"It is important to realize that in physics today, we have no knowledge what energy is."

− Richard Feynman

E=mv^2 is not a definition of energy, it is a quantification of it relative to mass.

I have not read Coopersmiths work but the way I read your question and a quick internet search of the topic, the 4/3 scaling law is similarly an attempt to quantify energy in a multi-dimensional form; x, y, z + motion.

As motion is a function of change in the three linear dimensions relative to delta time, then time becomes your fourth dimension.

Dimensional pressure is a concept outside of my understanding.

But to answer your exam question, I see energy as the transfer of photons, through space, between matter (atoms etc.) and from matter to space as a result of changes in time or object relative position (the three linear dimensions).

If you then ask me to define a photon other than a quanta of change in those four basics dimensions then my capabilities fail at that point.

(Edit - did my last sentence just define what you call dimensional pressure?)

Here energy is looked at from another perspective.

Universe also includes presence of Life or consciousness. This is considered sentient part of the universe. The insentient part is made up of energy. So the entire universe is made up of Consciousness and energy. The connection between the two is discussed in following article.

Consciousness and energy

Chapter In: The Big Bang: Theory, Assumptions and Problems

Editors: J.R. O’Connell and A.L. Hale, pp.275-284,

ISBN 978-1-60456-802-8; March 2012, Nova Science Publishers, Inc. N.Y.

Chapter download at: https://www.researchgate.net/publication/262494186_Consciousness_and_energy

https://www.novapublishers.com/catalog/product_info.php?products_id=21109

http://www.amazon.co.uk/The-Big-Bang-Assumptions-Problems/dp/1613245777/ref=sr_1_5?s=books&ie=UTF8&qid=1333360764&sr=1-5

http://www.amazon.com/The-Big-Bang-Assumptions-Problems/dp/1613245777

"Energy" as a concept is of our own (human) creation as a commodity of exchange that allows for a dynamic rather than a static Universe. The dynamic is manifest in the Universe as ongoing change with time of the states of being of the physically observable entities (stars, atoms and chipmunks, for example) that populate the Universe. Energy exchange, with advancing time as parameter, drives the ongoing change that we perceive as the evolving Universe. Energy is being continuously released into the Universe through the conversion of mass into ultimately radiant energy (photons) emitted from the surfaces of the stars in which the thermonuclear conversion of mass to energy occurs. (Accelerated expansion of the Universe is a (dark) energy-driven global change of the state of being of the Universe as a whole.) Energy thus emerges as a convenience for charactering quantitatively the processes of progressive change within the Universe. (Please see my recently submitted RG preprint "Pondering the Imponderables.")

In one of its oldest forms, "energy" is the ability to do work, i.e. to set something in motion (relative to something else). I believe this definition still stands. Other definitions, such as Coopersmith's, can be adopted, but she will need to make a pretty good case in order to redefine something that stood the test of time.

Just a extremely useful mathematical concept to describe Physical events ,if they are described by DETERMINISTIC Differential equations of any sort (Hamiltonians and integral invariants of motion equations ).Note that the complex motion equations of Classical Mechanics can be ALWAYS in principle be solved by the FIRST ORDER PDE of Hamilton Jacobi (specially numerically in a super computer for intercontinental ballistic artillery ...rsrsrs ) .From this landmark is that comes the Hamiltonian mathematical function as an extremely useful tol to solve Newton Equations (the Hamiltonian mathematical concept so useful in Quantum Mechanics when in formulated in the Operatorial Von Neuman formalism) .Of course Total Energy (surely an integral invariant of Newton Motion Equations ) may not coincides with the Hamiltonian ,when in presence of constraints.But do not forget , Motion equations come first from Energy concepts in Physics !.

A very importantly and good question.

I dont think anybody can explain what energy means.

My best offer is:

Energy is a dynamic expression connected to alteration and through this alteration all creation is possible .

Kurt Wraae

I think that energy is what the Coulomb force adibatically induces in charges as a function of the distance separating them.

To be or not to be! In all logic and beyond the specific manifestations in which it appears, energy is apparently only a concept and a mathematical tool for quantifying the conservation of matter and movement, in exchanges, at all scales. A question that seems to me more fundamental, and perhaps unfathomable, is the foundation of inertia.

I don't think that the foundation of inertia is unfathomable.

Energy is defined as a force moving through a distance, or the potential to produce such force moving through distance. It is a scalar dot product of two vectors.

dE = F*dr

Energy may also be defined in terms of velocity and momentum.

F = dp/dt

v = dr/dt

dE = v*dp

It is a scalar dot product of two other vectors. In this example just one direction vector r is defined to simplify the math.

Notice that F and p are in the same direction, but not necessarily in the same direction as r and v.

The two forms of dE are exactly equivalent in all respects as long as the definition of v is unchanged. These are fundamental concepts and can be used in most situations.

Jerry,

(As a discussion point not a personal attack) your answer is a typical circular response. The question is about understanding what energy is, rather than what energy does. Your response, excellent as it is in defining the effects of energy in a mathematical sense, goes nowhere to helping understand what it is.

From my point of view, the question is the most fundamental in physics and developing a detailed knowledge of exactly what energy is will kick start a completely new science.

Dear Ian,

You write: "From my point of view, the question is the most fundamental in physics and developing a detailed knowledge of exactly what energy is will kick start a completely new science."

I agree with you on this issue, but this requires reconsidering the axiomatically established "principle" that energy is an emergent property of motion or velocity.

We have definite clues that the energy of which all massive elementary charged particles are made could be a physically existing substance.

The first clue we got dates back to Walter Kaufman's experiments in 1903:

http://www.hep.princeton.edu/~mcdonald/examples/EP/kaufmann_ngwg_291_03.pdf

in which he demonstrated that the inertia of the momentum kinetic energy of electrons moving at relativistic velocities can be measured in addition to the inertia of the electron relativistic mass when measured translationally, but could not be measured transversally, which means that the kinetic energy that propels an electron is insensitive to transverse interactions.

Contrariwise, the inertia of the energy making up the rest mass of the electron and its relativistic mass increment can be measured as well longitudinally as transversally, which means that it is sensitive to interaction coming from all directions.

This means that "mass" can be defined as "omnidirectional inertia", that is resistance to change of state of motion from interaction coming from any direction.

In 2003, Paul Marmet discovered that the energy that makes up the relativistic mass increment that Kaufman measured corresponds to an increase of the electron magnetic field energy.

http://www.newtonphysics.on.ca/magnetic/index.html

From these discovery, the carrying energy of moving electrons can finally be clearly defined as being electromagnetic in nature, and also that the energy making up the electron mass is of electromagnetic nature. It seems not to be possible that this energy would not physically exist as a real "substance":

http://file.scirp.org/Html/17-7503469_84158.htm

Best Regards

André

Apologies for all the copies of less than a third if my answer to André bit that is the result of trying to post in the deep wilds of Catalonia. On my return I will clean up and finish the post.

Photons are known to only interact with each other when they are dimensionally and temporally (frequency) aligned.

This property is also observed in the way photons interact with matter. Photons interact directly with matter by energising the electrons surrounding atomic nuclei, which are constrained in a spherical path and are thus often aligned with the free travelling photons.

As André states energy and electrons do not interact when their paths are significantly misaligned

Thus all matter can be accelerated, heated, etc by the specific application of energy.

If we assume that photons = energy, then it is simple to calculate the effects of energy on matter. This is clearly demonstrated by Jerry's response above.

However, with our current level of knowledge, the assumption that energy and photons are one and the same is not fully validated. It may be there are different forms of energy that are not photon based. Gravity is one such possibility.

"It is important to realize that in physics today, we have no knowledge what energy is."

− Richard Feynman

To Eric Lord

You are right and You could ad - what it it means.

Kurt Wraae

What is energy?

Energy is “motion” in all its known and as yet unknown manifestations. Motion and matter together are the primary modes of existence in abstract space and time – which are the forms. There can be no motion (energy) without matter and no matter without motion.

The essence of motion "is to be the immediate unity of space and time; it is time really persisting through space, of space which is only made truly distinct through time….. Only in motion have space and time reality.” G.W.F. Hegel, Naturphilosophie, E., p.65.

Hi Abdul

The concept that energy is motion is full of some fundamental flaws. Consider these?

If I look at a stone, stationary on the ground, has it then zero energy?

It may be stationary relative to the ground but if I dig away the soil around it then falls to the bottom of the hole I have dug. So where did it get the energy to fall if I had not moved the stone but only the soil?

Similarly, the stone may be stationary with respect to the Earth, but the planet is both spinning and moving relative to the sun in a large ellipse and if somehow that stone and the Earth were to separate, the stone would appear to fly off at an incredible speed.

Can we still say that the stationary stone has no energy?

I could accept that it has zero relative energy with respect to the Earth but it still has large amounts of energy somehow built within it.

Einstein gave us the concept that all matter has an energy content equivalent to its mass times the speed of light squared and that this energy content is intrinsic to the mass concerned I.E. it is not relative to anything.

Einstein's assumption was that E=mc^2 is a maximum limit, I suspect that it is, in fact, a minimum energy level and motion, heat, radiation etc. are additive to the mass energy.

None of this tells us what energy is. It simply describes various effects of energy.

Ian

Dear Ian,

You cannot find any matter without motion. This includes, kinetic, vibrational, rotational motion in the three state of matter. You have Zero-Point vibrational energy even at absolute zero Kelvin mandated by the Uncertainty Principle. At very low temperature you have the Bose-Einstein condensate with peculiar motion including creeping up the walls of a container. You have motions within atoms, where the electrons and the nucleus are moving, within the nucleus the neutron, protons, etc. are moving; within the neutron and protons the quarks and gluons are moving. Even in absolute vacuum (if at all possible), you have the quantum vacuum, where virtual particle flick in and out of existence! In the cosmic scale you know that evrything is moving! So from the cosmic macrocosm to the tinniest of microcosm, you can find rest nowhere. Quantum dynamics and dialectics forbid absolute rest, for both unceasing motion is absolute and forever anywhere in the universe!

energy should not be only explained by equations . it is source of every thing in the universe, in different forms like it has its many forms in nature.

Abdul

As I responded to Jerry, your response equates an observed effect to a cause without justification.

To say that because we can see motion at both microscopic and macroscopic levels wherever there is energy therefore energy must equal motion is simplistic and does not allow us to understand what energy is at the fundamental level.

A better response to the observed motion is "energy is present wherever there is motion therefore energy is a cause of motion". This allows us to investigate further.

Are there instances where energy is present without motion? Certainly. In the field of a charged capacitor. Gravity is a perfect example. The field of a permanent magnet is another.

Energy was defined above as the capacity to do work. Movement is the result of work. The two are not the same.

It is important to distinguish the many effects of energy from the definition of what energy is, only then can we investigate and understand.

Ian

Ian> "A better response to the observed motion is "energy is present wherever there is motion therefore energy is a cause of motion".

This “better response” of you invoking causality unfortunately leads to the question: what is the “cause of energy” and more questions! In other words it leads to a “first cause” - the “effect” of a “cause”, which is unknown and hence a mystery. In a real sense you are invoking “The unmoved mover” or God; which is no scientific explanation at all!

Kant showed rather conclusively that beyond ordinary life experience, causality leads to contradictions, to antinomies. So, to preserve the revered rationalist notion of causality; Kant rejected objective reality instead; as "unknowable thing-in-itself"! Einstein rejected the "evil quanta" for the same reason as that of Kant! But the fact is that the objective reality is there, in spite of the wish or will of a Kant or a Einstein!

As we know now after Hegel and after the recognition of the quantum phenomena, causality is no guide to positive knowledge; it only leads to mystery. The reality is that the universe, Nature, Life, society and thought are mediated not by cause and effect (as the old world view presumed) but by the dialectical contradiction and the interplay of “chance and necessity” together. The old notion “ex nihilo nihil fit” and the laws of thermodynamics are violated by quantum dynamics and the uncertainty principle. “Chance and necessity" rules from the microcosm of the virtual particles of the quantum vacuum to the macrocosm of the of galaxies, their clusters and super-clusters etc.

Hegel very vaguely anticipated quantum dynamics with his primary dialectical contradiction of “Being-Nothing” – an ontological category. I have attempted an interpretation of this ontological category (“Being-Nothing- Becoming”) of Hegel in terms of the virtual particles of the quantum vacuum in some of my books and journal articles, mainly:

Article: "Ambartsumian, Arp and the Breeding Galaxies: Article Ambartsumian, Arp and the Breeding Galaxies*

Book, “The Philosophy of Space-Time: Whence Cometh Matter and Motion?”

https://www.amazon.com/Philosophy-Space-Time-Whence-Cometh-Matter/dp/984041884X

You cite examples of “energy without motion” i.e., “the field of a charged capacitor and Gravity”. But you must know that any energy (at all) is mediated through either real or virtual (bosonic) “exchange particles” like photons, magnetons, gluons, gravitons etc., which are in constant motion!

Dear Eric,

Note that Feynman's "today" was 50 years ago.

He also wrote:

"There are difficulties associated with the ideas of Maxwell's theory which are not solved by and not directly associated with quantum mechanics...when electromagnetism is joined to quantum mechanics, the difficulties remain."

Ref: Feynman, R.P., Leighton, R.B. and Sands, M. (1964) The Feynman Lectures on Physics. Addison-Wesley, Boston, Vol. II, 28-1.

But in our "today" we may have sufficient knowledge about both energy and electromagnetism to start moving forward again:

http://file.scirp.org/Html/17-7503469_84158.htm

Best Regards

André

Dear Ian,

You write: " However, with our current level of knowledge, the assumption that energy and photons are one and the same is not fully validated. It may be there are different forms of energy that are not photon based. Gravity is one such possibility."

With our current level of knowledge we know with certainty that the energy of which electromagnetic photons and the rest mass of elementary particles is the same "substance".

We know this because "massless" electromagnetic photons of energy 1.022 MeV or more can convert to pairs of massive electron-positron and that such pairs of massive electron and positron metastabilizing in positronium configuration readily convert back to massless electromagnetic photon state.

We also know that the electron's carrying energy (de Broglie's pilot wave) is electromagnetic in nature and is also made of the same substance, since Marmet discovered that part of this carrying energy provides the relativistic mass increment that can be directly related to the corresponding increment of the moving electron magnetic field.

If you read the paper I just referred Eric to, you will understand why.

All supporting formal references are provided.

Best Regards

André

Energy is a ratio that describes the possibility of change in the phenomenon between two spatially defined regions. Energy is equivalent with time and mass. Energy settles down to the least action equilibrium.

Noether’s theorem (https://en.wikipedia.org/wiki/Noether's_theorem) shows that conservation laws in physics are a consequence of symmetries of dynamical equations. Homogeneity of time (ie, the “unchangingness” of fundamental physical laws with the passage of time) is a symmetry. One could define energy as “that which is conserved on account of the homogeneity of time”.

Dear Robert Shour,

Could you please precise what the factor 4/3 is exactly representing?

Has it to do with a mismatch between observation and theory in certain experiments?

With best regards,

Thierry De Mees

Thierry De Mees.

The 4/3 law so called has been used to characterize work by Lewis Fry Richardson who in 1926 found 4/3 scaling in connection with wind eddies. It is not a mismatch between theory and observation; Lewis Fry Richardson, Max Kleiber and Stefan all based their results on observation. Clausius (1857 to 1860 articles); Kolmogorov to account for Fry's result(1941) ; West Brown and Enquist to account for Kleiber's data (1997) and Lawler et al. (2001) based their results on theory and mathematical calculation.

By connecting these different 4/3 and 3/4 patterns, and attempting to find a basis for them, I arrived at a scale invariant universal 4/3 law. I think there is such a law but I think the basis for it needs work. I don't know of any articles besides mine that suggests that there is a scale invariant universal 4/3 law. If the 4/3 scaling law was universally recognized, it would not be possible for 3/4 metabolic scaling to still be considered to be in issue or for the 1997 explanation of West Brown and Enquist to be used instead of the 4/3 scaling law.

I posted an article on a preliminary history of dark energy and the 4/3 law, and also an article on 3 scaling laws, both on RG, that give background. The law is an attempt to explain what happens when energy from a system with X dimensions is transmitted into a system with X - 1 dimensions. An 1838 article by Sarrus and Rameaux first looked at that issue in connection with the production of heat by organisms with constant body temperature. If we take X to be 4 (3 dimensional space plus 1 dimensional motion), then one arrives at 4/3 scaling. Since it looks as though the energy content of 4 dimensions after transmission or change of reference frames is occupying 3 dimensions, it seems to look like a kind of dimensional pressure: 4 dimensions squeezing into 3. This 4/3 scaling law appears to account for 3/4 metabolic scaling. Intriguingly, it also appears to connect to dark energy.

Clausius found that moving gas molecules have 3/4 the mean distance between the same gas molecules when they are not moving; looking at moving gas molecules as being in a 4 dimensional space, the same 4/3 scaling law occurs. The 4/3 scaling law occurs in a preliminary step in the derivation of Stefan's Law (Planck in his book on the theory of heat is easier to follow than Boltzmann who first came up with the derivation.) In my view, the most remarkable anticipation of the 4/3 scaling law occurs in an 1845 paper of John James Waterston. The 4/3 scaling law also arises in connection with the fractal envelope of Brownian motion.

Another aspect of the 4/3 scaling is that 4/3 scaling in a single reference frame is impossible. 4/3 degrees of freedom implies a ratio of the dimensions of 2 systems because a single dimension cannot have a fractional number of degrees of freedom.

Of the attempts I have made to outline these issues, I think the preliminary history I posted on RG s a good place to start. The connection between Minkowski's space time and the 4/3 scaling law is in another article on RG and supplements the preliminary history.

Thank you for your question.

Bob Shour

Thank you, Bob, do you have links to litterature that found the 4/3 law?

Regards,

Thierry De Mees

Thierry de Mees

Thank you for your request.

I found links for a few articles, for the rest I included cites.

In a few overviews I posted on RG, the references include the cites. The Sarrus and Rameaux article and the Clausius 1860 article have no direct links; I attached the originals of those two articles to articles I posted on RG.

My attempted overviews on RG include:

Working Paper A preliminary history of dark energy and the 4/3 law

Working Paper Minkowski space-time, the 4/3 law and dark energy

.Working Paper The Theory of Dark Energy: Introduction

Preprint Three scaling laws: an overview

I have an older (2015) article on 3/4 metabolic scaling arXiv mostly a struggle with the 4/3 law. The articles above on RG are more succinct.

For some source articles:

h. Article (Sarrus1838) Sarrus, F. & Rameaux, J. Rapport sur une mémoire adressé á l'Académic royale de Médecine Bull Acad R Med, 1838, Paris 3, 1094-1100

i. Working Paper Sarrus and Rameaux, 1838, translated into English

i. Waterston

i. Article (WaterstonAbstract1843-1850) Waterston, J. J. On the Physics of Media That Are Composed of Free and Perfectly Elastic Molecules in a State of Motion (abstract only) Proc. R. Soc. Lond., 1843-1850, 5, 604

ii. Article (WaterstonOrig1845in1892) Waterston, J. J. On the Physics of Media that are Composed of Free and Perfectly Elastic Molecules in a State of Motion Philos T Roy Soc A, 1892, 183, 1-79

(1) http://rsta.royalsocietypublishing.org/content/183/1

j. Clausius

i. Article (Clausius1860) Clausius, R. On the Dynamical Theory of Gases Philos. Mag., 1860, 19, Fourth Series, 434-436

ii. Working Paper Clausius’s 1860 article on mean path lengths

k. Stefan’s law

i. Article (Stefan1879) Stefan, J. Über die Beziehung zwischen der Wärmestrahlung und der Temperatur. Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften, 1879, 79, 391-428

ii. Article (Boltzmann1884) Boltzmann, L. Ableitung des Stefan'schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie Ann. Phys., 1884, 22, 291-294

iii. A good explanation of Boltzmann’s derivation is in: Book (Planck1914) Planck, M. The Theory of Heat Radiation (Translator Masius, M.) --- Second Edition P. Blackiston's Son & Co., 1914

l. Richardson:

i. http://rspa.royalsocietypublishing.org/content/110/756/709

m. Kolmogorov deriving Richardson 4/3 law

i. Article (Kolmogorov1941a) Kolmogorov, A. N. The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers (translation of 1941 article in Russian by V. Levin) Proc. R. Soc. Lond. A, 1991, 434, 9-13

(1) http://www.ams.jhu.edu/~eyink/Turbulence/classics/Kolmogorov41a.pdf

ii. Article (Kolmogorov1941c) Kolmogorov, A. N. Dissipation of energy in the locally isotropic turbulence (translation of 1941 article in Russian by V. Levin) Proc R Soc A, 1991, 434, 15-17

(1) http://www.ams.jhu.edu/~eyink/Turbulence/classics/Kolmogorov41c.pdf

n. Kleiber’s law (3/4 metabolic scaling)

i. Article (Kleiber1932) Kleiber, M. Body Size and Metabolism, Hilgardia, 1932, 6, 315

(1) http://hilgardia.ucanr.edu/fileaccess.cfm?article=152052&p=VOWQRB

ii. Article (Kleiber1947) Kleiber, M. Body size and metabolic rate Physiol Rev, 1947, 27, 511-541

iii. An attempt to account for 3/4 metabolic scaling: Article (WestBE1997) West, G. B.; Brown, J. H. & Enquist, B. J. A General Model for the Origin of Allometric Scaling Laws in Biology, Science, 1997, 276, 122-126

o. 4/3 fractal dimension of Brownian envelope: Article (LawlerDimBrowian2001) Lawler, G. F.; Schramm, O. & Werner, W. The Dimension of the Planar Brownian Frontier Is 4/3 Math Res Lett, 2001, 8, 401-411

i. http://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2001/0008/0004/MRL-2001-0008-0004-a001.pdf

Best wishes.

Bob Shour

"What is energy?" This question has been raised many times along with the questions "What is space?", "What is time?", "What is mass?" and so on. One response among the many contributions above is "In one of its oldest forms, "energy" is the ability to do work, i.e. to set something in motion (relative to something else). I believe this definition still stands. "

This "definition" , of course, doesn't work because it requires the definitions of the two terms "ability" and "work" in the context of "energy" which only leads to circularity without ever arriving at a true definition in terms of primitive concepts. Like "space" and "time", "energy" is its own primitive, given as an aspect of the Universe emergent from the origin of the Universe. Alternatively it is an ad hoc construction to provide gains and losses by which to allow dynamic (time-dependent) processes to occur by which the Universe evolves from state of being to state of being with advancing time. "Energy" is quantifiable and can arrive in parcels (photons) or it can flow as a fluid as in the transfer of heat. The basic property of "energy" is that it can appear in many forms - light, heat, motion - to suit whatever be the problem at hand. So it may be that I don't know how to define "energy" but I do recognize it when I see it.

Dear Robert,

Thank you for the information.

I have looked at Richardson's paper. It is about the diffusivity constant K for wind, which was found by Fick as being 0.2 cm²/s and 10^5 cm²/s for wind, by several authors.

At the page 724, it is clear that K is everything but a constant, and that it differs with the length scale.

The 4/3 exponent in fig 8 is an empirical curve fit in the drawing itself. Indeed, the value l^(4/3) is just a figure without any physical meaning, given for lengths in centimeters, and values between 1 m and 10 kilometers. Indeed, centimeters to the power 4/3 doesn't mean anything.

To me, the whole thing should be analysed in terms of kinetic energy related to temperature, which may add a number of parameters to the equations (at least the particles' velocities).

It is possible that the number of degrees of freedom differ between the motions inside a lab equipment, the motion near the Earth's surface (2 to 2.5 degrees), and the motion in the far atmosphere (close to 3 degrees).

That might explain why the curve on fig 8 is not linear. Also remark that there are few reference measurements, and only one for small distances.

As a conclusion, instead of adding much mystery, I would rather treat this 4/3 subject with the required scientific care...

With best regards,

Thierry De Mees

Thierry De Mees

Thank you for your most recent post. I am glad that you took the time to read Richardson's paper. It seems to me that your reading of Richardson’s 1926 paper applies a greater scientific background than I can muster; when I read it I was struck by the 4/3 exponent. I agree that a skeptical approach (scientific care) is required.

For Richardson’s 4/3, Kolmogorov arrived at a 4/3 exponent based on a mathematical approach using first and second moments in his 1941 papers, which I find difficult (so far) to follow. Korner in the Pleasures of Counting showed Richardson’s 4/3 law using dimensional analysis, consistent with 4/3 as a ratio of dimensions.

My history with the 4/3 law goes back to April 2008. May 2007 I found that the mean path length scaled a network of words. I did not believe it, because of implications and I could not believe it had been overlooked and found by someone as unknowledgeable as me. But it seemed to work and I submitted a paper on lexical growth to arXiv, still skeptical.

In March 2008 I read about 3/4 metabolic scaling and then read (many times) West Brown and Enquist (WBE 1997). If mean path length scaling was valid, then on applying it to metabolic scaling 3/4 might pop up. But not so. First, the log of a network of size n grows in size as n increases, while the 3/4 power of mass in metabolic scaling is constant. After adapting mean path length scaling, I applied the adapted scaling to a circulatory system and got a 4/3 exponent not 3/4. After a week of adding backwards and forwards, feeling algebraically incompetent I realized that the circulatory system grew by a 4/3 power while the organism’s mass scaled linearly. I thought (mistakenly) I had merely gotten the same result as WBE.

In mid April 2008 I read in Allen and Maxwell’s text on heat (p. 742) their derivation of Stefan Boltzman Law and 4/3 entropy in an intermediate step. That suggested that the 4/3 inadvertently stumbled on in 3/4 metabolic scaling might be universal in isotropic (black body) energy distribution. In August 2008 I had what I considered a crazy notion that 4/3 scaling might relate to cosmological expansion.

In the fall of 2010 I read two articles by Kornazewski and Kozlowski critiquing WBE 1997. Around the fifth reading I began to think they were right. I went back to my 2008 paper on 3/4 metabolic scaling and found that the math leading to the 4/3 was inexplicable; I could not figure out where the extra 1/3 in 4/3 came from or how to fix the mathematical inconsistency leading to it. It took me until 2015 and 2016 to use degrees of freedom and dimension for a possible explanation. My arXiv article on allometry went through 17 versions from 2008 to 2015 most of the later versions trying to figure out the source of the 4/3 (in addition to similar efforts in a paper on energy scaling).

In July 2011 I was reading Coopersmith on Energy the subtle concept and she mentioned Clausius’s work on mean path lengths (but no cite). I found the English translation in the Philosophical Transactions. Clausius stated and later proved (1859, 1860) using trigonometry and calculus that the mean path length of moving particles is 3/4 that of the same particles where only one is treated as moving. His article supported the idea of 4/3 scaling, because four 3/4 length paths fit into a path 3 units long. In other words, in a 3 degrees of freedom system a length from a 4 degrees of freedom system is 4/3 as long. Clausius’s result was not included in the Brush compilation of papers on kinetic theory, probably because Maxwell showed it did not work to give an accurate answer for specific heat.

I first read Sarrus and Rameaux 1838 around January 2016, Waterston in January 2017. If there is a 4/3 scaling law, then Waterston’s paper may be the first paper (in 1845) that invokes it, in a completely idiosyncratic way.

I read Richardson’s 1926 paper around July 2016. When I read Hunt’s article on Richardson around the same time that was the first time I saw the 4/3 law being used as a descriptor.

When I consider my years of skepticism, it seems to me reasonable that someone reading a paper for 15 minutes or so is skeptical.

The 4/3 scaling law and related ideas (if anthropomorphized and if valid) have the misfortune not to have a more capable and eloquent advocate. So far I have explored many implications of what might be a non-existent 4/3 scaling law. I am inclined to think it is not entirely wrong nor delusional but only the eventual collective opinion counts.

Of my articles that attempt to present the 4/3 scaling law til now (May 2018), I like the preliminary history most, but I entirely discount my opinion.

Thank you again.

Bob Shour