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A "power law", or "scaling law", is an equation of the form \[y=kx^n,\] where \(k\) is called a constant of proportionality and \(n\) is called the power, or "elasticity". Elasticity is frequently used in economics and ecology to describe the relationships between two variables, and can be calculated using the formula \[n = \frac{x}{y} \frac{dy}{dx}.\] The elasticity can be intuitively interpretation of the percent change in \(y\) per percent change in \(x\).
While a power law \(y= k x^n\) is not a linear equation, it can be transformed to a linear equation by taking logarithms of both sides, yielding \[\log y = \log k + n \log x.\] If the relationship between \(x\) and \(y\) is close to a power law, then the proportionality constant and the elasticity can be estimated using linear least squares.
Applications of power laws were probably repeatedly and independently discovered throughout history. The general scientific importance of scaling relationships was finally recognized and discussed by Galileo Galilei in Dialogues Concerning Two New Sciences (1638). Galileo proposed the "squarecube" law to describe how bones sizes scale as animals get bigger. The naive theory is that if you double the mass of a person or animal, and you double mass of the skeleton correspondingly, everything will work out just fine (As suggested in Gulliver's travels, for instance.) But Galileo argued that it isn't so  the skeleton's mass would have to increase more quickly to support this much weight.
One argument goes like this. Assume standard geometric scaling of lengths, areas, volumes, and mass holds: Mass is proportional to volume \(M \propto V\)), volume is proportional to the cube of the length scale (\(V \propto L^3\)), and area is proportional to the square of the length scale (\(A \propto L^2\)), for all systems. The strength of a bone in a body will be proportional to the crosssectional area (\(A\)) of the bone. Since the bones have to support the full mass of the animal (\(M_{body}\)), \(A \propto M_{body}\). Now, the crosssectional area of bone is proportional to the square of diameter, and the mass is proportional to the cube of the diameter, so the crosssectional area must be proportional to the to the skeletal mass \(M_{skel}\) raised to the 2/3rds power (\(A \propto M_{skel}^{2/3}\)). It now follows that \(M_{skel} \propto M_{body}^{3/2}\).
While Galileo's general argument against isometric skeleton mass scaling with body mass seems sound, it is only a hypothesis. We can test Galileo's hypothesis by comparing it's predictions to data. One such data set for a few common mammals was published by Kayser and Heusner in 1964. We can fit a powerlaw to this data using the logarithm transform described above.
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# Data extracted from Figure 7 of
# "Scaling in Biology: The consequences of size"
# by Knut SchmidtNielsen
# Journal of Experimental Zoology, 194: 287308, 1975.
# https://dx.doi.org/10.1002/jez.1401940120
#
# Data originally published by Kayser and Heusner, 1964
#
# Data rows are sequentially, Mouse, Rat, Rabbit, Cat, Dog, Man, Elephant
# Columns ...
# Body mass in kilograms, Skeleton mass in kilograms
0.0202, 0.00104
0.198, 0.0118
2.13, 0.176
3.18, 0.302
20.0, 3.76
73.2, 9.23
7721., 1558.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 

We find that the best fit power law does not have an elasticity of 3/2's as Galileo hypothesized, but instead has an elasticity of 1.12, a value very close to the linear scaling Galileo argued against. Other studies ( Prange, Anderson, and Rahn, 1979, Dumont, 2010 ) support this result. They consistently find elasticities that are slightly larger than 1, but not as large as the 3/2's Galileo predicted.
The study of how size effects shape, anatomy and physiology of life is called "allometry". Allometry has revealed many unexpected relationships in biology which remain difficult to explain. One example  bird egg size  is given below, along with others in the Exercises, but there are many more that have been discovered. While the data seem to clearly indicate the existence of allometries, they continue to inspire conversation and debate as to their causes.
The egg sizes of birds provides a simple and less controversial example allometry than Galileo's. In 1945, Brody published some observations on the relationship between adult bird mass and egg size. The data are plotted below, along with a powerlaw model fit using least squares. As you would expect, the egg size increases with the mass of it's parent. This increase is well described by a power law with elasticity \(0.77 \approx 7/9\). The data and code for creating the plot are given below.
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# mass of adult, mass of egg
# Brody, 1945, partially aggregated
# http://archive.org/download/problemsofrelati033234mbp/problemsofrelati033234mbp.pdf
113380,1700
4536,165.4
3629,94.5
1020,34
283,14
3.6,0.6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 

Power laws appear in all kinds of data sets. Perhaps one of the most surprising is how well a power law explains the sizes of wars in human history. In a 2003 article, Cederman measured the number of deaths in wars across the globe between 1820 and 1997, and used these to estimate the probable size of a war. It seems like the size of wars should be something that is wholely unpredictable without allot of extra information about the historical environment and the parties involved. Never the less, the surprising result was that the probability distribution of war sizes was welldescribed by a powerlaw distribution.
Another family of models that can be fit using linear least squares are exponential models of the form \[y = e^{f(x)}.\] Taking the logarithm of both sides, \[\log y = f(x).\] If \(f(x)\) can be written as a weighted sum of terms depending on \(x\), then linear least squares can estimate these weights.
A classic example of this is population growth. Naturalists have long observed that when a small population becomes established in a resourcerich environment, the number of individuals \(N\) often grows exponentially according to \[N = N _ 0 e^{r t}\] where \(N _ 0\) is the initial population size, \(r\) is the growth rate, and \(t\) is the time elapsed since initiation. Logtransformed, \[\ln N = \ln N _ 0 + {r t}.\] For example, from 1700 to 1850, the population of the United States
grew at a steady exponential rate according to census estimates.
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# year, number of people
#
# estimated sizes of the population of the United States
# of America, 1610  2010.
2010,308745538
2000,281421906
1990,248709873
1980,226542199
1970,203302031
1960,179323175
1950,151325798
1940,132164569
1930,123202624
1920,106021537
1910,92228496
1900,76212168
1890,62979766
1880,50189209
1870,38558371
1860,31443321
1850,23191876
1840,17063353
1830,12860702
1820,9638453
1810,7239881
1800,5308483
1790,3929214
1780,2780369
1770,2148076
1760,1593625
1750,1170760
1740,905563
1730,629445
1720,466185
1710,331711
1700,250888
1690,210372
1680,151507
1670,111935
1660,75058
1650,50368
1640,26634
1630,4646
1620,2302
1610,350
Based on linear least squares, the population grew according to \[\ln N \approx 38.859 + 0.030184 t, \; \text{and} \; N \approx 2^{(t1287)/22.96}.\] So, the data indicate the population has been doubling every 23 years.
Another exponential law that least squares can fit is Arrhenius's law. Arrhenius's law is a model that proposes the rate or frequency \(k\) of an event depends on the absolute temperature \(T\) according to the equation \[k(T) = e^{ac/T}\] where \(a\) and \(c\) are empirically determined constants. The constants of Arrhenius's law can be estimated using linear least squares, by taking the logarithm of both sides of the equation, \[\log k = a  \frac{c}{T}.\] Thus, any data set obeying Arrhenius's law should appear as a straight line when the logarithm of the rate is plotted against the reciprocal of the temperature  we call this an "Arrhenius plots". One particular catch with Arrhenius's law is that it relates rate to absolute temperature. Thus, relative temperature scales like Celsius and Fahrenheit which allow for negative temperatures will not work. Instead, we have to measure temperature using a scale like degrees Kelvin, where \(0\) is the absolute minimum temperature.
Svante Arrhenius (18591927) originally proposed this formula for applications in chemistry, but it appears to apply to some biological phenomena as well. In 1915, Charles and Aleida Snyder observed and recorded the flashing rates of male fireflies in Maryland. Their data relating temperature to flashing rate is shown below.
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#Temperature (C), flashing rate (per minute)
#
# Snyder and Snyder, 1920 data on
# @article{charles1920flashing,
# title={THE FLASHING INTERVAL OF FIREFLIES—ITS TEMPERATURE COEFFICIENT—AN EXPLANATION OF SYNCHRONOUS FLASHING},
# author={Charles, D and Snyder, Aleida v't H},
# journal={American Journal of PhysiologyLegacy Content},
# volume={51},
# number={3},
# pages={536542},
# year={1920},
# publisher={Am Physiological Soc}
# }
#
19.4,8.1
22.3,9.9
22.6,10.0
23.2,11.1
24.1,11.5
26.0,12.6
26.5,12.1
28.3,15.0
28.8,15.4
Just from looking at the data, we can see that the rate of flashing increases as the temperature increases, but perhaps not linearly (though see the exercise below). It has been proposed in Laidler (Journal of Chemical Education, 1972) that the relationship between temperature and flashing rate actually obeys the Arrhenius Law.
Using linear leastsquares and an Arrhenius plot, we see that the Snyder data are indeed wellexplained by the Arrhenius law \[k = e^{9.5  2510/T}.\]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 

As a final preliminary application of the leastsquares method, let us consider the chambered nautilus. The chambered nautilus is one of the most charismatic "living fossils" on the planet. At first glance, they appear to be some unusual combination of snail, fish, and jellyfish. But the chambered nautili are actually cephelopods, like squid, cuttle fish, and octopi. Nautili have been around since the Cambrian explosion, 500 million years ago and look much the same today as they did then. The famous documentary series "The Undersea World of Jacque Cousteau" memorably filmed this undersea creature, and in 2015 the "fuzzy nautilus" Allonautilus scrobiculatus was finally also filmed.
One of the many mysterious things about a chambered nautilus is the shape of its shell. A nautilus shell coils around itself, and when cut in half, reveals a spiral with the regular partitions that gave rise to the moniker "chambered". (Unfortunately, the nautilus is again threatened with extinction, in part because of our human attraction to this shell). The spiral itself is so attractive that one almost can not help but ask if it might be revealing some deeper law of nature. One way of getting at this question is asking what equation, if any, describes this spiral shape. And fortunately, the math of polar coordinates is just the thing for drawing spirals.
A spiral is a curve that winds a path around a center in a regular manner. One of the oldest definitions of a spiral comes from Sicily.
"If a straight line one extremity of which remains fixed be made to revolve at a uniform rate in a plane until it returns to the position from which it started and if, at the same time as the straight line is revolving, a point move at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane."  Archimedes, On spirals, circa 250 BC
This is what we now call an "Archimedean" or linear spiral. If we let the symbols \(t\) represent time, \(\theta\) be the angle of the line, and \(\rho\) be the distance of a point out from the center, then Archimedes' definition is equivalent to the parametric polar system \[\theta(t) = a t, \quad \rho(t) = b t \] where \(a\) and \(b\) are the velocities of change of the angle and the radius respectively. By solving the first equation for \(t\) and then substituting into the second, we find that this means the Archimedean spiral has its radius proportional to its angle, specifically \[\rho = \frac{b}{a} \theta .\] Archimedes wrote a treatsie on this spiral and its uses.
A little more thought about polar equations reveals that spirals come in many forms. A spiral might grow quadratically (\(\rho = b \theta^2\)), exponentially (\(\rho = e^{b \theta}\), sometimes called "logarithmic spiral" when written \(\ln \rho = b \theta\).), or according to some other crazy function (e.g. \(\rho = 2 \theta + \sin(6 \theta)\)).
Are any of these spirals is a good description of the spiral we see in a nautilus shell? One crude approach to this is to measure the width of each sequential revolution along a common ray from the center of the nautilus, and look for a pattern in how the width changes. For an Archimedean spiral, each revolution should have the same width. When we measure, what we actually find is that the radius between the first to the second spiral is twice the radius of the first spiral, and the radius between the second and third spiral is about 4 times the radius between the first and the second. So the shape of a chambered nautilus shell a closer match to a logarithmic spiral than Archimede's spiral. More systematic curve fitting supports this.
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#xcoord,ycoord
0.528897,0.0648674
0.525538,0.0469049
0.508178,0.206924
0.492023,0.140459
0.47521,0.294637
0.422145,0.270158
0.421131,0.394498
0.334304,0.345421
0.327787,0.497499
0.258484,0.558131
0.237464,0.399509
0.187465,0.0507932
0.187291,0.00660813
0.18009,0.609728
0.160457,0.111298
0.153812,0.088078
0.143745,0.420374
0.107665,0.637137
0.100245,0.180966
0.0660265,0.145215
0.0634161,0.00925503
0.045472,0.0512221
0.0322192,0.664555
0.0198792,0.42
0.0179807,0.0483927
0.0157719,0.220496
0.00559501,0.148053
0.00582319,0.0725251
0.0372118,0.682892
0.0541885,0.0636078
0.0607875,0.120662
0.062759,0.226776
0.0646392,0.395575
0.0905606,0.0244428
0.0938829,0.0752451
0.105794,0.0178073
0.112722,0.689162
0.150444,0.202871
0.179186,0.689363
0.182253,0.325733
0.210976,0.166801
0.233439,0.268177
0.260766,0.686588
0.265539,0.106542
0.275561,0.210648
0.296481,0.135056
0.302085,0.00997609
0.308337,0.0594914
0.336339,0.671711
0.40588,0.653794
0.460324,0.632811
0.520828,0.605803
0.57227,0.578768
0.62978,0.542688
0.672195,0.503541
0.720651,0.464413
0.793432,0.373998
0.851107,0.283538
0.905807,0.177963
0.930213,0.099487
0.951689,0.0092094
0.9611,0.123985
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 

Both the quadratic and logarithmic spirals fit the Nautilus shell's shape reasonably well, but the logarithmic spiral's fit is about twice as good in terms of square error, and visually seems to be a more satisfying fit. At small angles, the quadratic spiral folds over itself  an effect that is inconsistent with physiological and physical intuition of how the nautilus shell works. If we dismiss the quadradit spiral, then the best fit logarithmic spiral has approximate formula \[\rho = e^{0.172 \theta 3.34}.\]
We have shown with these examples that some very simple coordinategeometry curves  namely power laws and exponentials  can sometimes be very good models. This is fortunate, as science would be much more difficult if such simple formulas were never available. On the other hand, we can not help but ask ourselves why these simple curves explain the data. In some cases, such as population growth, there is a clear mechanism to justify the model (constant percapita reproduction and death rates). But in others, like the distribution of wars, mechanism seems to escape the imagination. It is quite reasonable to wonder if the ubiquity of these models is more a symptom of our inherent mental bias toward finding patterns in data, even when there are none (like animals in the clouds). But atleast in some contexts, there are good reasons to expect some data sets to behave so nicely. In the next 3 lectures, we'll explore some of ideas people have proposed as mechanisms for powerlaw models in particular.
In the examples above, the method of linear least squares can be used to help us study problems where the phenomena of interest is not actually linear. A word of caution should be raised though, to point out that we are abusing the linear least squares method. Improved statistical models for each of the problems above would account for nonlinear effects in the distributions of results and errors. For example, standard linear least squares implicitly assumes all errors are drawn from the same distribution. For power laws, however, it is very common to see situations where the errors scale with size in ways that do not correspond with constant variation after logarithmic transformation. We are fortunate that, for many problems, such nonlinear effects do not fundamentally change the results, and that more complicated statistical models often give essentially the same results as those obtained by a very fast and simple linear least squares analysis.
The oldest known example of the application of a powerlaw is found in the history of military engineering. In the 4th century BC, the construction of seige engines and artillary was beginning to have a noticable impact on warfare. Ancient writtings tell of the importance played by artillary firing spears and stones in the seiges of Halicarnassus and Tyre by Alexander the Great and the later Roman seige of Syracuse. But seigeengine construction was a demanding engineering task, particularly because of the tools and materials available at the time. According to Philon of Byzantium's book on the design of siege engines in the cities of Alexandria and Rhodes in the 3rd century BC, engineers had determined that the dimensions of their catapults should not be in proportion to the weight of the stones being thrown, but in proportion to the cuberoot of the weight of the stones. This formula was so important, Philon even presented a sophisticated geometric technique for accurately calculating this cube root.
Use geometric modelling to derive Philon's rule.
Based on Philon's rule, how much should then lengths in the engine be increased if the projectile weight is doubled?
The argument against isometric skeletal scaling we present above is a clumsy one. For one thing, it assumes isometric scaling of a body implies isometric scaling of the skeleton also. Suppose instead, the crosssectional area of bones scales proportionally to body mass, while the length of the bones scales proportional to the cube root of body mass. What bodytoskeleton mass relationship would this imply?
Kepler's harmonic law of planetary motion (his 3rd law) states that the square of the orbital period is proportional to the cube of the characteristic distance of the planet from the sun. Test this by fitting a power law to the data below. Give your answer to 4 decimal places.
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#period (days), distance (AU)
#
# Data supporting Kepler's third law
#
# distance is measured relative to the earth's mean distance,
# which is about 9.26e7 miles and 1.4960e8 meters
# Planets, in order: Mercury,Venus,Earth,Mars,Jupiter,Saturn
# from Timoshenko and Young, page 92
#
87.96,0.3871
224.6,0.7233
356.2,1.000
686.9,1.523
4332,5.200
10759,9.540
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# Body mass (kg), Stride frequency (per minute), type of animal
# Trotgallop transition
# Data extracted from figure 12 page 304 of SchmidtNielsen, 1975
# Scaling in Biology: The Consequences of Size
# https://dx.doi.org/10.1002/jez.1401940120
#
0.0114147294105,527.712615249,mice
0.0119388305755,543.270061821,mice
0.0324501676971,456.358342072,mice
0.0405911949976,436.896483228,rat
0.398121423131,278.480604293,dog
1.14914465892,262.759459364,dog
9.57038373087,208.262936379,dog
28.0199093282,158.029501577,dog
30.6447987596,155.750346704,dog
694.018672869,119.912471239,horse
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# length (cm), period (seconds)
10.0,0.560
20.0,0.860
30.0,1.050
40.0,1.280
50.0,1.390
60.0,1.520
70.0,1.640
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# magnetomotive force, maximum magnetization, hysteretic loss (ergs per cm cubic)
#
# Ewing's observations of hysteretic energy loss in soft iron wire
# according to Steinmetz, 1890, "Note on the law of hysteresis"
#
# maximum magnetization measured in lines per square centimeter
# hysteretic loss measured in ergs per cm cubic
#
01.50,01974,00410
01.95,03830,01160
02.56,05950,02190
03.01,07180,02940
03.76,08790,03990
04.96,10590,05560
06.62,11480,06160
07.04,11960,06590
26.50,13700,08690
75.20,15560,10190
Table 2, Page 45 of J. Huxley’s 1932 book Problems of Relative Growth provides a table of red deer antler statistics relative to body mass. Use python to recalculated a scaling relationship between the deer mass and the antler mass from this data. In your answer, present the data in a new table, your parameterization for the scaling law, your fitted parameter values, and a plot comparing the data to the fitted scaling law.
Theropods were an order of dinosaurs that included Tyranosaurous Rex and where also the ancestors of birds. Below is a data set containing records of length and mass of a variety of species of theropods. Fit a powerlaw model to these data. Estimate the parameters to 3 decimal places.
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# "Dinosaur Species","Body Length (m)","Body Mass (kg)"
"Staurikosaurus pricei",2.08,19
"Coelophysis bauri",2.68,15.3
"Coelophysis rhodesiensis",2.15,13
"Elaphrosaurus bambergi",6.2,210
"Liliensternus liliensterni",5.15,127
"Dilophosaurus wetherilli",6.03,283
"Ceratosaurus nasicoris",5.69,524
"Eustreptospondylus oxoniensis",4.63,218
"Metriacanthosaurus? sp.",3.8,130
"Allosaurus fragilis",7.4,1010
"Allosaurus atrox",7.9,1320
"Gorgosaurus (Paul's Albertosaurus ) libratus",5.8,700
"Gorgosaurus (Paul's Albertosaurus ) libratus",8.6,2500
"Daspletosaurus (Paul's Tyrannosaurus ) torosus",9,2300
"Tarbosaurus (Paul's Tyrannosaurus ) bataar",5.8,760
"Tarbosaurus (Paul's Tyrannosaurus ) bataar",7.7,2100
"Tyrannosaurus rex",10.6,5700
"Deinonychus (Paul's Velociraptor ) antirrhopus",3.06,45
"Deinonychus (Paul's Velociraptor ) antirrhopus",3.43,73
"Velociraptor mongoliensis",2.07,15
"Ornithomimus edmontonicus",3.3,110
"Ornithomimus brevitertius (=edmontonicus)",3.66,144
"Gallimimus (Paul's Ornithomimus ) bullatus",2.15,27
"Gallimimus (Paul's Ornithomimus ) bullatus",6,440
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# # of Revolutions, weight W (oz.)
#
0.25, 02.875
0.50, 04.000
0.75, 05.706
1.00, 08.901
1.25, 12.437
1.50, 14.700
1.75, 19.062
2.00, 26.50
2.25, 33.75
2.50, 40.00
2.75, 52.00
3.00, 76.00
Looking at Snyder's original firefly data relating temperature to flashing rate, you might suspect that the data can be explained just as well or better with a straight line as with the Arrhenius law. Try this and make a case for which provides a better explanation of the data.
An old reference book on ballooning includes the data below to describe the amount of hydrogen gas a balloon would need to reach various altitudes. Find an equation summarizing this data. (Bonus: What was book where these data come from?) (This result will reappear in our study of ballistic motion)
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# Altitude (meters), Volume of hydrogen (cubic meters)
12900, 1.25
18400, 10
29500, 640
42300, 80000
49700, 1250000
As mentioned above, several groups of researchers have collected data on how skeleton mass actually scales with body mass. Let's try to replicate their results.
The data of Dumont, 2010 is here. Use the data to produce a corrected version of figure 1, giving the power laws for each fit.
Extract the data from Prange, Anderson, and Rahn, 1979 as a csv file, and reproduce figure 1 of the paper. Do you get the same fit?
Find an equation to describe the scaling of a lightbulb's brightness as a function of the wattage applied to the bulb, based on the historical data below.
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# Luminoscity (candle power), Power (Watts)
#
# Luminosity characteristic of 40watt tungsten incandescent lamp.
# Steinmetz, 1911, page 254
# https://archive.org/details/matheengineering00steirich
2,12.25
4,16.33
8,21.35
12,25.60
16,28.91
20,31.64
24,34.55
28,37.29
32,39.26
36,41.47
40,44.14
44,45.42
48,47.05
64,54.31
96,65.73
128,76.77
192,95.24
256,109.0
291,118.2
320,123.1
382,135.6
460,145.2