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Scaling Laws

March 31, 2016

While the Ides of March, March 15th on the Roman calendar and notoriously the day of the assassination of Julius Caesar, might seem the most appropriate due date for income tax filing, the later date is actually near the Ides of April. The Ides of April, unlike that of March, is the 13th of the month, while tax day is usually on the 15th.

We can't equate our dates with the dates on the Roman calendar, since the Roman calendar had just ten months. Those months were not lengthened to fill out the year; rather, these had a mix of 30 and 31 days, like our own, with the period outside those being ignored. It would be nice if we, too, could ignore the winter days between December 31 and March 1. We decided, however, to create January and February to fill that span.

Months of the Roman Calendar
 MonthDaysIdes MonthDaysIdes
 Martius3115 Sextilis3013
 Aprilis3013 September3013
 Maius3115 October3115
 Iunius3013 November3013
 Quintilis3115 December3013

Although I spend some time collecting my tax information during January, I don't really get interested in the process until March; and, as March progresses, I start to get more concerned. Such procrastination is common to most people, and it was recently quantified by Tomasz Durakiewicz, a program director in the National Science Foundation Division of Materials Research, in a recent letter in Physics Today.[1]

Durakiewicz noticed that most proposals are submitted right before deadline. He analyzed a dataset of submissions for more than a thousand annual proposers, and he found that the data conform to a modified hyperbolic function, as shown below. This law that proposal submissions increase as an inverse function of the remaining time is a scaling law; that is, it's valid in this case no matter the length of the submission window.

Hyperbolic law of proposal submissions

Scaling law for proposal submissions.

Proposal submissions increase as an inverse function of the remaining time.

(Graphed using Gnumeric.)

One famous scaling law is the one about the size of animal bones that Galileo Galilei deduced in his Two New Sciences (1638). In this book dealing with mechanics and the mechanical properties of materials, Galileo realized that a body's volume increases at a much faster rate than its size, a principle known as the square-cube law.

What this means is that a King Kong could never be just an enlarged ape, since its bones (the load-carrying cross-section of which scaling as an area) couldn't carry its weight (which scales as the volume). An actual King Kong would be a big-boned, clumsy fellow, unlikely able to scale buildings.

Detail from a 1933 King Kong movie poster

Detail from a 1933 King Kong movie poster.

The 1933 film, King Kong, has many commendable features, including a score composed by Max Steiner, who wrote the score for the 1939 film, Gone with the Wind.

(RKO Radio Pictures/Roland Coudon poster, from "Doctor Macro," via Wikimedia Commons.)[2])

Scaling laws can give us useful information about physical phenomena. Consider, for example, how the size of the crater formed by an explosion can give us information about the energy released in the explosion. Since the volume of material ejected from a crater should be directly proportional to the explosive energy, it's easy to conclude that the diameter of a crater scales as the cube root of the energy.

It's possible to estimate the energy released in meteor impacts through use of such a scaling law and some calibrating experiments. The calibrating experiments are the craters left by nuclear explosions of known energy. The data for four terrestrial craters are shown in the following graph.

Meteor impact energy scaling with diamter

Energy scaling with the cube of crater diameter for Brent crater, Deep Bay crater, Boltysh crater, and Clearwater Lake

(Graphed using Gnumeric from data in ref. 3.)[3])

Kleiber's law states that an animal's metabolic rate scales as the 3/4-power of the animal's mass. One argument in support of this law is that the waste heat of metabolism needs to be removed at the interface between the animal and its environment; that is, at its surface area. This argument is similar to that for Galileo's scaling law for animal bones. It also appears that larger animals live longer and they travel further.

Adrian Bejan, a professor of mechanical engineering at Duke University thinks that the idea that larger things travel farther applies also to all objects and systems in motion, such as turbulent eddies in water and air, and rolling stones.[4-5] Bejan has been considering these issues for about a decade. One of his findings was that all animals should have roughly the same number of breaths per lifetime.[5] His new studies demonstrate that rolling stones and eddies have the same number of revolutions in their lifetimes.[5] Says Bejan,
"These three characteristics—life span, life travel and the constancy of the number of breaths or revolutions of bodies that move mass—unite the animal, the eddy and the rolling stone... Traditional camps believe that evolution is only biological and has already been explained to the hilt. I'm showing that evolution is actually based in physics and that it is simply design change over time. To the origin of life in non-living matter, abiogenesis, rolling stones and turbulence add the physics of evolution."[5]

Bejan's unconventional idea is that evolution is a concept that transcends whether a system is living or not. Anything that has a continuous change in a discernible direction over time can be described by the simple physical law of flow that any flowing system will trend toward an architecture that allows for an easier flow. For rivers, and for vascular systems, the resultant architecture involves a few large channels feeding into smaller branches.[5]

Figure caption

Adrian Bejan.

Bejan is co-author with J. Peder Zane of the book, Design in Nature: How the Constructal Law Governs Evolution in Biology, Physics, Technology, and Social Organization (Doubleday, January 24, 2012).

(Duke University photograph.)

Flow processes have been moving objects across the surface of the Earth for billions of years. Rolling stones, by becoming rounder with time, are evolving to have less friction so that they can travel further.[5] Some simple physics demonstrates that the time spent moving and the distance traveled by a rolling stone will increase with its mass. Likewise, larger turbulent eddies will have a longer lifetime and larger traveling distance.[5]

For massive bodies, these effects are small. For rolling bodies, the lifetime t and travel L increase only as the body mass M raised to the 1/6 and 1/3 powers, respectively;[4] that is,
t ∝ M1/6
L ∝ M1/3
For turbulent eddies, the lifetime evolves as the eddy mass M raised to the 2/3 power, while the travel increases as M2/3 times the bulk speed of the turbulent stream carrying the eddy.[4]


  1. Tomasz Durakiewicz, "A universal law of procrastination," Physics Today, vol. 69, no. 2 (February, 2016), p. 11.
  2. King Kong, Merian C. Cooper and Ernest B. Schoedsack, Directors, 1933, on the Internet Movie Database.
  3. Impact Processes: Meteor Crater, Arizona, Keyah Math Project, Arizona State University.
  4. Adrian Bejan, "Rolling stones and turbulent eddies: why the bigger live longer and travel farther," Scientific Reports, no. 6, Article no. 21445 (February 17, 2016), doi:10.1038/srep21445. This is an open access article with a PDF file available here.
  5. Ken Kingery, "Rolling Stones, Turbulence Connect Evolution to Physics," Duke University Press Release, February 17, 2016.
  6. Constructal Theory Web Site.                    

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