From Galileo’s simple case to universal 4/3 scaling

Abstract

Generalizing Galileo's strength of materials scaling implies 4/3 scaling is a universal physical law, arising in brain weight / body weight scaling, metabolic scaling, Peto's paradox, black body radiation, the fractal envelope of Brownian motion, the expansion of space in cosmology and so on.

Full text

1
From Galileo’s simple case to universal 4/3 scaling2
3
4
Robert Shour5
6
23rd April, 20237
Toronto, Canada8
Abstract9
Generalizing Galileo’s strength of materials scaling implies that 4/310
scaling is a universal physical law, arising in brain weight / body weight11
scaling, metabolic scaling, Peto’s paradox, black body radiation, the frac-12
tal envelope of Brownian motion, the expansion of space in cosmology and13
so on.14
Keywords Allometry, scaling, universal laws15
1 4/3 scaling16
1.1 Hypothesis17
4/3 scaling is a universal law of physics.18
1.2 Problem solving methods19
Pattern seeking and inductive reasoning preceded a deductive approach. De-20
duction guides that apply include:21

The gist and kernel of mechanical ideas has in almost every case grown22
up in the investigation of very simple and special cases of mechanical23
processes (Mach, 1907, Preface to the First Edition).24

The fun in science lies not in discovering facts, but in discovering new25
ways of thinking of them (Bragg, 1959).26

. . . as if an independent life and a reason of its own dwelt in these mathe-27
matical formulas; as if they were wiser than we were, wiser even than their28
discoverer; as if they gave out more than had been put into them (Hertz,29
in (Boltzmann, 1893)).30
The simple and special case here, per Mach, is Galileo’s: weight Wscales31
differently than area Asupporting W. The new way of thinking of Galileo’s32
case, per Bragg, is dimensional. The change to a dimensional perspective gives33
more than put in, per Hertz; then extrapolate (Packer et al., 2018).34

Universal scaling laws 2
2 Galileo’s simple case35
2.1 Galileo’s kernel36
Galileo asks how increasing the size of a cylinder affects its ability to support its37
weight. For length L, volume V∝W∝L3, weight Wis borne by cross-sectional38
area A∝L2. With increased length, Wscales by 3/2 — disproportionately —39
relative to A.40
2.2 Galileo’s allometric example41
Galileo gives a biological example: a heavier similar animal must have thicker42
bones to avoid fracture if the composition and strength of animal bone is in-43
variant for all sizes of animal.44
Since 1936, study of disproportionate organism scaling ‘of a part at a dif-45
ferent rate from that of body as a whole or of a standard’ has been called46
allometry (Huxley and Teissier, 1936; Gayon, 2000). Allometry has since 193647
been generalized to include scaling of processes such as metabolism.48
3 Principles implicit in Galileo’s 3D:2D case49
3.1 Dimension more fundamental50
Dimension does not derive from scaling; scaling derives from dimensions. Per51
Bragg above, change the conceptual framework from scaling to dimension.52
3.2 Change of problem in changed conceptual reference53
frames54
In Galileo’s simple case, the problem is to determine how the sizes of Wand A,55
with dimensions 3 and 2, change relative to each other. Scaling is necessary to56
find the increased sizes of Wand A.57
In metabolic scaling mentioned below, the problem is to determine how58
subsystems with dimensions 4 and 3 change relative to each other.59
In the change to a dimensional conceptual reference frame, having regard60
for dimension being more fundamental than scaling, the problems are then:61
1. How does the increase in Wproduced by scaling occupy each of the dif-62
ferently dimensioned Vand A?63
2. Can we usefully generalize from dimensions 3 and 2 to dimensions 4 and64
3 if we answer the first problem?65
3. What are the systems that have dimensions 4 and 3?66

Universal scaling laws 3
3.3 Notation67
Let Drepresent a number of dimensions (Banavar et al., 2002). Subscripts68
denote the number of dimensions, as in D1≡1D. Refer to a particular instance69
as, for example, a D3≡3D.70
Let vscale volume V,aarea A,γlength L. Then v=a3/2,a=v2/3,v=γ3,71
a=γ2,γ=v1/3,γ=a1/2(based on West et al. (1997)).72
3.4 Generalizations73
Generalize74
1. Dimensional principles of a system which has parts that are a Dand a75
D+ 1.76
2. By adding flow to a D+ 1.77
3.5 Simplification78
Simplify by insulating the system from heat and absenting friction and viscosity.79
3.6 Principles80
1. D+1
Dis fractal on a size increase of D+ 1.81
2. For the case of the cylinder, the ratio of dimensions of Wand Ais82
logL(L3)
logL(L2)=3
2.83
3. The logarithm ratio connects to the logarithmic expression of entropy:84
D+ 1 has D+1
Dtimes the entropy of D.85
4. D+ 1 has dimension D+1
Dthat of D.86
5. On an increase in size, D3increases its capacity to supply or transmit87
weight by D3
D2relative to D2’s capacity to support or use it.88
6. If W/A — weight per unit area — is invariant, then Wscaling by D3
D2
89
relative to Arequires Ato increase in size by D3
D2scaling. Generalize:90
to maintain an invariance based on D+ 1 and D,Dincreases in size91
disproportionately when D+ 1 increases in size, to offset D+ 1’s larger92
dimension.93
7. Generalize: differential — allometric — scaling preserves a biological in-94
variance.95
8. No summing a geometric series of all hierarchical sub-parts of Wand Ais96
required to model increasing size. D3
D2alone explains Galileo’s differential97
scaling.98

Universal scaling laws 4
9. Generalize: denote D+1
Ddimensional pressure of D+ 1 onto a D.99
10. Generalize: the capacity of D+ 1 is D+1
Dgreater than D’s. If the energy100
per dimension is constant D+ 1 has more capacity to contain energy than101
D. Denote this the principle of dimensional capacity: a system’s capacity102
is linear to its dimension.103
11. Let a given length L∝energy Elie in D+ 1’s reference frame. D+ 1104
has D+ 1 degrees of freedom relative to L.Dhas Ddegrees of freedom105
relative to L×D+1
D. In D’s reference frame, the same Eresults in a length106
D+1
Dlonger than in D+ 1’s.107
12. Degrees of freedom and length invert: relative to D,D+1 has D+1
Dgreater108
dimension and length D
D+1 smaller; relative to D+1, Dhas a D
D+1 smaller109
dimension and length D+1
Dlonger.110
13. Generalizing, the abstract capacity of D+ 1 to contain energy Eis D+1
D
111
greater than D’s.112
14. The exponent of D— its dimensional capacity — can increase with size113
and with time.114
15. Homologous D+1 and Dare distinct reference frames for the same energy.115
Energy and length per dimension is D+1
Dgreater in Dthan in D+ 1.116
16. Shifting reference frames from D+ 1 to Dfor the same energy entails no117
loss of energy. Nothing can be more efficient than transmitting energy118
from a D+ 1 reference frame to a Dreference frame.119
D+ 1 and Dexisting contemporaneously implies two distinct contempora-120
neous reference frames, and that the same energy produces a different length in121
the context of each reference frame. Weird (my view) but unexpectedly math-122
ematically sound.123
3.7 Dimensional capacity124
The concepts of dimensional capacity and dimensional pressure arise from a125
dimensional point of view of Galileo’s scaling.126
4 Generalization 1127
4.1 Add flow128
To a D3add flow of time or, per unit time, heat, energy, breaths, heart beats,129
information, signals, distance.130

Universal scaling laws 5
4.2 Biological examples: breathing and metabolism131
Consider an animal as a D3. Assume that its rate of breathing is proportional132
to its metabolism. Then the same scaling applies to rates of breathing and133
metabolism.134
4.3 Sarrus and Rameaux, 1838, breathing rate in 3D135
Bigger animals breathe more slowly than smaller animals. To explain why,136
Sarrus and Rameaux use scaling analogous to Galileo’s, though they do not137
mention him (Sarrus and Rameaux, 1838). They assume that to maintain a138
constant body temperature, animal skin surface D2must entirely dissipate heat139
produced by corresponding D3. The capacity to generate heat scales by 3
2
140
relative to A, which would overheat the animal unless the rate of breathing141
scales by 2
3relative to mass M. Their analysis uses a D3and a D2, and heat142
moving from body mass or volume to skin surface.143
Since 2/3 scaling is based on skin surface area, it is called a surface law.144
5 Metabolic scaling in 3D145
5.1 Surface law146
The surface law implies that metabolism Y∝M2/3.147
Max Rubner’s available measurements implied that M’s exponent b= 2/3148
(Rubner, 1883, 1902).149
Max Kleiber concluded that b= 3/4, (Kleiber, 1932): Kleiber’s Law; his150
contemporaries, 0.73 (Brody et al., 1934).151
5.2 Surface law doubts152
Feathers, folds, and stretching render skin surface area immeasurable and the153
surface law incapable of verification (Brody, 1945). Some still support 2/3154
(White and Seymour, 2003); others doubt 3/4 (Agutter and Wheatley, 2004),155
(Hulbert, 2014). Modern scholarship favors b= 3/4 (Savage et al., 2004;156
Ahluwalia, 2017).157
6 Generalization 2: add a dimension158
6.1 Fourth dimension perspectives159
Without a D1to represent energy flow, the D3
D2generalization for metabolic160
scaling is deficient. If in Kleiber’s Law Dis 3, what system is D+ 1?161
Is D+1 = 4 time (Blum, 1977)? Or time required for reproduction (Ginzburg162
and Damuth, 2008)?163

Universal scaling laws 6
Let C4denote animal circulatory system including D1blood flow propor-164
tional to energy flow.165
6.2 Flow as an additional dimension166
D1flow is orthogonal to a D3volume; a rate of flow cannot be the linear167
composition of D3’s spatial basis vectors. Spatial D3+ flow D1=D4.168
6.3 Generalizing from 3 to 4169
Generalize principles applicable to D3volumes of Galileo and of Sarrus and170
Rameaux to D4.171
1. Assume that intracellular chemistry of mammals are invariant regardless172
of size. Then energy use per unit cellular volume is invariant. Which173
implies mammals have about the same body temperatures.174
2. C4has 4/3 the entropy of empty circulatory system volume D3.175
3. D4has 4
3the dimension of D3.176
4. Generalizing from D3above: On uniform scaling of D4relative to D3, a177
distance in D3is 4
3of what it is D4. Absent constraints such as invariant178
intracellular energy use, a D3distance is 4/3 of the corresponding distance179
in D4.180
5. D4relative to D3creates 4/3 dimensional pressure on D3.181
6. On an increase in size, C4increases its capacity to supply energy by 4/3182
relative to D3’s capacity to use it.183
7. No summing of hierarchical tubes as in (West et al., 1997) is required.184
8. Shifting the same energy from a D4reference frame to a D3entails no loss185
of energy, more efficient than Carnot’s ideal heat engine.186
6.4 Degrees of freedom and distance187
Let a given length L∝Elie in D4’s reference frame. Then in D3’s reference188
frame, the same Eresults in a length 4
3longer. Lin D4gives 4 degrees of189
freedom relative to L. For the same E,L×4
3in D3gives 3 degrees of freedom190
relative to L×4
3.191
Degrees of freedom and length for D4are respectively 4/3 and 3/4 of corre-192
sponding values in D3, respectively 3/4 and 4/3 for D3compared to D4.193

Universal scaling laws 7
6.5 WBE 1997194
WBE 1997 (West et al., 1997), in addition to other shortcomings (Dawson, 1998;195
Kozlowski and Konarzewski, 2004, 2005; Banavar et al., 2010):196

Has the number of tubes from level to level of the circulatory system scale197
by n, indirectly scaling tube volume proportional to energy carried by198
blood. Tube radius rscales by β. WBE has β=n−(1/2), but it must be199
that β=n−(1/3) since ris a D1.200

Conflates a D1flow across a D2tube cross-section, leading to β=n−(1/2)
201
instead of β=n−(1/3). Blood flow D1is omitted. D1flow through D2is202
aD3, how conflation arises.203

Has γβ2in the denominator of their equation (5). γβ2=n−4/3, not204
express in WBE 1997. 4/3 is right but the mathematics is not. The 4/3205
of γβ2is a fortuitous clue. The extra 1/3 in 4/3 actually arises from206
the 4/3 dimension of D4compared to D3. Coincidentally, 1/3 is also the207
magnitude of n’s exponent for γ.208

Has an erroneous extra 1/6 in β=n−1/2’s exponent (1/2 instead of 1/3),209
which when squared gives 1/3 more than 1 in the 4/3 of their equation210
(5).211
7 Biological 4/3 scaling212
7.1 Biology213
In the case of metabolism, D4’s 4/3 higher scaled capacity is itself scaled by a214
3/4 power to maintain an invariance, the amount of energy required by a cell.215
7.2 Kleiber’s Law216
Invariant intracellular energy requirements results in metabolic rate decreasing217
disproportionately — scaling by a 3/4 power of mass M— when C4increases218
in size. The 3/4 in M3/4scales the 4/3 scaled increase in C4’s supply capacity219
relative to M’s D3. To analyze the effect, use subscripts bfor bigger and sfor220
smaller.221
n([C4]s)(4/3)ko3/4=([C4]b)k3/4=Mk
b
3/4,(1)
and the expression on the right implies Yb∝[C4]b∝Mb. Hence Kleiber’s law.222
The 3/4 scaling of Mindirectly reveals a scaling of C4’s 4/3 disproportion-223
ately increased scaled capacity.224

Universal scaling laws 8
7.3 Brain weight scaling225
A brain controls the body it is in. One might suppose that bigger brains scale226
linearly relative to their bigger bodies. But, Snell inferred empirically, brain227
weight scales as M3/4(Snell, 1892), now well supported (Eisenberg, 1981, p.228
275-283), (Armstrong, 1983; McNab and Eisenberg, 1989), (van Dongen, 2008,229
p. 2101). 4/3 scaling suggests why.230
D1signals transmitted though nervous system D3conduits are a D4, while231
the body controlled is a D3. The information capacity of the nervous system232
scales by 4/3, so brain weight need only scale by a 3/4 power of Mto control a233
larger body.234
7.4 Peto’s paradox235
A larger animal has more cells that can become cancerous and more routes for236
cancer cells to travel in. Cancer incidence in old age should be much greater in237
large animals than in small ones. But cancer is invariant in old age for differently238
sized species: Peto’s paradox (Peto, 1977). Unknown evolutionary mechanisms239
(Nunney, 2020) are unnecessary. Proliferation of cancer likely occurs in a D4;240
cancer has 4/3 greater scaled capacity to spread as animal size increases. The241
4/3 greater capacity of cancer proliferation is scaled by metabolism’s 3/4 scaling,242
leading to the observed invariance.243
8 Physics 4/3 scaling244
8.1 J. J. Waterston 1845245
Waterston observes (Waterston, 1892, p. 23) that on an increase in molecular246
energy (vis viva), between the two cases of constant pressure — a D4— and247
constant volume — a D3— the energy ratio is 4 : 3.248
8.2 Clausius and mean gas molecular path lengths, 1860249
Clausius compared the distance between gas molecules all moving — a D4—250
to the distance of all molecules still — a D3(Clausius, 1858, 1859). Using251
calculus, he showed that the mean path length of moving molecules is 3/4 that252
of stationary molecules, consistent with 4/3 scaling.253
8.3 Stefan and black body radiation, 1879254
Stefan concluded that the rate of radiant energy varies temperature by T4
255
in degrees Kelvin (Stefan, 1879), applicable to black body radiation. Planck256
(Planck, 1913, 1914; Allen and Maxwell, 1948; Longair, 2003) presents Boltz-257
mann’s derivation of Stefan’s Law(Boltzmann, 1884).258
Black body radiation in a sealed cavity corresponds to a D4. Planck has259
∂S
∂V T=4u
3T. In other words, black body radiation’s D4has 4/3 the entropy260

Universal scaling laws 9
of the cavity volume, consistent with 4/3 scaling. Radiation occurs at all scales261
and distances up to the cosmological. It follows that so does 4/3 scaling. Black262
body radiation is the exemplar of 4/3 scaling.263
8.4 Brownian motion envelope, 2001264
Particles buffeted by water molecules exhibit Brownian motion (Brown, 1828),265
aD4. The fractal dimension of Brownian motion is 4/3 (Lawler et al., 2001).266
Brownian ratchets (Peskin et al., 1993) may extend C4’s 4/3 energy distribution267
scaling to the intracellular level.268
8.5 Emergence269
Since the addition of radiating energy to D3causes D4to grow by 4/3 relative270
to D3, dimensional pressure may explain growth and complexity, and emergence271
of the universe, life, intelligence and language.272
9 Cosmological 4/3 scaling273
9.1 Cosmological distance274
Redshift distance, determined by light moving, is a measurement consistent with275
it occurring in D4.276
A luminosity distance of a celestial object is found by comparing the known277
distance and luminosity of the same kind of celestial object — a standard candle.278
Type 1A supernovas are, consistently, very bright standard candles (Kowal,279
1968). Luminosity distances are consistent with D3. The inverse square law280
does not apply to distances in D3; a radiation cone does not exist in a D3.281
It would follow that distances in D3space are 4/3 of those in D4.282
Gold’s hypothesis that light radiation in D4causes space to grow (Gold,283
1962) is consistent with 4/3 scaling.284
9.2 Astronomy285
D3luminosity and D4redshift distances should not be equal. A D3luminosity286
distance (4/3)Lshould correspond to D4redshift distance L. Luminosities287
should be 25% less than they would be if the D3distance were equal to the D4
288
redshift distance, as has been observed (Cheng, 2010, p. 259).289
Energy density of D4compared to D3should be ( 1
4/3)3=0.7033
0.2967 . Measure-290
ments aligns: for example, energy density 0.295 (Betoule et al., 2014) is <0.6%291
different than 0.2967.292
For energy densities distant from Earth, cosmology holds that for matter m,293
ϵm(a)= (ϵm)0/a3, and for radiation γ,ϵγ(a)= (ϵγ)0/a4(Ryden, 2003, p. 24).294
Thus distant energy density is 3/4 as much for matter compared to radiation.295
Treat matter as residing in D3and radiation in D4.296

Universal scaling laws 10
9.3 Allometry versus expanding space297
In C4, energy supply scales by 4
3; metabolism scales the supply capacity by 3
4
298
to maintain invariant intracellular metabolism.299
For cosmological space, unlike metabolism, D4’s dimensional pressure is not300
scaled by 3
4scaling. D3’s increase is unbounded due to 4/3 dimensional pressure301
relative to D3.302
9.4 Cosmic constant and inflation303
4/3 scaling implies that D3space has expanded relative to D4in a constant304
ratio, without invoking cosmological inflation. The cosmic constant then would305
not vary with time.306
10 Hidden connections307
10.1 Mysteries of the fourth dimension308
How can 1D flow be similar to a spatial dimension?309
The fourth dimension convincingly models special relativity (Speiser et al.,310
1911),(Minkowski, 1918).311
The nature of time (Barbour, 2009) in particular and of flow generally as a312
fourth dimension remains for now inscrutable. But if 4/3 scaling based on light313
radiation creates or increases D3, then all 4/3 scaling may be the shadow cast314
by the creation of the universe.315
10.2 The 3/4 head fake316
The 4/3 scaling in Stefan’s Law appears to be unconnected to 3/4 metabolic317
scaling. Black body radiation and metabolic scaling are assumed to be distinct318
unrelated phenomena. As a result, the generality of 3/4 metabolic scaling for319
physics has been unnoticed. But if 3 /4 metabolic scaling responds to 4/3 scaling320
of energy distribution capacity, it connects to black body radiation. Black body321
radiation and metabolic scaling models both scale the distribution of energy322
from a point.323
10.3 Dual reference frames324
The idea that D4and D3are distinct reference frames departs from the tradi-325
tional assumption that the universe has one reference frame. But dual reference326
frames are in evidence, in supply and demand not only in economics but also327
in biology (Banavar et al., 2002). And in Clausius’s mean path lengths and in328
black body radiation.329
Redshift distances and luminosity distances to the same supernovae are dif-330
ferent. That may reveal not just expanding space but also the universe’s dual331
reference frames (Arp, 2017).332

Universal scaling laws 11
10.4 Novelty333
Extraordinary claims attract extraordinary skepticism and scrutiny. The least334
oversight of a new hypothesis can sink it.335
4/3 scaling is a very ordinary generalization of Galileo’s strength of materials336
scaling that leads to ideas that overturn routine assumptions but also provide337
plausible explanations for diverse physical and biological phenomena. Let’s at338
least have a look.339
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