# 3/4 and Kleiber’s Law – Numberphile

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
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.

## 99 thoughts on “3/4 and Kleiber’s Law – Numberphile”

• ### Lawrence Sarsam Post author

Spherical Chicken joke! 😀

• ### IMortage Post author

Unfortunately, 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.

• ### Milkymalk Post author

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

• ### musikSkool Post author

why is the blue whale waving?

• ### TraianusMaster Post author

Your body is spherical? Maybe if you're American 😉

• ### Phie Post author

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

• ### Manny Post author

this 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).

• ### Manny Post author

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

• ### IMortage Post author

But 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.

• ### vantarinitel Post author

That 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 Willy Post author

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

BORG CUBES!

• ### xxXMathisX6x6x6 Post author

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

• ### Wand2Fishes Post author

In 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

• ### dojokonojo Post author

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

• ### Henry O'Keeffe Post author

I think you will find that most people do, and it's not working!
*save data corrupted*

• ### Albert Aparicio Isarn Post author

This is engineering xD

• ### Filip Wolf Post author

That's acctualy already known and people are moving to towns 🙂

• ### Filip Wolf Post author

cities* sorry

• ### akatsukiawsome13 Post author

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

• ### Guodlca Post author

The mouse he drew looks a bit like a male reproduction organ.

• ### Ray Chung Post author

actually 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.

• ### Newt32 Post author

Same as in big scale economics.

• ### Alec Brady Post author

Oh, how I wish I lived in a hug city!

• ### Brian Cramer Post author

Male reproductive organ ? What are you twelve? Say penis

• ### rm5 Post author

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

• ### Nathan Aaron Post author

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

• ### Nathan Aaron Post author

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

• ### H Sunset Post author

superlinear 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

• ### pjoden Post author

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

• ### Sam Rossman Post author

So 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)

• ### alan Post author

looks like the whale is waving at us

• ### AIO Reference Post author

Can'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 buger Post author

I 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 Dubs Post author

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

• ### JDobsable Post author

Who else thought that mouse looked like a nut sack?

• ### Aros of Kernow Post author

Maths in biology, perhaps it's finally catching up with chemistry and physics…

• ### Robeon Mew Post author

either he explained it wrong, or he doesn't understand what he's trying to explain

• ### JLConawayII Post author

An 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 Moore Post author

Interesting how he says "cah-PILL-or-ees"
Everywhere I've heard, it's "CAH-pill-air-ees"

• ### Jaap Willem van Randen Post author

Loved this one Brady! One of the best Ive seen so far!

still one of my favourite videos on the channle

• ### Robbie Jakob-Whitworth Post author

I've met this guy. Sat next to him at a meal. Woo! 😀

• ### YtubeUserr Post author

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

• ### DasIllu Post author

the whale looks like he wants to highfive the elephant.
can't be unseen 😛

• ### Billy Fairbank Post author

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

• ### Denny Chen Post author

"Spherical body"
Spot the physicist

• ### John Lord Post author

hmm, 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 author

At least it's not the 2/3 rule…

• ### Ronald de Rooij Post author

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

• ### m3ntalcas3 Post author

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

• ### ameba 2804 Post author

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

• ### Noah Smits Post author

the Chef Sperm has the highest mass

• ### Kevin Fegan Post author

I 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 Machado Post author

The 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 Elmore Post author

500th comment!Ps. Those are some very phallic animal drawings.

• ### Annoy-o-tronino Kripperino Post author

that mouse does not look well… 0:38

• ### abdel halim shehata Post author

UR THE MAN!!!!!!!!!!

• ### Steppenkater Post author

An 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!

• ### dan Post author

the mouse looks interesting (le lenny)

• ### Matthew Randolph Post author

this is offensive! Identify as a sphere! uughh /s

• ### Hector Nonayurbusiness Post author

does this work for reptiles?

• ### GabeLucario Post author

Are the axes of the graph logarithmic or linear?

• ### Gorzoid Post author

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

• ### KaizokuKevin Post author

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

• ### Qupid VOneOhOne Post author

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

plus 9

• ### Hi Eil Post author

0:39 if you see the legs as fur it's a thing very different of a mouse

• ### Inky Scrolls Post author

A blue whale? Somebody get Alan!

• ### TheNintendoFanatic Post author

Is mass in grams?

• ### PG Post author

Wait till you see my elephant…….OHHHHHH

• ### justin h Post author

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

• ### misa misa Post author

Platform nine and 3/4

• ### Alexandria González Post author

I kinda had a gut feeling towards 2/3 at the very beginning.

• ### BethShep1998 Post author

"This is very important in biology."
stops video

• ### Yerrik Post author

One note: metabolic rate has units of power— not energy

• ### Alejandro Tello Post author

All that sophistication and they couldn't make a proper estimate for the relationship between mass and volume in biological systems?

• ### njack1994 Post author

I 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 Gregor Post author

Wow how wonderful. Darwin?? How does it work with Darwinian theory?

• ### SorrirBoy Post author

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

• ### Simon Ruszczak Post author

Actually 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 Latterell Post author

Mammals 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 BlockBros Post author

THAT HANDWRITING OMG

• ### Klaus Ole Kristiansen Post author

When he said 3/4 and biology, I thought of Mendel's law.

• ### Piotr Czary Post author

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

• ### Mateo De los Angeles Martinez Post author

Well, I think this law wouldn’t work on The Hulk😂

• ### Warping Nawrwhal Post author

Numberphile should get another side channel called colo(u)rphile.

• ### Venkatesh babu Post author

Mass 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.

• ### Lothar Freitag Post author

this 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 author

A 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 Bjerno Post author

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

Interesting

• ### Cheeseburger Monkey Post author

nah it's reverse
that's proved by kurskesagt

• ### bowlchamps37 Post author

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

• ### blaszlob13 Post author

Why 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 Langosz Post author

So this is why physists use spherical cows.

0.75.

THANKS sir