

MilkymalkPost authorIt's the biological equivalent of "Let's treat Pi as 3.2" 😀

musikSkoolPost authorwhy is the blue whale waving?

TraianusMasterPost authorYour body is spherical? Maybe if you're American 😉

PhiePost authorit's supposed to be water coming out of it's snout -_-

MannyPost authorthis says that you should use fractals to model most system and create more efficient systems. witch can actually be more efficient than villages(doesnt mean that they are).

MannyPost authoranimals arent spheres neither, but in science you can sometimes simplify a system to a sphere, to make the math easier to solve.

IMortagePost authorBut animals are functionally very close to spheres, their volume increases faster than their height/width, which is not the same for cities, as those are essentially flat / 2D objects.

vantarinitelPost authorThat actually holds true, and not just for the reasons you've mentioned. Shipping costs go down, transportation costs go down, effects on immediate ecosystem quantities goes down (as in, while the square footage that used to be Manhattan is screwed permanently, cramming more people into just Manhattan instead of Manhattan plus Nico, Oklahoma and Galen, Montana will help the mini ecosystems in Nico and Galen.) ^ ^

Uranium WillyPost authorHe not telling us something to simplify things, warm blooded animals use much more energy than a equivalently sized cold blooded animal.

The SeekerPost authorBORG CUBES!

xxXMathisX6x6x6Post authorgreater size/amount/population produces greater complexity, which due to evolutionary processes, inevitably becomes more efiicient.

Wand2FishesPost authorIn Sweden, a mouse is also what americans calls a "beaver". But that mouse looked more like its male counterpart…..I really have a dirty mind

dojokonojoPost authorYou think it's crazy, but that's what we physicists do!

Henry O'KeeffePost authorI think you will find that most people do, and it's not working!

*save data corrupted*

Albert Aparicio IsarnPost authorThis is engineering xD

Filip WolfPost authorThat's acctualy already known and people are moving to towns 🙂

Filip WolfPost authorcities* sorry

akatsukiawsome13Post authorNOOOOOO!!!! it was 69 and on impulse I liked it… sorry world…

GuodlcaPost authorThe mouse he drew looks a bit like a male reproduction organ.

Ray ChungPost authoractually what you've said is actually supported by some factions of urban planning and is the major argument against urban sprawl. The argument basically states that large cities like New York and London are more eco-friendly than the urban sprawls of Los Angeles. Of course, the other factor is commuting. Besides putting people together, if you also integrate and localize, people wouldn't have to drive as far to do the stuff they do everyday.

Newt32Post authorSame as in big scale economics.

Alec BradyPost authorOh, how I wish I lived in a hug city!

Brian CramerPost authorMale reproductive organ ? What are you twelve? Say penis

rm5Post authorsuperlinear, wtf? Either it's linear or it is not, in which case it may follow, for example, power, logarithmic/exponential laws etc.

Nathan AaronPost authorI swear I'm going to start saying "naught" instead of "zero" because of these videos and freak my math teacher out.

Nathan AaronPost authorSuperlinear is simply a flavor of linear. When watching a video made by mathematicians, assume they know more words than you.

H SunsetPost authorsuperlinear indicates the slope is greater than the related line. Yes,it is a curve, but he wasn't finished taking us through the solution so the the word "superlinear" serves to comment on the relationship of reality to linearity

pjodenPost authorThe elephant must be extremely much more than thousand times heavier than the mouse.

Sam RossmanPost authorSo then why does this relationship hold true for organisms that expend no energy heating themselves (i.e. ectotherms, the vast majority of species on this planet)

alanPost authorlooks like the whale is waving at us

AIO ReferencePost authorCan't it be that since volume of a sphere is 4/3πr^3 and surface area is 4πr^2 so its effectively 3 r^2/r^3 = log 3 * 2/3 which is almost 3/4???

bug bugerPost authorI dont get it if it keeps curving then it would go all the way down and Godzilla would need as much energy as a mouse??

Dan DubsPost authorWhen 2 atoms combine energy is released because 2 combined atoms need less energy then 2 single atoms. ? 3/4 energy?

JDobsablePost authorWho else thought that mouse looked like a nut sack?

Aros of KernowPost authorMaths in biology, perhaps it's finally catching up with chemistry and physics…

Robeon MewPost authoreither he explained it wrong, or he doesn't understand what he's trying to explain

JLConawayIIPost authorAn elephant 1000 times more massive than a mouse? That's either an enormous mouse, or a very small elephant. Also, I'm intrigued by the whale/broccoli hybrid there. I'm guessing that's supposed to be a water spout?

Michael MoorePost authorInteresting how he says "cah-PILL-or-ees"

Everywhere I've heard, it's "CAH-pill-air-ees"

Jaap Willem van RandenPost authorLoved this one Brady! One of the best Ive seen so far!

linkviiiPost authorstill one of my favourite videos on the channle

Robbie Jakob-WhitworthPost authorI've met this guy. Sat next to him at a meal. Woo! 😀

YtubeUserrPost authorThis is clearly pseudo-math (the maths counterpart of pseudo-science).

DasIlluPost authorthe whale looks like he wants to highfive the elephant.

can't be unseen 😛

Billy FairbankPost author"Your body is more or less spherical" is probably not something you should tell your wife (even if it's true)

Denny ChenPost author"Spherical body"

Spot the physicist

John LordPost authorhmm, so which city structure is more efficient (lower metabolic rate), a US style square street plan, or a more evolutionary city like Rome, Paris, or London?

Jake T.Post authorAt least it's not the 2/3 rule…

Ronald de RooijPost authorI don't know about this one…. I have done environmental science, but this reasoning seems a bit too dodgy for my taste.

m3ntalcas3Post authori buy the explanation kinda but need more vids on it to fully understand what he means

ameba 2804Post authorJust for the record: an elephant isn't 1000x bigger than a mouse. A 10 yo human is. Elephant is rather 200 000 times.

Noah SmitsPost authorthe Chef Sperm has the highest mass

Kevin FeganPost authorI don't know. I'm not convinced. Unless there's something more to it that wasn't said. It all seems so arbitrary. I mean it seems that knowing the MR is (statistically) related to a power of 3/4 they just found something that was "about" 3/4, and even closer to 3/4 if we adjust for this other thing. Lots of other things could have a 3/4 relationship and without "reason", you could just pick anything. Say, the distance from the mouth to the nose is 3/4 of the distance from the nose to the eyes so that must be where the 3/4 came from. Or some 2 constructions in DNA have a 3/4 relationship, …

Ricardo MachadoPost authorThe 3/4 was reached by doing a log-log plot, and the slope they found was aprox. 3/4. But there is much more to it. Birds, mammals, reptiles, invertebrates have different requirements: a mammal and a bird with the same body weight have different metabolic rates. That is why there is a constant called Kleiber's constant to modify the basic law.

Dee ElmorePost author500th comment!Ps. Those are some very phallic animal drawings.

Annoy-o-tronino KripperinoPost authorthat mouse does not look well… 0:38

abdel halim shehataPost authorUR THE MAN!!!!!!!!!!

SteppenkaterPost authorAn elephant is about 200,000 times heavier than a mouse. Or just say 100,000 but not 1000. Just saying 😉

But nevermind. Still a great video!

danPost authorthe mouse looks interesting (le lenny)

Matthew RandolphPost authorthis is offensive! Identify as a sphere! uughh /s

Hector NonayurbusinessPost authordoes this work for reptiles?

GabeLucarioPost authorAre the axes of the graph logarithmic or linear?

GorzoidPost author"wait till you see my elephant" this guy just loves drawing dicks doesn't he

KaizokuKevinPost authorWait a second so instead of using one of those shitty calorie counters can i just put my weight into that equation?

Qupid VOneOhOnePost authorI chocked on my breakfast when he told me to treat animals as spheres. "You need to have your cows radiating milk isotropically." 😂

Hi EilPost authorplus 9

Hi EilPost author0:39 if you see the legs as fur it's a thing very different of a mouse

Inky ScrollsPost authorA blue whale? Somebody get Alan!

TheNintendoFanaticPost authorIs mass in grams?

PGPost authorWait till you see my elephant…….OHHHHHH

justin hPost authorDid anybody else look for 1/4, 2/4 and 4/4 before actually starting to watch this and realizing its a fraction? lol

misa misaPost authorPlatform nine and 3/4

Alexandria GonzálezPost authorI kinda had a gut feeling towards 2/3 at the very beginning.

BethShep1998Post author"This is very important in biology."

stops videoYerrikPost authorOne note: metabolic rate has units of

power— not energyAlejandro TelloPost authorAll that sophistication and they couldn't make a proper estimate for the relationship between mass and volume in biological systems?

njack1994Post authorI do not think this is very accurate. What does size have to do with efficiency of energy consumption if you are not factoring in proportional workloads. Ever heard an ant can lift like 50 times it's own weight. Not looking so proportional to me.

William GregorPost authorWow how wonderful. Darwin?? How does it work with Darwinian theory?

SorrirBoyPost authorI know another rule that involves numbers 3 and 4 ( ͡° ͜ʖ ͡°)

Simon RuszczakPost authorActually using a sphere, or a cube, or any other 3D shape makes no difference !

As the radius (scaling) doubles, the surface area to volume (A/V), halves, (^ 2/3).

Ryan LatterellPost authorMammals are blackbody radiators. Assume energy expended on consumption and travel is essentially zero compared to energy lost from blackbody radiation (mechanical hunting and gathering << thermal heat loss).

Heat loss due to blackbody radiation is proportional to the square of an animals dimension (surface area or radius^2) and also proportional to the cube root of the animal's mass (equation for a sphere or cylinder is cubic) >>> surface area scales at a 2/3 power with mass. Heat loss is proportional to K-boltzman*((T,body-T,environment)^2)*(Surface area), this might end up giving a relationship between mass and energy loss that aligns mathematically with this 2/3 power multiplied by another factor T^2, possibly giving a 4/3 power law relationship for warm blooded mammals if you work things out. Assume also that body mass at the center of any limb (r=0) requires no energy to heat and is essentially getting a free ride from surrounding tissue. Body mass at the surface loses heat according to temperature difference with environment and basic blackbody thermo equations. These will serve as boundary limits for the differential equation that tells us how metabolic rate scales with mass.

Use paper and pencil to figure the rest out. Rods (limbs) will radiate differently than spheres (rabbits, mice). Fix math where needed and make the necessary assumptions??

Cobalt BlockBrosPost authorTHAT HANDWRITING OMG

Klaus Ole KristiansenPost authorWhen he said 3/4 and biology, I thought of Mendel's law.

Piotr CzaryPost authorthe bigger animal, the slower rate of loosing heat because on average the larger distance from heat spot to outside

Mateo De los Angeles MartinezPost authorWell, I think this law wouldn’t work on The Hulk😂

Warping NawrwhalPost authorNumberphile should get another side channel called colo(u)rphile.

Venkatesh babuPost authorMass balance through center of gravity. A sphere with 120 elements has hydrogen and electrons to distribute itself into weighing balance. And lower quark switch. 10 electrons split in group of two five to join two quark to one with two free electrons. Two cross two gives four virtual worlds. Two real worlds. Mass exchange is diametric transfer. So 20 out of one hundred and twenty are extreme dark holes.

Finlay HutchinsonPost authorWhat about surface area

Lothar FreitagPost authorthis only applies to warm blooded animals i guess. well while i know that reptiles need way less food compared to their weight because of being coldblooded, i am not so sure about insects, since you mentioned the mayfly. They are also considered coldblooded, though they don't have the capillary system of vertebrates and therefore no "blood", but i believe they might use a lot of energy for flying, though even if they use more energy because of that, i would wager that you need a different constant for coldblooded and another one for warmblooded.

Would it not also make a huge difference if you lived, say in the savanna compared to the icy waters around the poles, or is the isolation most of those animals have bringing it to about the same values?

wat am i doing?Post authorA farmer asked a physicist for help because his chickens where dying out. The physicist then came back and said "I can fix your problem, but it only works with spherical chickens in a perfect vacuum"

Lau BjernoPost authorCould you please make an in-depth video on this topic? This was extremely superficial.

FOXTrot3Post authorInteresting

Cheeseburger MonkeyPost authornah it's reverse

that's proved by kurskesagt

bowlchamps37Post authorI have never seen a 19kg light elephant. 1:21

blaszlob13Post authorWhy is the metabolic rate inversely proportional to the volume? I mean, he wrote MR ∝ r^(2/3) = r^2/r^3, so MR ∝ 1/r^3.

Ged LangoszPost authorSo this is why physists use spherical cows.

Kuusik 100Post author0.75.

Aliyahbhat BhatPost authorTHANKS sir