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Scaling  andScale Invariance

I have multiplied visions and used similitudes
by the ministry of the prophets
.
Hosea 12:10

Dimensionless Parameters  by  Eric W. Weisstein
Note on the history of the Reynolds number  (pdf, 828 kB)  by  N. Rott.
Looking out for number one: Benford's Law, by Walthoe, Hunt & Pearson

ScalingOn the sizes of Men, Fleas and Quarks

(2006-01-03)   Scaling according to Galileo
From ants to elephants, following the father of modern physics.

For a given material, the strength of a structure depends on its various cross-sections and is thus proportional to the square of the overall size.  As weight is proportional to the cube of size, such a structure would therefore collapse if scaled-up beyond a certain size.  This subject was first discussed by Galileo Galilei (1564-1642) who put the following words in the mouth of Salviati, on the "second day" of the  Dialogues Concerning Two New Sciences  (1638).

You can plainly see the impossibility of increasing the size of structures to vast dimensions [...]  If his height be increased inordinately, he would be crushed under his own weight.

We may reasonably expect the dynamic forces which make a creature jump to be proportional to the cross-section of its muscles, just like static forces are expected to be proportional to the cross-sections of its bones.  At the very least, this is a much better starting point than the popular misguided assumption discussed in the following article, which would have us believe that humans built like scaled-up fleas could jump over skycrapers!

(2006-01-03)   On the jump of a flea
Are fleas really much better jumpers than people?

Les puces peuvent sauter 135 fois leur taille.  C'est comme si
un homme sautait aussi haut qu'un immeuble de 65 étages.

CaramBar Info  [inside candy wrapping]

French kiddy sensationalism notwithstanding, the performance of creatures having vastly different sizes should not be compared using the linear scaling implied by the above comparison between men and fleas...

Taking the above at face value, a jumping creature would be expected to release an energy roughly proportional to its volume  (limbs apply a force proportional to the square of the size, along a launching trajectory proportional to the size).  This mechanical energy is thus expected to be proportional to the creature's volume or its mass [since all living tissues have roughly the same density].  Neglecting air resistance, this would mean that all jumping creatures are expected to jump to about the same height,  not  very different heights proportional to their sizes...

Fair comparisons of the jumping performances of various animals are best based on the ratio of the aforementioned mechanical energy to the mass of the creature  (this ratio is equal to half the square of the speed reached at liftoff).

People can jump up only slightly more than a foot in height.  Gifted athletes can do significantly better, but an athlete who clears a bar several feet off the ground does so partly because his center of gravity is already about 3 feet high to begin with, and also because the center of gravity of his bent body may stay under the bar...

The human flea (pulex irritans) is commonly quoted as being able to jump about a foot in length, or a few inches in height.  This is commensurate with human performance, as predicted.  The size of the flea is essentially irrelevant...

(2012-12-07)   Kleiber's Law   (1932)
A  ¾  power law.

Max Kleiber  (1893-1976)  graduated from the ETH Zürich in 1920 as an agricultural chemist and went on to obtain a doctorate  (1924)  with a dissertation on the  Energy Concept in Nutrition.  He joined the  Animal Husbandry Department  of  UC Davis in 1929, where he constructed respiration chambers and researched the energy metabolism in animals of various sizes.

In 1932, Max Kleiber concluded experimentally that the metabolic rate of animals varies roughly as their mass raised to the power of  3/4.

Naively, one could have expected the exponent of that  power law  to be to  2/3  as would be the case is energy was simply produced proportionally to the mass  (or volume)  of a warm-blooded animal and lost at a rate proportional to the surface area of its outer skin.

The experimental law obtained by Kleiber implies mathematically that the circulatory and/or respiratory system of animals has a fractal structure.  This is, of course, consistent with anatomical observations.  A simple justification of Kleiber's original exponent  (k=3/4)  would be obtained if energy was produced by the bulk of a  d-dimensional  system  (d being 3  or less)  and lost through a fractal object of dimension  k.d  which has to be  2  or more.  That would give the inner surface of the lungs and/or the circulatory system below the skin an  effective  combined fractal dimension  2.25  (9/4)  which seems about right...

The body of a  14.4 g grey mouse is approximately  60 mm  long and  20 mm  wide  (with a 70 mm tail  of 2 mm  width).  A black rat  would be roughly 3 times as big and 20 times more massive.

A male adult African savanna elephant typically weighs  5500 kg, and is thus about  400000  times as massive as a common mouse  (or 75 times as big, take your pick).  A  30 m  blue whale  would weigh about 180000 kg  (33 times the mass of an elephant or 3 times its size).  In the video quoted in the footnotes below, Pr. Woolley  (Oxford University)  doesn't seem to have any clear quantitative idea of the sizes of the animals he uses as examples...

Kleiber's Law   |   Max Kleiber (1893-1976)
3/4 and Kleiber's Law  by  Thomas Woolley  (Numberphile series, filmed by Brady Haran).

(2007-08-03)   Drag coefficient & Reynolds number
On the resistive force exerted by a fluid on a sphere at constant velocity.

In 1883, Osborne Reynolds (1842-1912)  introduced a dimensionless parameter as he investigated the transition from laminar to turbulent flow for fluids in pipes.

That parameter  R  was first called "Reynolds number" by Arnold Sommerfeld as he used it in what's now known as the  Orr-Sommerfeld equation  which he introduced in the paper entitled "Ein Beitrag zur hydrodynamischen Erklärung der turbulenten Flüssigkeitsbewegung" presented in Rome in 1908, at the 4th International Congress of Mathematicians  (3, 116-124).

The uniform motion of a sphere through a fluid involves the following quantities:

• The mass of the sphere:  m.
• The radius of the sphere:  r.
• The dynamic viscosity of the fluid:  h.
• The density of the fluid:  r.
• The speed of the sphere relative to the fluid:  v.
• The resistive force:  F.

Those form four relevant quantities:  r  (in m),  v  (in m/s),  F/m  (acceleration, in m/s)  and  h/r  (kinematic viscosity, in m2/s).

As two units are involved, there must be two dimensionless parameters which are functions of each other.  One is the  drag coefficient  (C)  the other is the aforementioned  Reynolds number  (R).  Other such pairs of parameters would be acceptable, but this is the traditional choice which we do retain.

(2016-04-14)   Benford's Law :
The leading decimal digit is  n  with probability  log10 (1+1/n).

This probability distribution occurs for numbers whose logarithms have uniformly distributed  mantissae.  That's what happens when quantities are reported in random units.

In 1881, it was the American astronomer Simon Newcomb (1835-1909) who first stated this law, prompted by his own observation that the beginning of his log tables was used more often than the end.  This was rediscovered in 1938 by the American physicist Frank Benford (1883-1948) who was instrumental in popularizing the law now named after him.

Floating-point numeration   |   0.30103     Wikipedia :   Benford's law