Radical By Scale

Nature’s Quarter Power Scaling Laws

Why isn’t average human lifespan a million years? Why do organisms stop growing, age, and die?

A curious child may drive her parents crazy with such questions. Innocuous as they sound, they hide profound secrets. In the fourth installment of Radical Series, let us explore universal scaling laws that have a radical of three quarters (exponent = ¾) and the magic number 4!

Photo by marek kizer on Unsplash

Folklore is rife with tales of immortality and the elixir of life. Philosophical works expound on the topics of life and death. Religious texts are replete with professed answers stemming from revelations. Illuminated by newly discovered knowledge, they seem to grow quaint and unsatisfactory by the minute. We are well into the twenty-first-century technology of editing genes to fight disease, studying the gene responsible for aging, and neural implants to augment our primate brain.

If snake oil explanations fall woefully short, the burden of providing sane alternatives that reveal the truths about biological life, aging, and death falls on the collective shoulders of the scientific community. It may yet allow the eradication of our dogged beliefs and intuitions threatening to destroy civilization.

Given physical laws govern everything, there must be a quantitative basis for understanding biological systems. In the late eighteenth century, Immanuel Kant rallied for chemistry to be grounded in mathematics. For over two centuries, we persevered. We assembled the tools of genetics, genomics, simulation, AI, and big data. We tilled the soil of knowledge. We may very well have reached a critical juncture for planting the science of biology firmly into the fertile mathematical ground, fulfilling his wishes. In 2000, Stephen Hawking presciently remarked this is the century of complexity.

I think the next century will be the century of complexity — Stephen Hawking, Jan 23, 2000


We are well into the roaring twenties since his famous prophecy. Even if one subscribes to his idea that humankind is nothing but chemical scum, there is no denying its extraordinary complexity — whether one subscribes to cosmic cynicism, worldly romanticism, or otherwise. A genetic blueprint is mind-blowing in its faithful rendition of an organism. From ingredients and processes to the end product, it scales over twenty-one orders in magnitude. That is complexity in a nutshell.

Novel Approach

Clever physicists may have gotten away with spherical cows. That approach fails to get us off the ground in describing what is arguably the most complex phenomenon in the universe. It is called life. Unlike the regularity and continuity of Euclidean geometry or even the smoothness of Riemannian curvature, biological life is inherently jagged, irregular, and discontinuous — a fractal!. Despite their great successes in revolutionizing our understanding of the fundamental nature of reality, it’s about time we graduated from Euclid and Riemann to develop a comprehensive quantitative basis of all biological life. Not to be construed as a rallying cry for abandoning cosmic pursuits, but an appeal to elevate understanding life’s complexity with the gravitas it deserves.

Self-similarity at scale: A fractal Image Source: Benoit Mandelbrot’s 96th Birthday Google doodle


It is undeniable this pursuit is a tall order but not unattainable. Great strides have occurred in the field of complexity. It comes under a topical heading of scale. Of the many meanings of the word scale, one that is pertinent to our discussion is a transformation that either enlarges or shrinks something. For instance, a one-to-two (1:2) scaled-down version of an object is proportionally half its size. Companies merge to benefit from economies of scale. Nature obeys quarter-power (1/4) scaling laws — therein emerges a radical!

To grasp scaling laws, we must first acknowledge our shortcomings. When it comes to natural phenomena, our intuitions are unreliable at best and hazardous at worst. As if to add insult to injury, schools drum Euclidean geometry into young minds, further skewing their intuition. Despite that, there is hope for a brave young explorer of the mathematics of complexity that permits a closer inspection into scaling laws. But first, let us begin with a real story where intuitions went awry, resulting in a fatal consequence.

Fatal Trip

Immunologists and virologists routinely develop vaccines and their variants to combat pathogens. Pharmacologists sweat details about drugs, dosage, and their effects. We take this well-oiled machinery for granted because it all works as intended. It wasn’t always the case. And there is always collateral in the pursuit of knowledge.


Photo by George Pagan III on Unsplash

Around the mid to late fifties, Lysergic acid diethylamide (LSD) was not yet a recreational psychedelic. It was widely known to hold great potential in psychotherapy and was readily available for research. Proponents of LSD claimed a mild acid trip could alter mood, boost creativity, or with an amplified dosage, even induce a temporary delirium/psychosis. A mild dose of LSD ~ 0.004 mg per kg of body weight induces hallucinations, and a higher dosage of 0.02 mg/kg causes temporary madness. A milligram is a thousandth of a gram. These are tiny dosages compared to 6 mg/kg of aspirin, say, for treating a mild headache.


Photo by Wolfgang Hasselmann on Unsplash

Contrary to wrongly held popular belief, elephants can only be tamed in captivity to accept human contact but never domesticated like our other beloved pets. The majestic beast remains wild. Male elephants (bulls) are known to undergo a rough period of heightened testosterone (hormonal surge) during which they go berserk, experience intense sexual urges, and display highly aggressive behavior. We have known this for at least a century, even having coined a Persian/Urdu word, musth, that captures this intoxicated bout of madness.

Linear Thinking

The Easy Street of linear intuition ends in the Tragic Alley! — VK

If you are wondering what could be the connection between LSD and an elephant, wonder no more! Let’s say one chose a teenaged male elephant for conducting LSD research. Now, as to why one would choose an elephant instead of a lab rat or a guinea pig is a valid point, but a logical line of reasoning may occur along the following trajectory:

  1. Elephant and human are both mammals.
  2. A teenage bull is a perfect specimen to study musth.
  3. LSD induces madness, albeit temporarily.
  4. We could induce madness and study the drug’s effects all in one fell swoop. Brilliant!

To induce madness in a ~3000 kg elephant is not for the faint-hearted! Once it’s decided, the only question that remains is the determination of dosage to administer. There may be a handy study that shows 0.1 mg/kg was non-lethal in a domestic cat (~5 kg).

Q: What dosage should we administer to the elephant?

Perhaps you came up with 0.004 mg per kg × 3000 kg = 12 mg or 0.02 mg per kg × 3000 kg = 60 mg based on human dosage. Or better still, you went with the cat study and calculated 0.1 mg per kg × 3000 kg = 300 mg. All of the above are wrong answers!


That’s what happened. We’ll never know what exactly occurred in the minds of psychiatrist Louis Jolyon West and his sidekick Chester Pearce. A fatal dose of 297 mg got administered into Tusko’s rump with a dart fired from a rifle. It occurred with the help of Warren Thomas, the director of Oklahoma City’s Lincoln Park Zoo, in the fall of 1962. Their research appeared in the prestigious Science journal. The U.S government banned LSD.

Image Source: bizarrepedia.com

In an exam, some points may get deducted for linear thinking. But in the exam of real life, linear extrapolation may cause harm, or worse, prove fatal if acted. It makes all the difference how one reasons, especially if you are a professional, a researcher. This story highlights fallible human intuition that resulted in a tragic comedy of errors.


Enter the world of non-linearity. Exponential and power laws are a recurring theme that governs life’s machinery, infrastructure, cities, companies, and even the global economy. And they contain a magic number. Life’s dimensionality is the magic number, 4 — three spatial (3D) dimensions plus one fractal (1F) dimension. Sorry numerologists!

It is a consequence of the hierarchical, deeply nested, fractal networks that fill an organism. An organism could be any complex system — an animal, a tree, or a transportation grid in a city, a supply chain, or a social network. Networks (respiratory system, circulatory system, access roads, roots, and branches in trees) organically emerge subject to universal scaling laws that are non-linear. They optimally distribute energy/resources to the fundamental units (cells/homes/individuals). That is where the rubber (biology) meets the road (physics) to build the organism (an elephant, a sequoia tree, or New York City). Some physical quantities emerge to be scale-invariant — meaning they don’t vary between a cat and an elephant or any other organism in-between.

Supply Vs. Demand

Life and death have everything to do with the supply and demand of energy. Growth splits the difference between the demand for maintenance and supplies in stock. The split is uneven. As an organism (animal/city) grows, the lion’s share gets used to build new units (cells/homes) and the remainder for maintenance. The script gets flipped as the animal stops growing.

But why? Why does growth slow down and eventually stop? The answer lies in the structure of the network that supplies nutrients to terminal units. The energy supply network can only scale sub-linearly, exponent = 3/4 even though it’s already optimal by nature — it fills all available space in 3D and then some (1F). However, here’s the kicker — the demand scales linearly exceeding the supply, exponent = 1. It’s as if nature cruelly, but thankfully, sowed seeds of destruction within the apparatus of supplying energy that is required to keep us alive. So while we get busy not growing, we get busy dying.

Illustration of linear vs Sub-linear scaling laws. Log-linear plot showing blue sub-linear curve representing network supplying energy. It always grows slower than the linear curve in black representing the demand.

The illustration above shows a linear vs. sub-linear scaling relationship. Notice y = x¹ (demand curve) scales linearly while y = x^¾ (supply curve) will never catch up by design (¾ < 𝟣). Conquering death requires outsmarting nature’s law (mathematics)!

Kleiber’s Law

Max Kleiber, Professor of Animal Physiology, University of California, Davis (1929–1960).

Metabolism is the absorption of energy by an organism. It is a complex biochemical process occurring in molecular units called respiratory complexes within a cell. The process that breaks down carbohydrates, lipids, and proteins fuels the cells.

Max Kleiber was no stranger to animals. His doctoral thesis titled The Energy Concept in the Science of Nutrition is from ETH, Zürich, which proudly boasts John von Neumann and Albert Einstein as its alumni. In 1929, he ended up in the Animal Husbandry department of U.C Davis to study how energy gets absorbed in animals. His invaluable contribution to society is in two simple but profound statements. For any animal:

  • Metabolic rate scales as ¾ power of body-weight (sub-linear scaling)
  • Energy utilization efficiency is independent of body size (invariant)

These results dating back to 1932 get captured in his beautiful book titled The Fire of Life.

Log-Log plot showing ¾ (three-quarter) power scaling over 27 orders of magnitude from unicellular forms (green) to mammals (blue) and even extending to cells and mitochondria (red). Source: Physics Today

Resurrecting Tusko

With the knowledge of non-linearity under our belt, we can revisit the question we posed earlier. Unfortunately, we can only resurrect Tusko in a scientific spirit. If researchers had paid attention to a crucial detail that dosage scales sub-linearly with an exponent ¾, the animal could have survived their reckless experiment.

Using human dosage, we get 0.004 mg per kg × 3000^¾ kg = 1.6 mg or 0.02 mg per kg × 3000^¾ kg = 8 mg. Or if we are feeling adventurous, we take the cat study and calculate 0.1 mg per kg × 3000^¾ kg = 40 mg. The catastrophic dose was two orders in magnitude higher than the safe limit.

So What?

It’s hard to fathom life emerged radically non-linearly— by natural selection, energy conservation, optimizing the efficiency of its utilization that humans can only dream of, and non-linear scaling laws. Fractal networks that supply nutrients to invariant terminal units have optimized their delivery mechanism by filling space in four dimensions.

The exponents are:

  • three-quarters (¾) for growth rate, brain sizes, gray matter, cross-sectional areas of aortas, and tree trunks!
  • a quarter (¼) for life spans, lengths of aortas and genome, and heights of trees.
Amazing Scale: A blue whale 100 ft in length near a Mr. Tanakit captured by his mate! Photo Source: NeverDry Expeditions, 2019

A blue whale is a non-linearly scaled version of a human. A giant sequoia shares more in common with a human than s/he can fathom. In deference to her brilliance, let’s strive to better shepherd mother nature. The emergence of mathematical biology with the discovery of non-linear universal scaling laws is exciting. This century is indeed the one of complexity required to understand the chemical scum on this moderate-sized planet. The magic equation of life is 4 = (3d + 1f)!

Thanks for reading.


  1. The Fourth Dimension of Life: Fractal Geometry and the Allometric Scaling of Organisms — Geoffrey West, James Brown, Brian J. Enquist, 1999
  2. Body Size And Metabolic Rate — Max Kleiber, Physiological Reviews, 1947
  3. Lysergic Acid Diethylamide: Its Effects on Male Asiatic Elephant — Louis Jolyon West, Chester M. Pierce, Warren D. Thomas, Science, 1962
  4. A simple practice guide for dose conversion between animals and human — Anroop B. Nair, Shery Jacob, Journal of basic and clinical pharmacy, 2016

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