(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 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...
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...
(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:
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.
(2016-04-14) Benford's Law (Newcomb 1881, Benford 1938)
The leading decimal digit is d with probability log10 (1+1/d).
Equivalently, the first digit is less than d with probability log10 (d).
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.
("Uniform mantissae") :
The mantissae
(fractional parts of the logarithms) are uniformly distributed in the
interval [0,1[.
("Scale Invariance") :
If a positive constant u and the base b aren't powers of the same integer,
then the leading digits of X and u X form two
identically distributed random variables.
("Leading-digits law") :
The probability that the leading radix-b digits
are the radix-b digits of the integer n is equal to logb (1+1/n).
("First-digit law") :
In any base of numeration b ≥ 3
the most significant digit is d (1≤d≤b-1)
with probability logb (1+1/d).
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.
Uniform mantissae (1) implies (2), (3) and (4) criteria:
Equivalence of the single and multiple-digits criteria :
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 )
Joint probability (%) for the two leading digits, according to Benford's law :
%
0
1
2
3
4
5
6
7
8
9
All
1
4.139
3.779
3.476
3.218
2.996
2.803
2.633
2.482
2.348
2.228
30.103
2
2.119
2.020
1.931
1.848
1.773
1.703
1.639
1.579
1.524
1.472
17.609
3
1.424
1.379
1.336
1.296
1.259
1.223
1.190
1.158
1.128
1.100
12.494
4
1.072
1.047
1.022
0.998
0.976
0.955
0.934
0.914
0.895
0.877
9.691
5
0.860
0.843
0.827
0.812
0.797
0.783
0.769
0.755
0.742
0.730
7.918
6
0.718
0.706
0.695
0.684
0.673
0.663
0.653
0.643
0.634
0.625
6.695
7
0.616
0.607
0.599
0.591
0.583
0.575
0.568
0.560
0.553
0.546
5.799
8
0.540
0.533
0.526
0.520
0.514
0.508
0.502
0.496
0.491
0.485
5.115
9
0.480
0.475
0.470
0.464
0.460
0.455
0.450
0.445
0.441
0.436
4.576
All
11.968
11.389
10.882
10.433
10.031
9.668
9.337
9.035
8.757
8.500
100%
The probability that a given digit d occurs at position k ≥ 2 is the sum:
bk-1-1
Pb (d) =
å
logb ( 1 + 1/(bp+d) )
p = bk-2
For all practical purposes, that's just 10% when k is 3 or 4 or more:
Probability (%) of a digit occurring in kth position, according to Benford's law :
%
0
1
2
3
4
5
6
7
8
9
1st
0
30.103
17.609
12.494
9.691
7.918
6.695
5.799
5.115
4.576
2nd
11.968
11.389
10.882
10.433
10.031
9.668
9.337
9.035
8.757
8.500
3rd
10.178
10.138
10.097
10.057
10.018
9.979
9.940
9.902
9.864
9.827
4th
10.018
10.014
10.010
10.006
10.002
9.998
9.994
9.990
9.986
9.982
5th
10.002
10.001
10.001
10.001
10.000
10.000
9.999
9.999
9.999
9.998
...
10.000
10.000
10.000
10.000
10.000
10.000
10.000
10.000
10.000
10.000
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 .
(2006-01-03) On the jump of a flea
