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Flowchart for deriving 4/3 scaling

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Abstract

A flowchart display of the logical steps. A reasoning flowchart shows the pivotal extension from 3D to 4D in the argument for universal 4/3 scaling.
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Flowchart for deriving 4/3 scaling2
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4
Robert Shour5
6
17th July, 20237
Toronto, Canada8
Abstract9
A reasoning flowchart shows the pivotal extension from 3D to 4D in10
the argument for universal 4/3 scaling.11
Keywords analogy, dimensional capacity, scaling12
1 4/3 scaling a universal law?13
1.1 Relevance of 4/3 scaling14
If 4/3 scaling is a universal law of physics, then it provides plausible explanations15
for16
Kleiber’s Law that metabolism scales by a 3/4 power of mass (Kleiber,17
1932).18
Peto’s paradox about the spread of cancer (Peto, 1977).19
Brain weight scaling (Snell, 1892, 2021), (Eisenberg, 1981, p. 275-283),20
(Armstrong, 1983; McNab and Eisenberg, 1989), (van Dongen, 2008, p.21
2101).22
The 4/3 fractal envelope of Brownian motion (Lawler et al., 2001).23
Expansion of cosmological space due to an unknown dark energy (Schmidt24
et al., 1998; Perlmutter et al., 1998; Riess et al., 1998).25
Since these problems and likely others remain unresolved, of interest to the26
physics community and appear amenable to resolution if 4/3 scaling applies27
to them, it is helpful to identify key components in the arguments in favor of28
universal 4/3 scaling.29
1.2 Flowchart advantages30
A flowchart display of the logical steps, not the evidence, leading to 4/3 scaling,31
has advantages, including:32
Universal scaling laws 2
Visual display of the argument, as an alternative and supplement to a33
verbal or written description of the argument.34
To exclude extraneous steps which distract from the essential argument.35
To reduce the argument to essential steps.36
To reveal assumptions necessary to make universal 4/3 scaling credible.37
To identify those steps in the argument that must be established to sustain38
it, or to at least render it plausible.39
1.3 This flowchart origin40
This flowchart was devised in connection with an article under way relating41
reasoning, analogies and logical steps leading to universal 4/3 scaling. Since42
the flowchart seems a helpful concise summary of the logical steps, it might be43
helpful to add it as a resource pending completion of the article of which it is a44
contemplated part.45
It can be considered to be appended to articles recently posted on Research-46
Gate relating to 4/3 scaling including:47
From Galileo’s simple case to universal 4/3 scaling48
Is biology’s quarter scaling universal in physics too?49
A scale factors derivation of Kleiber’s law.50
Why scaling and not dimension, Galileo?51
as well as other and earlier articles.52
1.4 Reasoning flowchart53
The flow of reasoning in the chart is essential to making the case for 4/3 scaling.54
The first line of the flowchart with boxes with white backgrounds displays55
headings. The second line represents a dimensional conceptual reference frame.56
The third line represents a scaling conceptual reference frame.57
2 Required for validity58
Box 1a refers to Galileo’s treatment of the strength of beams (Galilei, 1638) in59
Two New Sciences. He applies those observations to weight WL3and the60
cross-sectional area of weight bearing bone AL2.61
Box 1b refers to Galileo’s explanation of why heavier animals have thicker62
bones. The dimensional observation in box 1a leads to the scaling solution in63
box 1b.64
The transition from box 2b to box 2a relocates Galileo’s scaling argument65
back to the original dimensional reference frame. In the dimensional reference66
Universal scaling laws 3
Ref Frame
a. Dimension Galileo, W:A
L3:L2L3:L2
W per D Add 1D flow
L4:L3
b. Scaling
1: Dim scaling 2: Dim cap 3: Extend to 4D
Figure 1: Reasoning flowchart
frame, Galileo’s scaling solution is characterized as arising from the difference in67
the dimensional capacity of 3 dimensions compared to the dimensional capacity68
of 2 dimensions. Since Galileo reasoned from box 1a to box 1b to examine69
the effect of scaling induced by dimension, it is arguably justified to reverse70
the direction of reasoning, beginning from scaling and going back to dimension.71
This step implies the existence of a principle of dimensional capacity. Articles72
on ResearchGate discuss dimensional capacity.73
The transition from box 2a to box 3a is the critical step. If dimensional74
capacity is a valid or, at least, plausible principle, then extend it from 3D for75
volume and weight to 4D for volume plus 1D flow through 3D volume. Blood76
flow through circulatory system tubes and light or energy radiated into a 3D77
space are examples of 1D flow. If the extension from 3D to 4D is valid, then78
there is a logical basis for universal 4/3 scaling.79
Accordingly, the key question becomes: Is the step from 2a to 3a justified?80
Bibliography81
Armstrong, E. (1983). Relative brain size and metabolism in mammals. Science,82
220(4603):1302–1304.83
Eisenberg, J. F. (1981). The Mammalian Radiations. Univ. of Chicago Press.84
Galilei, G. (1638). Discorsi e dimostrazioni matematiche intorno ´a due nuove85
scienze attinenti alla meccanica & i movimenti locali. Appresso gli Elsevirii.86
Kleiber, M. (1932). Body size and metabolism. Hilgardia, 6:315.87
Lawler, G. F., Schramm, O., and Werner, W. (2001). The dimension of the88
planar Brownian frontier is 4/3. Math Res Lett, 8:401–411.89
McNab, B. K. and Eisenberg, J. F. (1989). Brain size and its relation to the90
rate of metabolism in mammals. The American Naturalist, 133(2):157–167.91
Universal scaling laws 4
Perlmutter, S., Aldering, G., Goldhaber, G., et al. (1998). Measurements of 92
and Λ from 42 High-Redshift Supernovae. arXiv:astro-ph/9812133.93
Peto, R. (1977). Epidemiology, multistage models, and short-term mutagenicity94
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Cancer, pages 1403–1428. Cold Spring Harbor Laboratory Press.96
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Supernova Search: Measuring Cosmic Decelerationand Global Curvature of100
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Snell, O. (2021). Otto Snell’s 1892 Hirngewichtes von dem orpergewicht, trans-105
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dem orpergewicht und den geistigen ahigkeiten). ResearchGate.107
van Dongen, P. A. M. (2008). Brain size in vertebrates. In R., N., ten Donkelaar,108
H. J., and Nicholson, C., editors, The central nervous system of vertebrates,109
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Article
Full-text available
We present spectral and photometric observations of 10 Type Ia supernovae (SNe Ia) in the redshift range 0.16 ≤ z ≤ 0.62. The luminosity distances of these objects are determined by methods that employ relations between SN Ia luminosity and light curve shape. Combined with previous data from our High-z Supernova Search Team and recent results by Riess et al., this expanded set of 16 high-redshift supernovae and a set of 34 nearby supernovae are used to place constraints on the following cosmological parameters: the Hubble constant (H0), the mass density (ΩM), the cosmological constant (i.e., the vacuum energy density, ΩΛ), the deceleration parameter (q0), and the dynamical age of the universe (t0). The distances of the high-redshift SNe Ia are, on average, 10%–15% farther than expected in a low mass density (ΩM = 0.2) universe without a cosmological constant. Different light curve fitting methods, SN Ia subsamples, and prior constraints unanimously favor eternally expanding models with positive cosmological constant (i.e., ΩΛ > 0) and a current acceleration of the expansion (i.e., q0 < 0). With no prior constraint on mass density other than ΩM ≥ 0, the spectroscopically confirmed SNe Ia are statistically consistent with q0 < 0 at the 2.8 σ and 3.9 σ confidence levels, and with ΩΛ > 0 at the 3.0 σ and 4.0 σ confidence levels, for two different fitting methods, respectively. Fixing a "minimal" mass density, ΩM = 0.2, results in the weakest detection, ΩΛ > 0 at the 3.0 σ confidence level from one of the two methods. For a flat universe prior (ΩM + ΩΛ = 1), the spectroscopically confirmed SNe Ia require ΩΛ > 0 at 7 σ and 9 σ formal statistical significance for the two different fitting methods. A universe closed by ordinary matter (i.e., ΩM = 1) is formally ruled out at the 7 σ to 8 σ confidence level for the two different fitting methods. We estimate the dynamical age of the universe to be 14.2 ± 1.7 Gyr including systematic uncertainties in the current Cepheid distance scale. We estimate the likely effect of several sources of systematic error, including progenitor and metallicity evolution, extinction, sample selection bias, local perturbations in the expansion rate, gravitational lensing, and sample contamination. Presently, none of these effects appear to reconcile the data with ΩΛ = 0 and q0 ≥ 0.
Chapter
Humanity and human intelligence are considered to be derived from the large human brain; therefore brain size is regarded as a relevant and interesting parameter. This chapter covers brain size in an evolutionary perspective. Such a starting point of course has an inherent limitation: attention is only paid to overall brain size, and not to the size of brain subsystems. Nevertheless, overall brain size is an interesting parameter.
The Mammalian Radiations
  • J F Eisenberg
Eisenberg, J. F. (1981). The Mammalian Radiations. Univ. of Chicago Press.
Discorsi e dimostrazioni matematiche intornoá due nuove
  • G Galilei
Galilei, G. (1638). Discorsi e dimostrazioni matematiche intornoá due nuove
The dimension of the
  • G F Lawler
  • O Schramm
  • W Werner
Lawler, G. F., Schramm, O., and Werner, W. (2001). The dimension of the
Brain size and its relation to the
  • B K Mcnab
  • J F Eisenberg
McNab, B. K. and Eisenberg, J. F. (1989). Brain size and its relation to the
Otto Snell's 1892 Hirngewichtes von dem Körpergewicht, trans-105 lated into English by Robert Shour (Die Abhängigkeit des Hirngewichtes von 106 dem Körpergewicht und den geistigen Fähigkeiten)
  • O Snell
Snell, O. (2021). Otto Snell's 1892 Hirngewichtes von dem Körpergewicht, trans-105 lated into English by Robert Shour (Die Abhängigkeit des Hirngewichtes von 106 dem Körpergewicht und den geistigen Fähigkeiten). ResearchGate.
  • S Perlmutter
  • G Aldering
  • G Goldhaber
Perlmutter, S., Aldering, G., Goldhaber, G., et al. (1998). Measurements of Ω 91 and Λ from 42 High-Redshift Supernovae. arXiv:astro-ph/9812133.