Now, the volume of the network is just the sum of the volumes of all of its vessels or branches, and these can be straightforwardly calculated from knowing how their lengths and radii scale, thereby connecting the self-similarity of the internal network to body size. It is the mathematical interplay between the cube root scaling law for lengths and the square root scaling law for radii, constrained by the linear scaling of blood volume and the invariance of the terminal units, that leads to quarter-power allometric exponents across organisms.
