Allometric Scaling Explorer

Discover why an elephant's heart beats slower than a mouse's: explore how metabolism, lifespan, and other traits scale with body mass through the power law Y = Y0 · M^a, from shrews to blue whales.

Worked example — Allometric Scaling Explorer

Quantity (Y)	Metabolic Rate
exponent	0.75

These are the values the calculator shows when the page opens. Change any input above to recompute them.

Loading interactive simulation...

A century of argument over one decimal place, worth a factor of two 🖖

The tool ships both candidate exponents as presets, so you can put the dispute side by side. Surface Area Scaling uses α = 0.67, the value you get if metabolism is limited by how much heat a body can shed through its skin. Kleiber’s Law uses α = 0.75, the value actually measured across species. Over the 10,000-fold mass range the tool spans, 0.67 predicts a 479-fold rise in metabolic rate and 0.75 predicts 1000-fold — the same data, a factor of 2.09 apart. That gap is why the exponent was fought over for most of the twentieth century: it is not a rounding detail, it is a doubling.

Why a power law becomes a straight line 🖖

This tool plots body mass against a trait like metabolic rate on log-log axes. Any power law Y = Y₀ · M^α turns into a straight line there, and its slope is the exponent α. That is why dragging the slider tilts the fitted line — you are reading biology's scaling exponent straight off the graph. Set α = 3/4 and data from shrews to blue whales snap onto one line spanning many orders of magnitude.

Every mammal's heartbeat budget 🖖

Heart rate falls as M^(−1/4) while lifespan rises as M^(1/4), so their product — total heartbeats in a lifetime — barely depends on size. A shrew and a blue whale each spend roughly 1.5 × 10⁹ beats before they die. Humans are the striking exception: modern medicine and safety stretch us to about 2.9 × 10⁹, nearly double the mammalian norm.

Problem solved in full

A 4,000 kg elephant and a 25 g mouse without food 5 steps
Work out how much longer an elephant can go without food than a mouse. This is Kleiber (BMR) with α = 0.75, using the pair the tool already compares: a 4,000 kg elephant and a 25 g mouse.
1. A power law has no scale of its own, so a ratio of outputs depends only on the ratio of inputs and never on the units either is measured in. The entire comparison therefore rides on one number.
  
  $\frac{Y _{2}}{Y _{1}} = (\frac{M _{2}}{M _{1}})^{α}, \frac{M _{E}}{M _{m}} = \frac{4000}{0.025} = 160000$
2. 160,000 is 20 to the fourth power, so the three-quarter power is a fourth root followed by a cube: 20, then 8,000. The answer is exact, and no calculator touched it.
  
  $160000 = 2 0^{4} ⟹ 16000 0^{0.75} = (2 0^{4})^{3/4} = 2 0^{3} = 8000$
3. Kleiber's law describes the whole animal; what one gram of it burns is that divided by mass. Subtracting one from the exponent turns three quarters into a negative quarter, and the elephant's gram runs twenty times slower than the mouse's.
  
  $\frac{B}{M} \propto M^{3/4 - 1} = M^{- 1/4}, \frac{8000}{160 000} = \frac{1}{20}$
4. Now give both animals a fuel tank. If stored energy is a fixed share of body mass it scales with the first power, while the burn rate scales with three quarters, so the quotient — how long the tank lasts — carries the difference.
  
  $E = f M ⟹ τ = \frac{E}{B} \propto \frac{M}{M ^{3/4}} = M^{1/4}, 16000 0^{1/4} = 20$
5. A quarter power flattens everything it touches, so endurance is the slowest-moving quantity in the comparison. Doubling it takes a sixteen-fold animal, and between a 60 kg and a 90 kg human it buys 11%.
  
  $\frac{τ _{2}}{τ _{1}} = 2 ⟺ \frac{M _{2}}{M _{1}} = 2^{4} = 16; (\frac{90}{60})^{1/4} = 1.11, (\frac{100 000}{0.025})^{1/4} = 2000 = 44.7$
Answer

The tool prints α = 0.75 and, for this pair, a mass ratio of 160,000 against a metabolic ratio of 8,000. The number it does not print is the one those two make between them: 160,000 ÷ 8,000 = 20, which is also the fourth root of the mass ratio. That single 20 is the elephant's advantage in fasting time and the mouse's penalty in feeding rate at once — the same quarter power, read in opposite directions. Because the exponent is so small, the effect is invisible inside a species and unmissable across a class: 11% between two adults, and 44.7× across the mouse-to-blue-whale span the chart plots. Nothing here required any biology beyond one exponent and the assumption that fuel is a fixed fraction of mass.

$B \propto M^{3/4} ⟹ \frac{B}{M} \propto M^{- 1/4}, τ \propto M^{1/4}$

References (4)

The 3/4 exponent, as originally measured: Max Kleiber, "Body size and metabolism." Hilgardia 6, 315–353, 1932.
The fractal-network argument for 3/4 — the explanation this tool deliberately does not assert as settled: G. B. West, J. H. Brown & B. J. Enquist, "A General Model for the Origin of Allometric Scaling Laws in Biology." Science 276, 122–126, 1997.
The case for 2/3 instead, which is why the exponent is still argued over: C. R. White & R. S. Seymour, "Mammalian basal metabolic rate is proportional to body mass^2/3." PNAS 100, 4046–4049, 2003.
A direct re-examination of the evidence for either value: P. S. Dodds, D. H. Rothman & J. S. Weitz, "Re-examination of the '3/4-law' of Metabolism." Journal of Theoretical Biology 209, 9–27, 2001.

By the 42Math team · Updated 2026-08-17

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Kleiber (BMR) - Kleiber's Law (BMR): How metabolic rate scales to the 3/4 power of body mass across mammals, explaining why larger animals are more energy-efficient.
Surface area - Surface Area Scaling: Geometric scaling showing how heat loss and surface-area-to-volume ratio scale to the 2/3 power of mass according to the square-cube law.
Heart rate - Heart Rate Scaling: Shows why elephants have a pulse of 30 bpm while mice tick at 600 bpm, scaling to the negative 1/4 power of body mass.
Lifespan - Lifespan Scaling: Explores the quarter-power scaling relation between body size and animal lifespan, showing how horse lifespan scales to 10 times that of a mouse.