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1
Is biology’s quarter scaling universal in physics too?2
3
4
Robert Shour5
6
31st March, 20237
Toronto, Canada8
Abstract9
Numerous instances of allometric quarter scaling occur in plants and10
animals. This article makes the case for quarter scaling and related 4/311
scaling as universal physical laws.12
Keywords Allometry, scaling, universal laws13
Contents14
1 Universal biological scaling, generalized 315
1.1 Metabolicscaling........................... 316
2 West, Brown and Enquist 1997 (WBE 1997) 417
2.1 Varying WBE 1997’s derivative . . . . . . . . . . . . . . . . . . . 418
2.2 WBE 1997 derivation doubts . . . . . . . . . . . . . . . . . . . . 419
2.3 Problems raised by WBE 1997 . . . . . . . . . . . . . . . . . . . 420
3 Biological quarter-power scaling 521
3.1 Examples ............................... 522
3.2 Quarter-power scaling and dimension . . . . . . . . . . . . . . . . 523
3.3 Fourth dimension: time? distance? other? . . . . . . . . . . . . . 624
3.4 Supplyanddemand ......................... 625
4 Non-biological analogs, numbers, logarithms and entropy 626
4.1 Fractality ............................... 727
4.2 Seekinganalogs............................ 728
5 Stefan’s Law 729
5.1 EmpiricalLaw ............................ 730
5.2 Planck’s derivation version . . . . . . . . . . . . . . . . . . . . . . 731
5.3 Planck’s terminology . . . . . . . . . . . . . . . . . . . . . . . . . 832
5.4 Implications of Planck’s derivation . . . . . . . . . . . . . . . . . 833
5.5 Longairversion ............................ 934
5.6 Flowasadimension ......................... 935
5.7 Degrees of freedom, dimension and length . . . . . . . . . . . . . 936
5.8 Stefan’s Law and 4/3 scaling . . . . . . . . . . . . . . . . . . . . 937
5.9 Efficiency ............................... 938
Universal scaling laws 2
6 Dimensional capacity 1039
6.1 Galileo’sexample........................... 1040
6.2 Generalizing Galileo from 3 and 2 to 4 and 3 . . . . . . . . . . . 1041
6.3 The fourth dimension mystery . . . . . . . . . . . . . . . . . . . . 1042
7 Stefan’s Law derivation and metabolic scaling 1043
7.1 Analogy................................ 1044
7.2 4/3 supply scaling and metabolism . . . . . . . . . . . . . . . . . 1145
7.3 Metabolic scaling algebra . . . . . . . . . . . . . . . . . . . . . . 1146
7.4 3/4 as Nature’s metabolic scaling head fake . . . . . . . . . . . . 1247
8 Other 4/3 biological scaling 1248
8.1 Cancer constancy in old age for different animals . . . . . . . . . 1249
8.2 Brain weight scaling . . . . . . . . . . . . . . . . . . . . . . . . . 1250
9 Physical examples of 4/3 scaling 1251
9.1 Introduction to this section . . . . . . . . . . . . . . . . . . . . . 1252
9.2 Black body radiation 1879 . . . . . . . . . . . . . . . . . . . . . . 1253
9.3 J. J. Waterston 1845 . . . . . . . . . . . . . . . . . . . . . . . . . 1354
9.4 Clausius1860............................. 1355
9.5 Richardson on wind eddies, 1926 . . . . . . . . . . . . . . . . . . 1356
9.6 Brownian motion’s 4/3 fractal envelope . . . . . . . . . . . . . . 1457
9.7 Social network information transmission . . . . . . . . . . . . . . 1558
10 Cosmological predictions based on 4/3 scaling 1559
10.1 Introduction to this section . . . . . . . . . . . . . . . . . . . . . 1560
10.2 Cosmological expansion . . . . . . . . . . . . . . . . . . . . . . . 1561
11 Astronomical observations 1762
11.1 Cosmological distances . . . . . . . . . . . . . . . . . . . . . . . . 1763
11.2 Implied luminosities . . . . . . . . . . . . . . . . . . . . . . . . . 1764
11.3 Cosmological energy densities . . . . . . . . . . . . . . . . . . . . 1765
11.4 Energy density variation with distance from Earth . . . . . . . . 1766
11.5 Cosmological horizon problem . . . . . . . . . . . . . . . . . . . . 1867
11.6 Special relativity and kcalculus................... 1868
12 Discussion of problem section 1869
12.1 Introduction to this section . . . . . . . . . . . . . . . . . . . . . 1870
12.2 Suggested answers . . . . . . . . . . . . . . . . . . . . . . . . . . 1871
12.34/3occurrences............................ 1972
13 Universal quarter scaling and 4/3 law? 2073
13.1 Further questions? . . . . . . . . . . . . . . . . . . . . . . . . . . 2074
13.2Connections? ............................. 2075
Bibliography 2076
Universal scaling laws 3
1 Universal biological scaling, generalized77
Study of disproportionate scaling in organisms is called allometry (Huxley and78
Teissier, 1936; Gayon, 2000). Pervasive allometric quarter-power scaling (Savage79
et al., 2004) has been called a universal law of biology (Brown and West, 2004;80
Ahluwalia, 2017).81
If quarter-power scaling were a universal law of physics:82
The same scaling principles in biological and non-biological systems would83
enable analysis from different perspectives. Which increases chances of84
identifying fundamental principles.85
Analogies based on scaled biological systems would inform us about scaled86
non-biological physical systems and vice versa.87
Principles common to biological and non-biological systems may illumi-88
nate attributes of emergence.89
1.1 Metabolic scaling90
Galileo in 1638 used scaling to explain why larger animals have thicker bones91
(Galilei, 1638; Galileo, 1914).92
Later, Sarrus and Rameaux used scaling to hypothesize why a large mammal93
breathes slower than a smaller one (Sarrus and Rameaux, 1838). A mammal dis-94
sipates body heat through skin surface area A. Dissipated heat must equal heat95
produced by animal mass Mto maintain constant body temperature; otherwise96
the animal would overheat and die. Slowing breathing of a larger mammal by97
exponent 2/3 maintains a constant temperature. The 2/3 expresses a surface98
law: animal skin surface area in contact with ambient air A∝L2relative to99
length L, while animal volume V∝M∝L3. Sarrus and Rameaux may have100
been first to model a physiological process using dimensions.101
Breathing supplies oxygen for metabolism. The breathing rate should be102
proportional to metabolic rate. If the surface law applies, metabolic rate Y∝103
Mb, with b= 2/3.104
From data, Max Rubner concluded that b= 2/3 (Rubner, 1883, 1902).105
From more data, Max Kleiber doubted the surface law and concluded that106
b= 3/4, (Kleiber, 1932): Kleiber’s Law. Brody and Proctor also concluded that107
b= 0.73 (Brody et al., 1934).108
Despite surface law defects (Hardy and DuBois, 1937; Brody, 1945, for exam-109
ple), some still argue b= 2/3 (White and Seymour, 2003); others doubt b= 3/4110
(Agutter and Wheatley, 2004; Hulbert, 2014). Modern scholarship, based on111
more abundant data, favors b= 3/4 (Banavar et al., 2002; Savage et al., 2004;112
Banavar et al., 2010; Ahluwalia, 2017).113
Universal scaling laws 4
2 West, Brown and Enquist 1997 (WBE 1997)114
2.1 Varying WBE 1997’s derivative115
West, Brown and Enquist in 1997 (WBE 1997) derived b= 3/4 geometrically116
(West et al., 1997).117
In WBE 1997, nis scale factor of the number of circulatory system tubes118
per level. As tubes decrease in size away from the aorta, β=n−1/2scales tube119
radius, and γ=n−1/3scales tube length. Tube length is also the radius of the120
corresponding spherical volume receiving blood. A different approach finds that121
at the kth level of the circulatory system,122
(β2)k×γk= (n−1/2)2k×(n−1/3)k=n−(4/3)k,(1)
while spherical volume scales by n−k. Supply scales by 4/3 : 1 compared to123
cellular use as animal size increases. This approach has advantages over WBE124
1997.125
2.2 WBE 1997 derivation doubts126
Volume invariance of capillaries required by WBE 1997 has been doubted (Daw-127
son, 1998). WBE 1997 assumes that the number of capillaries Nc∝Mb, which128
is inconsistent with WBE 1997’s Nc∝M; both cannot be true (Kozlowski and129
Konarzewski, 2004, 2005). The assumption in WBE 1997 that capillary size is130
invariant but also proportional to the radius of a service volume may be incon-131
sistent (Banavar et al., 2010). WBE 1997 defines scale factor βof circulatory132
system tube radius and n−1as scale factor of the number of tubes per level.133
The number of tubes is an indirect measure of volume.134
WBE 1997 states that β=n−1/2, which is impossible; radius being linear,135
βmust equal n−1/3. Despite these limitations, WBE 1997’s derivation provides136
problem solving insights.137
2.3 Problems raised by WBE 1997138
1. Problems implicit in WBE 1997 include:139
1.1. In terms of n−1:β2×γ=n−(4/3). Are WBE 1997 steps leading to140
3/4 valid?141
1.2. γβ2appears in the denominator of their equation (5). γβ2= 4/3,142
but 4/3 is not mentioned. Is 4/3 required in a derivation?143
1.3. Is WBE 1997’s assignment of level 0 to the aorta justified?144
1.4. How does the extra 1/3 in 4/3, relative to the volume receiving blood,145
arise? Parenthetically, in April of 2008, finding 4/3 instead of 3/4146
suggested that my mistaken algebra inverted 3/4. After a week’s147
effort that highlighted 4/3, an inverted assumption explained 4/3.148
Then 4/3 stood out a couple of weeks later on reading an instance of149
4/3 in Allen and Maxwell (Allen and Maxwell, 1948, p. 742).150
Universal scaling laws 5
1.5. Is the extra 1/3 in 4/3 due to γ=n−(1/3)?151
1.6. Is 4/3 a subtle but elusive clue to deriving 3/4 metabolic scaling?152
1.7. Is per level 4/3 scaling of a circulatory system tube capacity (supply)153
relative to the linear increase in service volume capacity (demand)154
different in kind from WBE 1997’s derivation, which uses a geometric155
series in their equation (4)?156
1.8. Does squaring β=n−(1/2) in the derivation obviate the need for β?157
Why find β’s square root if a step then squares it?158
1.9. What features of a sphere contribute to its role as a capillary service159
volume in WBE 1997?160
1.10. What is the 4 in 3/4 the dimension of?161
2. WBE 1997 approximates the volume of blood in the circulatory system162
in its equation (4). But a circulatory system has only a finite number of163
levels. Do the approximations in WBE 1997 undermine its validity? Can164
approximations be avoided?165
3. WBE 1997 assumes the circulatory system is “a space-filling fractal-like166
branching pattern” (West et al., 1997, p. 122). Are there underlying167
principles that imply circulatory system fractality?168
4. Animal body size varies by 21 orders of magnitude (West et al., 1997, p.169
122), mammals by 8 (Savage et al., 2004, p. 260). In allometry, each part170
and process has a similitude and so must their relationship in order for171
the animal to function. What similitude is preserved by the operation of172
a part in relation to a differently scaled process?173
3 Biological quarter-power scaling174
3.1 Examples175
A compilation (Savage et al., 2004) of biological quarter-power scaling includes:176
mass-specific metabolic rate, heart and respiratory rates, stride frequencies, life177
spans, times to first reproduction, muscle twitch contraction times.178
3.2 Quarter-power scaling and dimension179
The surface law is based on the relationship between the two dimensional surface180
area of an animal, and its three dimensional volume proportional to its mass181
and weight. Galileo successfully used scaling implied by dimension to analyze182
animal bone size in relation to animal weight. Assume that his use of dimension183
is valid.184
By analogy, infer that in 3/4 metabolic scaling, the numerator three relates185
to mass proportional to three dimensional volume, and that the denominator186
four is of a dimension. Physiological processes that vary by a 1/4 or -1/4 power187
may involve a fourth dimension.188
Universal scaling laws 6
3.3 Fourth dimension: time? distance? other?189
Jacob Joseph Blum, physiologist, suggested that the biological quarter power190
was based on time (Blum, 1977). (Ginzburg and Damuth, 2008) suggested that191
the fourth dimension represented time required for reproduction.192
Banavar et al. infer that supply-demand balance leads to exponent D/D + 1193
for dimension D(Banavar et al., 2002). Banavar and colleagues later infer that194
the fourth dimension relates to supply distance (Banavar et al., 2010).195
If Blum and Banavar et al. correctly infer a fourth dimension, where does196
the extra 1/3 in 4/3, that results in the 4 in the fraction, come from?197
3.4 Supply and demand198
Compare an adapted version of WBE 1997’s scaling of the circulatory system199
to scaling of spherical service volumes supplied with blood.200
Circulatory system tubes scale as β2γ=n−(1/2)2×n−(1/3) =n−(4/3).201
Energy receiving spherical volumes, similar at all scales, grow linearly.202
This derivation of the ratio of exponents, 4/3 : 1 has fewer steps than203
WBE 1997 and no approximations, but is still wrong.204
The extra 1/3 in 4/3 might seem to arise from γ=n−(1/3). But β=205
n−(1/2) in WBE can’t be right since βscales one dimensional radius, nec-206
essarily implying β=n−(1/3).207
Physical system analogs considered below imply that supply and use systems208
scale differently per level. If the same relative scaling between supply and use209
applies per level, the entire system or organism has the same scaling relation-210
ship as levels do. WBE 1997 and other studies to the contrary seek scaling at211
organism level, not per energy distribution level.212
4 Non-biological analogs, numbers, logarithms213
and entropy214
Mammalian metabolism scaling by a 3/4 power of mass implies other instances215
of quarter-power scaling (Savage et al., 2004), as mentioned above.216
Let’s find physical phenomena, modeled by similar mathematics, involving217
numbers 3, 4, 3/4 or 4/3 based on dimensions of subsystems.218
Logarithms are relevant since in scaling relative to length L, logL(L3) = 3219
and logL(L4) = 4. Fractional exponent 3/4 is a ratio of logarithms. Based on220
Boltzmann’s work, entropy is defined as a logarithm. We should be alert to221
physical analogs involving logarithms or entropy.222
Universal scaling laws 7
4.1 Fractality223
Fractality can be characterized by a fractional dimension. 3/4 and 4/3 suggest224
that physical phenomena with scaling analogous to quarter-power scaling are225
fractal.226
4.2 Seeking analogs227
Let’s start by looking for 3, 4, 3/4, and 4/3, logarithms and entropy related to228
distribution of energy.229
5 Stefan’s Law230
5.1 Empirical Law231
Derivation of Stefan’s Law is a consequential physical analogy because isotropic232
distribution of energy, as in black body radiation, is so generalizable, to ecolo-233
gies, steady temperatures at ground level, organisms, physical systems, the uni-234
verse, and to all scales.235
Josef Stefan’s law is based on on experiments by Dulong and Petit (Dulong236
and Petit, 1817) and Tyndall (Tyndall, 1869, p. 427). Stefan concluded that237
the rate of radiant energy varies with temperature T4in degrees Kelvin (Stefan,238
1879). Does exponent 4 represent a dimension?239
5.2 Planck’s derivation version240
Boltzmann derived Stefan’s Law mathematically (Boltzmann, 1884). Planck241
provides a more accessible derivation (Planck, 1913, 1914), often followed (Allen242
and Maxwell, 1948; Longair, 2003, for example).243
Planck treats Stefan’s Law as pertaining to black body radiation (Planck,244
1914, Ch. II). A sealed chamber with a perfect vacuum and a perfectly reflecting245
inner wall has a piston fitted so that no heat is lost or added to the cavity. The246
cavity is at an equilibrium temperature since radiation inside is isotropic — on247
average radiating the same in all directions. Energy density in the chamber248
therefore depends only on chamber volume less piston volume and temperature249
Tin degrees Kelvin.250
From his equation (74) Planck finds251
∂S
∂V T
=4u
3T,(2)
explained in section 5.3. The 4/3 in (2), as in (Allen and Maxwell, 1948, 742-252
743), connects to the unexpected 4/3 of metabolic scaling.253
Universal scaling laws 8
5.3 Planck’s terminology254
Sis the entropy of the cavity containing the radiation.255
Vis the volume of the cavity, or per Planck, of the vacuum (p. 59).256
uis energy density per unit volume. The energy contained in Vis Vu.257
Tis the absolute temperature of the cavity, in degrees Kelvin.258
5.4 Implications of Planck’s derivation259
Tis an indirect measure of energy density for a given cavity: energy ϵper unit260
volume. If the energy input into the cavity increases, temperature Tincreases.261
T’s magnitude depends on how one calibrates a thermometer.262
uis the number of units of calibrated absolute temperature degrees that263
indirectly measures the energy density per unit volume: u=nT for some n.264
Hence u/T is a number and does not have a theoretical role to play in (2).265
Parse (2) to interpret what the mathematics implies about the physics it266
describes:267
The left expression with Tas subscript shows the partial derivative is268
taken with respect to T.269
Entropy Sis the number of degrees of freedom — the dimension — of270
a four dimensional scaling system. Denote a four dimensional system271
D4. Radiation in the cavity scales by 4/3 relative to V, when the partial272
derivative is relative to T, as revealed on the right side of (2).273
If V(large V) scales by v(small v), then Sscales by (4/3)v.274
D4’s scaling exponent is 4/3 that of D3Radiation adds a fourth dimension275
to the empty cavity.276
Suppose a black body scenario but D4without walls. If D4were to scale277
up, then D3would also scale up. Radial length in D3would grow by 4/3278
the radial length increase in D4. Treating D3as analogous to space, space279
would grow.280
Put another way: model the flow — of radiation, molecules, light, time,281
energy, blood, information, liquid or air — as a fourth dimension added to the282
volume within which such motion occurs.283
When motion of some kind is added to a volume, the system with motion is284
four dimensional and scales by exponent 4 relative to the scaling of the empty285
volume which scales by exponent 3. Or we may say, relative to D3’s three286
dimensions of volume, D4scales by 4/3.287
Universal scaling laws 9
5.5 Longair version288
Longair’s equations (11.25) and (11.26) (Longair, 2003, p. 296) arrive, at page289
297 at:290
T
3∂ϵ
∂T V
=ϵ+ϵ
3=4ϵ
3.(3)
ϵ/3 in (3) is due to cavity pressure p— from motion — on an internal wall of291
the cavity; pressure is on average one third along each of the orthogonal axes of292
the cavity space.293
5.6 Flow as a dimension294
A fourth dimension models radiation flow in Stefan’s Law and blood flow in295
metabolism. How does a flow dimension differ from each of the three spatial296
dimensions?297
For a constant flow, the rate of motion is proportional to rate of transmission298
of energy or information and to the distance traveled per unit time. Moreover,299
time flows, at a rate determined by time itself, ‘equably’ (Newton, 1846, p. 77).300
Let subscripted length Lrepresent a dimension in {L1, L2, L3}. Let motion301
be defined as dL
dt ≡˙
Lfor time t. Then a system with motion is {L1, L2, L3,˙
L},302
a construct.303
5.7 Degrees of freedom, dimension and length304
Consider length 4Lin D4and its corresponding length in D3. 4Lin D4has four305
degrees of freedom relative to Land three degrees of freedom in D3relative to306
D3’s corresponding length (4/3)L. The system with larger degrees of freedom307
— 4 — has lengths 3/4 of those in the system with smaller degrees of freedom308
— 3. Compared to D3,D4has 4/3 the dimensions or degrees of freedom, and309
3/4 the magnitude of lengths.310
Length L in D4with energy unchanged has length (4/3)Lin D3. As if D4’s311
extra dimension is absorbed by D3growing radially by 4/3 as much.312
5.8 Stefan’s Law and 4/3 scaling313
4/3 scaling results in, and is more fundamental than, Stefan’s Law. Stefan’s314
Law is significant because it reveals universal 4/3 scaling.315
5.9 Efficiency316
Energy proportional to Lin D4and proportional to (4/3)Lin D3is undimin-317
ished when the reference frame changes from D4to D3and back. Changing318
reference frames is maximally energy efficient, more efficient than the heat cycle319
of an ideal heat engine. An energy efficient way for a universe to emerge.320
Universal scaling laws 10
6 Dimensional capacity321
6.1 Galileo’s example322
Galileo observed that weight proportional to volume scales by exponent 3 while323
cross-sectional area of supporting bone scales by exponent 2. The difference324
in dimension of weight and bone’s cross-section cause, on a change in animal325
size, the different scaling of weight and bone. Galileo does not need a geometric326
series.327
Change the reference frame from scaling implied by dimension to dimension328
itself. Dimension is more fundamental than scaling. With an increase in animal329
size, cross-sectional area that scales by exponent 2 must scale in size by 3/2, to330
support increased weight proportional to volume that scales by 3. The lower di-331
mensioned subsystem must scale by 3/2 in size because the higher dimensioned332
subsystem scales in dimension by 3/2.333
6.2 Generalizing Galileo from 3 and 2 to 4 and 3334
Generalize Galileo’s observations. Instead of dimensions 3 and 2, suppose sub-335
systems of dimensions 4 and 3. This seems to be absurd; there are only three336
spatial dimensions. Extend Galileo’s mathematics, regardless of our imperfect337
understanding of the fourth dimension, an approach frequently productive in338
mathematics and physics. Then, on an increase in size involving D4,D3must339
increase in size by 4/3. If D4increases by Lradially, then D3must increase by340
(4/3)L.341
6.3 The fourth dimension mystery342
Minkowski successfully invoked time as a fourth dimension in connection with343
special relativity (Speiser et al., 1911; Minkowski, 1918). A fourth dimension344
based on motion and flow is conceivable. Let’s treat the fourth dimension as345
a plausible mathematical and physical construct for now, subject to further346
investigation some other day.347
7 Stefan’s Law derivation and metabolic scaling348
7.1 Analogy349
Analogize observations in section 5.4 to metabolic scaling.350
Blood carrying energy to animal cells flows through hierarchically scaled351
three dimensional circulatory system tubes, a D4system. Denote a circulatory352
system with blood flow C4to distinguish it from the general D4case. On353
an increase in animal size, C4’s capacity to deliver energy to an animal’s cells354
grows — scales — by 4/3 as much as the animal’s three dimensional volume355
proportional to animal mass. Cells will overheat unless the rate of delivery of356
energy via blood flow slows by a 3/4 power of mass.357
Universal scaling laws 11
7.2 4/3 supply scaling and metabolism358
For a mammal to have constant body temperature, rate of energy supply must359
equal rate of energy use; simplifying, metabolism Y∝M. This applies at every360
scaled hierarchical level of the organism. If more energy is supplied to a cell361
than it needs, the cell will overheat, its intracellular processes will be impaired,362
and the cell will die.363
With an increase in size, C4scales by exponent 4 , mass Mscales by exponent364
3. References to C4and Min what follows refer to energy amounts proportional365
to their size.366
For C4and M, subscript sdenotes a smaller animal and ba bigger one.367
7.3 Metabolic scaling algebra368
For kiterations of cscaling of C4for bigger animal (b):369
([C4]b)kc = ([C4]s)4kc.(4)
Animal energy use capacity, proportional to mass M, scales as370
Mkc
b=M3kc
s.(5)
The exponents of scaling for the ratio supply : use, for an animal that is larger,371
are 4kc : 3kc = 4 : 3, relative to the smaller animal:372
([C4]b)kc
Mkc
b
=([C4]s)(4/3)kc
Mkc
s
.(6)
Assume intracellular physiology and biochemistry are the same for small and373
large animals. Then their cells have the same optimum temperature. Since,374
with increased size, energy supply compared to energy use scales by 4/3, the375
metabolic rate must slow supply capacity, represented by a 3/4 exponent for376
the numerator in (6):377
([C4]s)(4/3)kc3/4
Mkc
s
=([C4]s)kc
Mkc
s
.(7)
The expression on the right of (7) shows the bigger animal’s energy supply and378
use in the same balance as it was for the smaller animal.379
3/4 appears as the exponent of mass Min metabolic scaling because, energy380
supplied being proportional to both of (C4)band Mb, they scale as381
n([C4]s)(4/3)kco3/4=([C4]b)kc3/4=Mkc
b
3/4⇒[C4]b∝Mb.(8)
Conclude that 3/4 metabolic scaling reverses the 4/3 scaled increase in en-382
ergy supply capacity that occurs when an animal increases in size, to maintain383
invariant a constant intracellular optimal temperature.384
Universal scaling laws 12
7.4 3/4 as Nature’s metabolic scaling head fake385
The ample literature on 3/4 metabolic scaling did not consider that 3/4 is an386
inverse scaling in response to 4/3 scaling of C4. By analogy to head fakes in387
basketball, 3/4 as mass’s exponent is a head fake by Nature, so convincing388
that mentions of 4/3 scaling for over 100 years — discussed below — were not389
associated with metabolic scaling. It is difficult, if not impossible, to account for390
3/4 metabolic scaling without 4/3 scaling. The profound and subtle difficulty391
in metabolic scaling is not the mathematics but rather the distraction of 3/4392
scaling and the obscurity of 4/3 scaling.393
8 Other 4/3 biological scaling394
8.1 Cancer constancy in old age for different animals395
The risk of cancer in old age is almost constant for animals of all sizes (Peto,396
1977), called Peto’s Paradox (Nagy et al., 2007; Nunney, 2020). The para-397
dox is that larger animals have more places for cancer to start and more ways398
for it to spread. Cancer spread, a D4system, scales with size by 4/3. 3/4399
metabolic scaling exactly offsets cancer’s increased proliferation, similarly to400
how 3/4 metabolic scaling offsets 4/3 scaling of energy supply. Other than my401
ResearchGate postings, 4/3 scaling seems not to have been invoked for Peto’s402
paradox (Kempes et al., 2020).403
8.2 Brain weight scaling404
If nervous system connections, as a D4system, scale by exponent 4 when an-405
imal body mass scales by exponent 3, then the nervous system has 4/3 the406
control capacity relative to increased body size compared to the animal’s D3
407
requirements proportional to M. If so, animal brains scaling by a 3/4 power408
of mass enable neuronal control of animal mass to be the same for small and409
large animals. Measurements provide support that this is the case (Snell, 1892),410
(Eisenberg, 1981, p. 275-283), (Armstrong, 1983; McNab and Eisenberg, 1989),411
(van Dongen, 2008, p. 2101).412
9 Physical examples of 4/3 scaling413
9.1 Introduction to this section414
This section presents non-biological examples of 4/3 scaling. Cosmology has its415
own section following.416
9.2 Black body radiation 1879417
A cavity is D3. Radiation within and together with the cavity is a D4system.418
Absolute temperature Tis an indirect measure of linear (one dimensional) mo-419
Universal scaling laws 13
tion. D4scales by exponent 4 when energy is added to the cavity; so does one420
dimensioned T: Stefan’s Law. Black body radiation scales by 4/3 relative to421
D3volume.422
9.3 J. J. Waterston 1845423
Lord Rayleigh, in 1892 when he was secretary of the Royal Society, had Water-424
ston’s 1845 paper published (Waterston, 1892). Waterston observes (p. 23) that425
on an increase in molecular energy (vis viva), between the two cases of constant426
pressure (a D4system) and constant volume (a D3system), the ratio of the427
amounts of energy is 4 : 3. Waterston’s 4/3 ratio may be the first detection of428
4/3 scaling, in 1845.429
9.4 Clausius 1860430
Clausius considered the relative velocity of moving molecules. Maxwell in re-431
sponse showed that two particles with the same speed, v1=v2, have relative432
speed v=√2v1(Maxwell, 1860).433
Clausius’s approach was different. He concluded (in translation) that434
The mean lengths of path for the two cases (1) where the remain-435
ing molecules move with the same velocity as the one watched, and436
(2) where they are at rest, bear the proportion to one another of 3
4
437
to 1 (Clausius, 1858, 1859).438
Clausius subsequently showed (Clausius, 1860) that the average relative speed r439
of moving molecules was 4/3 that of an individual moving particle. He treated440
the second particle as stationary relative to the first moving particle, in effect441
comparing D4and D3. Accordingly, the mean path length of moving molecules442
is 3/4 that of stationary molecules. Clausius meant to show the relative velocity443
between moving molecules but instead found the lengths ratio of D4compared444
to D3.445
Treat molecules moving in space, as in the Clausius’s example, as a D4
446
system. A radial length of Lin D4is (4/3)Lin D3.447
Clausius’s 1860 article up to 2023 is not much noticed. The editor of448
Maxwell’s collected works described the mathematics in Clausius’s 1860 arti-449
cle as incorrect (Maxwell, 1890, Vol. 1, p.387). But Clausius, if I understand450
things, correctly solved a different problem. If so, that different problem antic-451
ipates the cosmological expansion of space.452
9.5 Richardson on wind eddies, 1926453
Richardson’s 1926 article on diffusion in atmospheric air (Richardson, 1926)454
finds a 4/3 exponent in an equation. The first list are reasons for suspecting a455
connection to 4/3 scaling, which was my impression these past few years, and456
the second list for doubting that. I leave the question of a connection to 4/3457
scaling as unresolved.458
Universal scaling laws 14
Reasons to suspect a connection to 4/3 scaling:459
Richardson used small free balloons and volcanic ash to stand in for air460
molecules (Richardson and Proctor, 1927) and measured their rate of sep-461
aration distance ℓ. To estimate how air molecules separated due to atmo-462
spheric change, he chose the square of the distance between two molecules,463
in the context of the equation assumed to apply with Kconstant:464
d(t)2=Kt +d(0)2.(9)
He reasoned that Kmight be a function of ℓ.465
By measurement he found466
F(ℓ)=0.4(ℓ)4/3,(10)
which seems to connect to 4/3 scaling in general, since ℓis a length (Frisch,467
1995, p. 102), (K¨orner, 1996, p. 176-192).468
G. I. Taylor remarked that Richardson:469
had the idea that the index was determined by something470
connected with the way energy was handed down from larger471
to smaller and smaller eddies. He perceived that this is a pro-472
cess which, because of its universality, must be subject to some473
simple universal rule (Taylor, 1959), consistent with the ther-474
modynamics of 4/3 scaling, and its possible universality.475
On the other hand,476
Researchers following Richardson expressed doubts that the exponent of477
ℓis 4/3 (Reiter and Lester, 1967; Grachev et al., 2013). Which weakens478
if not destroys the argument, if 4/3 is not the exponent.479
ℓis not a length but rather is a separation distance.480
The expression Richardson considered in (9) squared the distance, which481
is not the same as a one dimensional length appearing in two related482
reference frames.483
Kolmogorov in papers called K41 provided theoretical reasoning for Richard-484
son’s empirical diffusion results (Kolmogorov, 1991b,a).485
The reader might find Richardson’s 4/3 connects to 4/3 scaling here consid-486
ered but it may be the most one can say is that it is an open question.487
9.6 Brownian motion’s 4/3 fractal envelope488
Robert Brown, a botanist, observed pollen immersed in water moving to and489
fro (Brown, 1828). This kind of motion is called Brownian motion. Mandelbrot490
Universal scaling laws 15
conjectured that the fractal dimension of Brownian motion is 4/3 (Mandelbrot,491
1982, p. 243).492
In 2001, Lawler et al. announced and sketched proofs that found the fractal493
dimension of Brownian motion is 4/3 (Lawler et al., 2001) in four mathematically494
sophisticated papers (Lawler et al., 1999) (Lawler et al., 2000a,b,c).495
If particles moving on a water surface is a D4system, and the water is a D3
496
system, then the 4/3 fractal envelope follows from section 9.4.497
4/3 scaling may play a role in cellular metabolism, the energy of random498
motion being captured by Brownian ratchets Peskin et al. (1993). If so, 4/3499
scaling applies to the whole organism down to the intracellular level.500
9.7 Social network information transmission501
Let µbe the mean path length of a social network Nwith nnodes. Scale N502
hierarchically into levels with clusters of µi,ia natural number, and µk=n;k503
hierarchical levels. Then504
dµi
dt =µ(11)
for all i. That implies that every cluster scales by the natural logarithm e, since505
exis its own derivative: an isotropic network scales by the natural logarithm.506
The natural logarithm base of the logarithm (Jensen, 1906; Khinchin, 1957)507
maximizes the value of the function.508
Information transmission in a social network is a D4network transmitting509
to static nodes which form a D3network. This implies that µfor information510
transmission in a social network is (4/3)e= 3.624. Social network information511
µhas been measured as 3.65 (Watts and Strogatz, 1998, Table 1), consistent512
with a 4/3 scaling law.513
10 Cosmological predictions based on 4/3 scal-514
ing515
10.1 Introduction to this section516
Light moving in space, space + radiation, is a D4system. A length Lin D4
517
should be (4/3)Lin the corresponding D3space.518
Gold speculated that light motion expands the universe (Gold, 1962), con-519
sistent with 4/3 scaling expanding D3space.520
10.2 Cosmological expansion521
The following implications of 4/3 scaling are compared to astronomical obser-522
vations in a following section.523
Redshift is the Doppler effect applied to light moving. A redshift distance524
is a measurement in D4.525
Universal scaling laws 16
Luminosity distance of a celestial object at an unknown distance is found526
by comparing the known distance and luminosity of the same kind of527
celestial object — a standard candle. Type 1A supernovas — SN1A —528
are used for estimating large cosmological distances because they are so529
bright, and because their luminosity is relatively consistent (Kowal, 1968).530
Luminosity comparisons take place in a shared D3system.531
Redshift distances are not equal to luminosity distances because their ref-532
erence frames have different dimensions. A redshift distance to a standard533
candle uses light’s motion, and is not the same as the luminosity distance.534
Assuming that the universe has a single cosmological reference frame,535
instead of two cosmological reference frames, D3and D4, implies that536
redshift distance and luminosity distance to a standard candle should be537
equal, contrary to 4/3 cosmological scaling.538
Luminosity found using a standard candle that is 3/4 of luminosity ex-539
pected based on redshift distance occurs because the same distance is 4/3540
as much in D3compared to D4.541
The inverse square law for luminosity depends on the conical spread of542
radiated light, which does not apply to a D3luminosity distance estimate543
of a standard candle. The difference in luminosity between two different544
standard candles depends linearly on the D3distances only.545
If space had only one reference frame, D4, with redshift distances as the546
only measure of cosmological distances, then the inverse square law would547
apply to luminosities of standard candles at different distances. The 4/3548
theory would explain luminosity that is 3/4 of what is expected on the549
basis of redshift distance as arising because luminosity distance is 4/3 more550
than redshift distance. A redshift distance perspective together with the551
inverse square law would imply that the SN1A that is 4/3 farther away is552
showing luminosity that is553
p4/3≈1.154,(12)
or 15% less than expected, since luminosity decreases when distance in-554
creases; distance squared and luminosity are inverse to each other accord-555
ing to the inverse square law.556
Let Ebe an amount of energy. Then the ratio of energy density ϵin D4
557
for a length Lcompared to D3for corresponding length (4/3)Lwould be:558
ϵD4
ϵD3
=E/L3
E/ [(4/3)L]3=64
27 ≈0.7033
0.2967.(13)
Energy density of space ϵ, based on the distance from Earth, for a cosmic559
scale factor a, would vary by 1/a3in D4and 1/a4in D3; distances in D3
560
are stretched by 4/3 relative to those in D4.561
Universal scaling laws 17
Every radial expansion Lof D4is coincident with a radial expansion of562
D3of (4/3)L. The rate of expansion of the universe, based on the ratio of563
a length in D4to the same length in D3, is constant. The cosmic constant564
is 4/3 comparing D3to D4. Measurements that assume the existence of565
one only of D3and D4make it appear that the universe is stretching.566
Analogously to metabolic scaling, the 4/3 scaling occurs at each level of567
hierarchical scaling. As between the two reference frames, the scaling is568
constant at every level.569
D3distances recede from us faster than D4distances. D3distances out-570
pace light’s distances, which are in D4.571
11 Astronomical observations572
11.1 Cosmological distances573
1998 observations (Schmidt et al., 1998; Perlmutter et al., 1998; Riess et al.,574
1998) of SN1A measured luminosities an average 25% less than anticipated575
(Cheng, 2010, p. 259), that is, 3/4 of what the redshift distance would predict.576
These observations are consistent with 4/3 cosmological scaling predicting that577
redshift distance to a SN1A would be 3/4 of the luminosity distance to the same578
object.579
11.2 Implied luminosities580
1998 astronomical observations inferred that the distances of the high-redshift581
SN Ia average 10% - 15% farther than expected (Riess et al., 1998, abstract).582
That conclusion likely assumed a single cosmological reference frame, D4, and583
the application of the inverse square law to luminosities. 4/3 scaling would584
account for this 1998 observation.585
11.3 Cosmological energy densities586
In 2014, data on 740 Type IA supernovae (SN1A) gave Ωm= 0.295 (Betoule587
et al., 2014), <0.6% different than 0.2967 in (13). In 2018, data on a combined588
total of 1048 SNIA gave Ωm= 0.295 −0.319 (Scolnic et al., 2018, Table 14. p.589
22). 2019, combined SNIA data gave Ωm= 0.293 −0.369 (Abbott et al., 2019).590
0.2967 in (13) is consistent with Ωmin these surveys.591
11.4 Energy density variation with distance from Earth592
Based on the Friedmann and fluid equations, energy density ϵ, with origin sub-593
scripted 0, varies with distance (Ryden, 2003, for example, p. 64) as,594
for matter m,ϵm(a)= (ϵm)0/a3;595
for radiation γ,ϵγ(a)= (ϵγ)0/a4.596
Universal scaling laws 18
These theoretical implications are consistent with 4/3 cosmological scaling, if597
we identify the mass reference frame with D3and the radiation reference frame598
with D4.599
11.5 Cosmological horizon problem600
4/3 cosmological scaling implies that distances in D4are accessible at light601
speed, while the same distances in D3are not, which might address the horizon602
problem in cosmology: how do parts of space connect if they are too far apart603
to reach at light speed.604
11.6 Special relativity and kcalculus605
Hermann Bondi showed that scaling based on his kcalculus can be used to606
derive special relativity (Bondi, 1962).607
12 Discussion of problem section608
12.1 Introduction to this section609
Responses below use the same numbering as the section 2.3 problems:610
12.2 Suggested answers611
1. Regarding WBE 1997:612
1.1. Their equation (5) has γβ2which equals n−(4/3) but WBE 1997’s613
derivation is still defective; in addition to other defects, the exponent614
of βin terms of nin WBE 1997 is impossible.615
1.2. 4/3 scaling of blood’s energy supply compared to blood’s energy use616
can derive metabolic scaling.617
1.3. The first segment of a radiation cone away from a source of radiation618
is level 1. Level 0 is an energy source at the radiation cone apex.619
By analogy, the level 0 energy source in metabolism is food, external620
to the organism. The aorta should be assigned level 1, like the first621
segment of a radiation cone.622
1.4. The extra 1/3 in 4/3 in WBE 1997, relative to the recipient volume,623
arises because scale factor nof the number of tubes from level to624
level, indirectly scales volume. WBE 1997 has radius scale factor625
β=n−1/2instead of β=n−1/3;βscales one dimensional length626
while nscales three dimensional volume. The erroneous extra 1/6627
in β=n−1/2’s exponent, squared in WBE 1997, gives the the extra628
1/3. WBE 1997 conflates flow in the scaling of cross-sectional area,629
leading to the erroneous β=n−1/2. Missing from WBE 1997 is the630
assignment of a dimension to flow.631
Universal scaling laws 19
1.5. See above remarks on squaring β=n−(1/2).632
1.6. It seems so.633
1.7. Per level 4/3 scaling of a circulatory system tube capacity (supply)634
relative to linear increase in the service volume capacity (demand)635
is different in kind from WBE 1997’s geometric series. Using a ge-636
ometric series to obtain total blood volume has every level of the637
circulatory system contribute to total blood volume to find the scal-638
ing relationship. 4/3 scaling is based on dimensions of supply and639
receipt systems per level.640
1.8. Squaring the square root of the tube radius scale factor in WBE641
1997 shows taking the square root is unnecessary. Scaling the cross-642
sectional area is also unnecessary. Since 4/3 scaling is due to the643
different dimensions of supply and use, scaling of the sizes of tube644
radius, length, and cross-sectional area is unnecessary.645
1.9. Capillary service volume being represented by a sphere in WBE 1997646
is serendipitous, because a spherical distribution volume suggests647
isotropic distribution of energy. Isotropic distribution applies to cos-648
mic microwave background radiation (CMBR) at cosmological scales649
(Smoot et al., 1992; Fixsen et al., 1996). Our C4systems provide650
clues for cosmology.651
1.10. C4.652
2. The mathematics of 4/3 scaling makes approximations unnecessary. With653
a 1/12 difference between 3/4 and 2/3, approximations undermine and654
possibly invalidate derivations of b.655
3. Fractality is built into 4/3 scaling, as the ratio of logarithms of supply656
scaling compared to use scaling. Fractality is a consequence of 4/3 scaling,657
not an assumption as in WBE 1997.658
4. The relationship similitude preserved is energy supplied to unit volume of a659
cell, consistent with the invariance of cellular physiology and biochemistry660
in animals of different sizes.661
12.3 4/3 occurrences662
Exponents 4/3, 3, or 4 occur in 3/4 metabolic scaling, black body radiation663
described by Stefan’s Law, cosmological energy density away from Earth, and664
a problem solving network’s degrees of freedom. When the exponent increases,665
the number of ways for a process to occur increases. A parameter with exponent666
4 has more capacity than a corresponding parameter with exponent 3.667
A length in D4stretching by 4/3 when in D3is exemplified in Clausius’s668
1860 work and in cosmological expansion of space.669
Universal scaling laws 20
13 Universal quarter scaling and 4/3 law?670
13.1 Further questions?671
Foregoing is a synopsis of why we may suspect that both quarter scaling and672
the related 4/3 scaling are universal laws of physics.673
Here are some further questions:674
Are there other instances in biology and physics of quarter scaling and of675
4/3 scaling?676
How is the fourth dimension best understood?677
Can special relativity be derived using 4/3 scaling principles?678
How does the fourth dimension relate to emergent phenomena?679
Can the equivalence of different derivations, such Waterston’s, Clausius’s,680
Kolmogorov’s, etc., leading to 3/4 and 4/3 scaling, be demonstrated?681
If 4/3 scaling is a universal law, what are its implications, if any, for682
quantum mechanics and general relativity?683
Do the principles underlying 4/3 scaling relate to quantum entanglement,684
the cosmological horizon problem, cosmological inflation?685
Energy and 4/3 dimensional pressure occur at all scales. Are they related?686
13.2 Connections?687
How does metabolic scaling connect to black body radiation and cosmology?688
Thermodynamics and statistical mechanics. Thermodynamics is quintessen-689
tially multi-disciplinary.690
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