Roughly, surface area, A, of an animal scales as its length, L, squared |

A = L^{2} |

and the mass, M, scales as L cubed |

M = L^{3} |

Heat is dissipated across the surface, and metabolic rate is proportional to heat dissipation, so |

metabolic rate ≈ L^{2} = M^{2/3} |

and |

(metabolic rate)/(unit mass) ≈ M^{-1/3} |

Seems reasonable. Pulse rate is related to metabolic rate per unit mass, and we know smaller animals have faster heartbeats. |

However, extensive and careful measurements by Kleiber showed
that metabolic rate per unit mass scales as M^{-1/4},
instead of M^{-1/3}. Why should this be? |

More heat is lost through the lungs than across the skin surface, so the fractal structure of the lungs may account for this difference. |

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