3/4 and Kleiber's Law - Numberphile

okay so today we're going to be talking about the number 34 or .75 unusual I know because it's not a whole number like not 1 2 3 4 but this one is very important in biology so uh Max cerman in the 1930s ploted a graph of mass of animal against metabolic rate down here we have you know you'll have the mouse he'll have his little legs there little ears that's that's that's a mouse that's the worst Mouse I've ever seen wait till you see my elephant so somewhere up here A Thousand Times heavier you know you'll have you'll have an elephant let's have a let try this one that's not bad dude that's not bad well you know it's kind of like a big mouse to be honest and then all the way all the way up here you have your blue whale okay so he plotted the against the metabolic rate of of all the animals so Mouse elephant blue whale there was also May flies in there and as you might expect with the bigger animals they need more energy to survive and that's what metab metabolic rate essentially measures how much energy do you need to survive so smaller animals need a small amount of energy bigger animals need more energy so one one way it could happen is that since the elephant is a thousand times heavier the mouse it would need a thousand times much more energy and that if we 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 times heavier than the mouse so it would need a million times more energy or it could be superlinear so the elephant is a thousand 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 kber found was that it's neither of those the curve actually tapers off and so it's below the straight line what that means 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 your uh your spiders and all your insects they need very little energy but they 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 him the three quarters what Max found was that this curve is that the metabolic rate is proportional to the mass to power three quarters now statistically it was a it was around 3/4 so something like 0.74 or 0.76 but on average that's what biologists now take so this is C's rule or the 3/4s rule well when they found three quarters one thing we want to understand is where do that number come from is it just a coincidence that it's 34s or is there some reason behind that we can get at so originally when they were looking for this three quarters 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 is well your surface area the bigger your surface area the more you lose he 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 a sphere and we give it a radius R so that that's the body of the animal okay so even if you're a mayfly you'd be a tiny sphere if you're a blue whale you be an enormous sphere so the area the the surface area of this sphere that's proportional to R 2 as you increase R if you double R your surface area increases by four 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 the metabolic rate is proportional to these two guys you'd expect it to be metabolic rate is proportional to R 23 so you'd get a 23 power out so although 2/3 is quite close to 3/4 we want 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 3/4s 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 mouse to the elephant to the blue whale and that does spit out the 3/4 power L at the end of it so as I said yeah the the 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 got their own circulation system because they have roads that deliver uh cars to places they have uh 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're usually following a quarter structure so whatever this number may be on top you'll usually find a four on the bottom