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!
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...
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 if 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...
[画像: Come back later, we're still working on this one... ]
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).
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:
Those form four relevant quantities: r (in m), v (in m/s), F/m (acceleration, in m/s2 ) 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.
That the ten digits do not occur with equal frequency must be evident
to anyone
making much use of logarithmic tables, and noticing
how much faster the first ones wear out than the last ones.
Simon Newcomb (1881)
Mathematically, Benford's law is a property which may or may not apply to an infinite dataset (a random variable with infinitely many values). The qualifier Benford applies to datasets verifying said property, which is the case when one of the following four equivalent criteria is satisfied.
The first-digit law is often presented in the decimal case as a definition of Bendford's law. However, this restricted criterion is not equivalent to the other three. There are distributions which obey the first-digit law in a particular base but fail to do so in any other base.
We'll prove the fruitful equivalence of these criteria, using a few lemma:
Lemma : If q is irrational, then the fractional parts (np mod 1) of the sequence 0, q, 2q, 3q, 4q, ... is uniformly disributed in the interval [0,1[. (Therefore, the sequence of the powers an is Benford.)
Lemma : If the random variable Y is uniformly distributed in the interval [0,1[ then so is the random variable (q+Y) mod 1 when q is constant.
Come back later, we're still working on this one...
A sequence of q radix-b digits d1 ... dq may be uniquely identified by the integer formed by their concatenation using b as the base of numeration.
N = d1 bq-1 + d2 bq-2 + ... + dq-1 b1 + dq b0
N must be greater than or equal to bq-1 for this to represent a legitimate sequence of leading digits (d1 must be nonzero). The q radix-b digits so specified are the leading digits with the following probability:
P = logb ( 1 + 1/N )
The probability that a given digit d occurs at position k ≥ 2 is the sum:
For all practical purposes, that's just 10% when k is 3 or 4 or more:
Canadian-born American astronomer. Simon Newcomb (1835-1909) first stated what's now called Benford's law in 1881, after noticing that the first pages of tables of logarithms show more wear than do the last pages. log tables wore out faster than the end. The law is named after the American physicist Frank Benford (1883-1948) who popularized it as the law of anomalous numbers in 1938, while working at the GE Research Laboratories in Schenectady, NY. and .
Floating-point numeration
|
0.30103
Wikipedia :
Benford's law
"Note on the Frequency of Use of the Different Digits in Natural Numbers" by
Simon Newcomb (1835-1909).
American Journal of Mathematics, 4 pp. 39-40 (1881).
"Using digital frequencies to detect fraud". Mark J. Nigrini.
The white paper [which became "Fraud Magazine" in 2004] 8, 2, pp. 3-6 (April/May 1994).
A
Statistical Derivation of the Significant-Digit Law by
Theodore P. Hill (1943-)
Statistical Science 10(4): 354-363 (November, 1995).
DOI: 10.1214/ss/1177009869
Benford's Law
by Adrien Jamain
(ICL &
ENSIMAG,
September 2001).
Why does Benford's Law hold?
(Mathematics Stack Exchange, 2010年07月27日)
A basic theory of Benford's Law
by Arno Berger (PhD 1997) & Teodore P. Hill
Probability Surveys, vol 8, pp. 1-126 (2011).
Number 1 and Benford's Law (9:13) by
Steve Mould (Numberphile, 2013年01月20日).
Hank Warren's proof for Benford's Law
(Mathematics Stack Exchange, 2014年05月13日)
|
Hacker's Delight
Benford's Law (22:44,
26:52,
23"05) by Mark J. Nigrini (2018).
The Mathematics of Benford's Law
by Arno Berger & Ted Hill (Arxiv, 2020年04月22日).
