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Introduction

The dimensions and degrees of freedom of system underlie many natural phenomena. Dimensional capacity appears to play a role in biological scaling including metabolic scaling, in collective intelligence, in black body radiation, wind gust scaling, Brownian motion, dark energy and other types of systems. Models for diverse systems discussed in the various articles on RG rely on dimension and degrees of freedom. In particular, the principle of dimensional capacity is explicitly relied on.

**Skills and Expertise**

Education

September 1973 - June 1976

September 1969 - May 1973

## Publications

Publications (148)

Since 4/3 scaling, which applies to bounded, finite systems, as in metabolic scaling, Peto's paradox and brain weight scaling, also applies to unbounded expanding cosmological space, 4/3 scaling is probably valid and the mathematics of WBE 1997 is probably wrong.

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.

This paper intends to fill in a logical gap in a 2009 paper, A theory of intelligence. The mean path in a social network measures degrees of separation, or social distance, in steps or social connections, between pairs of people in the same generation --- horizontally. The principles underlying mean path length between contemporaries apply for cons...

4/3 scaling applies to Kleiber's Law and the expansion of cosmic space.

A one page (nutshell) derivation of Kleiber's Law based on degrees of freedom.

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.

PDF of some of the original emails. Printed using an outlook reader evaluation copy from https://www.coolutils.com/TotalMailConverter.

In July 2005, after studying inference and language, I hypothesized that average IQs increase because the compression of information in ideas increases. In September 2005, I emailed James Flynn, of the eponymous Flynn Effect, about this hypothesis. After the September exchange of emails, I tried to find evidence for the hypothesis and an equation a...

Numerous instances of allometric quarter scaling occur in plants and animals. This article makes the case for quarter scaling and related 4/3 scaling as universal physical laws.

Mathematics implicit in allometric and similar scaling is here made explicit. Explicit mathematical modeling implies the physics applicable to phenomena that scale. The mathematics is demonstrated using examples from Galileo's \textit{Dialogues Concerning Two New Sciences} and Sarrus and Rameaux's work on how rates of heart beating scales with size...

An equation based on network entropy predicts a national economic growth rate from the rate at which national average IQs increase.

A society's problem solving intelligence can be quantified using its mean path length $\mu$ and clustering coefficient $C$. Societies and their ideas are small world networks. Knowledge transmission, acquisition and exchange occurs in human societies and from stored written records. $\mu$ and $C$ together with the principle that network degrees of...

The speculation by some philosophers and physicists that consciousness is unsolvable using physics can be refuted by the statistical mechanics of problem solving.

Scaling ideas of Galileo and of Sarrus and Rameaux translate, using ratios of dimensions, into an algebra of compared scale factor exponents. That algebra reveals that the scaled circulatory system of a similar larger animal has dimensional capacity to distribute energy that is 4/3 greater than the animal's scaled dimensional capacity to use that e...

Scaling ideas of Galileo in 1638, Sarrus and Rameaux in 1838 and West Brown and Enquist in 1997, modified and generalized, derive Kleiber's Law.

Comparing scale factors may help analyze metabolic scaling. This approach implies circulatory systems distributing blood-energy-scale by a 4/3 power of body mass. Invariance of the rate of energy use per cell, regardless of mammal size, requires 3/4 scaling of 4/3 scaled energy distribution. Summary

By analogy to the findings of Swadesh's glottochronology, facial expressions used by members of society likely diverge from the expressions used by the ancestral society at the rate of 5.6% per thousand years. If so, that slow rate and the difficulty of detecting it would explain the suggestion that facial expressions relating to emotion are innate...

Estimates of Omega_m after Betoule 2014 are also consistent with a 4/3 ratio of dimensions accounting for dark energy.

A 4:3 ratio of dimensions can account for expansion of the universe. Supportive astronomical theory and data are outlined.
The article, Deriving Kleiber's law from its history, recently posted, lays foundations for this article and would be good to read first.

A succinct derivation of Kleiber's Law.
More detail can be found on RG, including in Deriving Kleiber's law from its history

Scaling ideas used by Galileo in 1638, by Sarrus and Rameaux in 1838 and by West Brown and Enquist in 1997 help derive Kleiber's Law.
The paper is less than 5 1/2 pages.
Ideas in this paper lay the foundation for the more recent: RG article, Dark energy modeled by scaling

4/3 laws are often based on a ratio of the degrees of freedom of two systems. Applicable phenomena include dark energy, 3/4 scaling of metabolism (Kleiber's Law) and of brain weights, 4/3 scaling of wind eddies, isotropically radiated energy, Peto's cancer paradox, information distribution in social networks. How should 4/3 laws be introduced? By g...

This is a critique of Hayflick’s limit and the allometry of mammalian aging DOI: 10.13140/RG.2.2.33994.49609/4.

This is a revision of an article first posted on RG in December 2020 on the individual’s rate of thought, clarified, reorganized, and (hopefully) improved.

Heinrich Hertz observed that an equation, once discovered, sometimes seems wiser than its discoverer. E. T. Bell's paraphrase of Hertz's observation is often cited. This article gives the provenance of Hertz's original remark: ``Man kann diese wunderbare Theorie nicht studieren ohn bisweilen die Emfindung zu haben wohne den mathemitschen als seien...

The degrees of freedom of a network that distributes energy measures network output capacity. The degrees of freedom of a network using that energy measures use capacity. The ratio of degrees of freedom --- 4 --- of a system distributing energy and the corresponding system using energy --- with 3 degrees of freedom --- gives a 4/3 ratio of capaciti...

Degrees of freedom of an energy transmitter quantifies its capacity to distribute energy. Degrees of freedom of an energy receiver quantifies its capacity to use energy. The degrees of freedom of a transmitter increases with size by 4/3 times the degrees of freedom of the receiver: there are economies of scale, because transmitting capacity grows w...

English translation from the original German of Ludwig Boltzmann's 1884 article deriving Stefan's Law.

If the Hayflick limit is approximately invariant for normal mammalian cells for all mammalian species, Kleiber's Law applied to the Hayflick limit suggests an allometric scaling applies to mammalian ageing. In particular, human longevity can be allometrically estimated based on mouse longevity.

Dimensional capacity and its corollary 4/3 scaling laws can resolve Peto's 1977 cancer paradox.

The same universal 4/3 law, itself a consequence of the principle of dimensional capacity, underlies 3/4 scaling of brain weights and metabolisms.

English translation from the original German of Otto Snell's 1892 article.

The average individual human problem solving rate contradicts existence of a language organ or instinct.
This article adopts and expands on ideas in a book, The Intelligence of language, in a 2008 article on Lexical growth, and in a 2009 article on a Theory of intelligence to ground an argument against the hypothesis of a language organ.

Society's rate of collective problem solving R can be approximated using outputs such as lexical creation and improvement, technologies and economic growth. Three factors generate society's collective problem solving capacity: networked brains, concepts and average individual problem solving capacity r. If multiplicative factors representing networ...

Average IQs since about 1970, lighting efficiency expressed in lumens per labor cost per hour from 1750 B.C.E. to 1992, and the English lexicon from 1657 to 1989 all increased at similar rates, about 3.41% per decade. PGA Tour average drive distances from 1980 to 2018 increased by about 3.66% per decade. Are the rates measuring the same effect? The...

Here outlined are possible problems with a 4/3 law based on dimensional capacity and possible tactics for addressing those problems. 4/3 scaling relates to metabolic scaling, wind eddy scaling, Brownian fractal envelope and astronomical observations denoted dark energy, among other phenomena.

A society's management of its problem solving resources is analogous to management of financial assets. And vice versa.

The principle of dimensional capacity may qualify as one of `other laws of physics hitherto unknown' referred to in Schrodinger's What is Life?

Inferences implying the existence of dark energy are based on astronomical observations. The elusiveness of an explanation is due to the current omission from physics of principles based on dimensional capacity. Those principles imply that the same energy appearing in four dimensions has 4/3 as much energy per dimension in three dimensional space.

Society having a whole generation educated in school to be informed might improve society's ability to respond to a pandemic and collectively solve problems a pandemic raises.

Metabolism is slower for larger organisms than for small ones. Since the 1880s, if not earlier, a fractional exponent $b$ of organism mass $M$ has mathematically represented this physiology as metabolism $Y \propto M^b$ with $b<1$. Researchers since then have attempted to determine what value $b$ has and why. Instead of starting from $M^b$, after l...

The concepts of network entropy and clustering coefficient may assist in modeling virus transmission within societies.

Mathematical concepts compress solutions to problems involving natural and other phenomena. Hundreds of generations of human society have tested those solutions for their accuracy and utility and have continuously refined them. Those concepts likely often reflect fundamental aspects of the universe including phenomena not yet modeled, perhaps not y...

Astronomy infers existence of an unknown force denoted `dark energy' accelerating the expansion of space. Astronomy assumes matter decelerates expansion. Instead, apply by analogy the method of inverting conceptual reference frames and treat expanding space as the inertial reference frame. Gravitation appears as an apparent force because matter coh...

An animal's circulatory system and system of alveoli by theory should both scale with size by a 4/3 exponent.

The network effect of sharing losses among economic actors might increase chances of keeping more of the economy intact until the COVID-19 epidemic has run its course. The concepts of network entropy and dimensional capacity assist analysis.

In poetry, sometimes there are parallel structures and patterns based on words, phrasings, appearance (thinking of e. e. cummings): rhyming words, syllables per line, word play, witty and unexpected word pairings and contrasts, and so on. In physics there are also parallel patterns and structures, but instead of rhyming words, there are general or...

This about someone's great idea: a heat map to track the whereabouts and potential whereabouts of the virus.

Cell phones are almost ubiquitous. Soon, COVID-19 might also be ubiquitous? Can we use cell phones to fight epidemics?

Identifying everyone who is infectious is arguably our best strategy now. Universal and ongoing identification with enforced isolation of infectious persons might allow resumption of social and economic life.

This article raises questions: is identification, detection and isolation superior to mass social distancing? And is it feasible at this advanced stage of epidemic spread?

The role of the formula for network entropy suggests a way for the concept of the soul to arise. The word `soul' reifies absence, upon death, of a person from a social network. The effect of a person after they are absent from a network persists: soul.

This article gives an over view of, and references, several papers that provide an explanation of dark energy. A constant $4:3$ ratio of dimensions for contemporaneous spatial reference frames can account for expansion of the universe.

Two assumptions, the rate of spread and when diagnosis is possible permit a simplified model of an epidemic. The simplified model facilitates consideration of policy options for containment of the epidemic. The simplified model implies that international travel restrictions for a virus like COVID-19 can help slow the spread of the epidemic.

Network effects arising from individual action raise hope that in those actions individuals collectively and emergently will help limit the spread of COVID-19.

Actions by individuals to inhibit the transmission of COVID-19 may have substantial collective effects. The concepts of dimensional capacity and network entropy play a role in reaching that conclusion. In the case of person to person or proximity disease transmission, increasing COVID-19 transmission path length might reduce the rate at which the d...

This article compares COVID-19 transmission means to the dissemination of information over the internet as a means of combating COVID-19, not from the perspective of a health professional, but from a network perspective.

This paper is intended to lay out a theory that accounts for the expanding universe and provide reasons for the plausibility of the theory. The theory is based on dimension. Justification in part arises from noting that the 4/3 scaling that accounts for cosmological expansion is a universal law of nature, based on dimensional capacity, a fundamenta...

The formula for network entropy enables calculation of the average rate of individual human problem solving, absent language and social networking, about 5.6\% per thousand years. This article discusses some implications of an individual rate that low.

A formula for network entropy is analogous to the formula for thermodynamic entropy. Network entropy relates the rate of collective problem solving to the average rate of individual problem solving. By analogy network entropy can be applied to the genius concept in two ways. The first analogy is based on the relationship of the component factors in...

The principle of dimensional capacity observes that the capacity of a system to contain weight, heat, energy or information is proportional to its dimension, and more generally, to the system's degrees of freedom. The principle of dimensional capacity provides an alternative explanation to the many worlds hypothesis. The capacity exists for many wo...

Clausius named entropy, a concept he derived, in 1865. Ideas that arose after 1865, namely (1) mean path length in steps, (2) Boltzmann's logarithmic characterization of entropy, (3) Jensen's inequality and (4) dimensional capacity, help simplify the concept of entropy as it was originally presented by Clausius. Simplification has conceptual and pe...

A problem in the modeling of 3/4 metabolic scaling is analogous to the problem addressed by cosmological inflation. Perhaps both are resolved by the principle of dimensional capacity.

The principle of dimensional capacity relates to Shannon's definition of the capacity C of a discrete channel in his 1948 article on the mathematical theory of communication.

In Clausius's definition of entropy, he divides heat change by temperature. Why? This article attempts to address that question. The reasons appear to be: more degrees of freedom increases the capacity of a system (the principle of dimensional capacity). By assumption, an ideal heat engine is optimally efficient. Jensen's inequality implies that a...

The principle of dimensional capacity is not part of physics at December 2019. However, the principle of dimensional capacity seems to underlie simple modeling of 3/4 metabolic scaling, the 4/3 fractal envelope of Brownian motion and (so-called) dark energy. Perhaps convenience afforded by the principle of dimensional capacity outweighs the skeptic...

A very short paper.
An analogy of the universe to a Carnot ideal heat engine without the piston, so that the "working substance" in the heat engine "chamber" expands in an unbounded way when energy is added to the heat engine chamber.

An average individual rate of problem solving helps estimate that language is about 150,000 years old. Estimates of the average individual problem solving rate derive from glottochronology, the rate of phonemic change, and, using a statistical mechanical method, indirectly from society's collective problem solving rate.
Earlier papers considering n...

Lengths, areas and volumes scale differently because of their different dimensionality. A dimensional point of view provides a perspective different than that of scaling on the effect of an increase in size of a system having length, area and volume. Dimension is more fundamental since it is dimension that induces scaling when length increases in a...

This article reviews different, but not all independent, ways to arrive at the role of a network's mean path length as scale factor for network distribution of energy or information. Quantitative and theoretical uses of mean path length are briefly summarized in a separate paper posted in August 2019 on ResearchGate.
This paper therefore sets out...

A recent study finds that, based on a sample of 17 languages, the average rate of transmission of information is the same for all languages. That average rate of transmission of information and the rate of English lexical growth both appear to be instances of a universal collective problem solving rate. This paper explores qualitative reasons for i...

Using the mean path length as a scale factor leads to a formula for network entropy. Mean path length scaling has uses both quantitative and theoretical. Quantitative uses are available because the mean path length can be measured and in some cases can be estimated based on theoretical considerations. Theoretical uses mostly relate to the idea of d...

Ever so slightly changing Galileo's conceptual reference frame for explaining the effect of increased animal weight on the cross-sectional area of weight-bearing bones leads to a possible explanation of dark energy.

This article is a follow-up to the article, Why scaling and not dimension, Galileo? Galileo used scaling to examine how animal size affects the thickness of animal bones. Scaling was sufficient for that purpose. Does a dimensional standpoint have advantages over scaling? Dimension might provide a conceptual reference frame sufficient to answer ques...

Using a scaling standpoint, Galileo explained why an increase in animal weight requires thicker weight-supporting bones. He could have adopted a dimensional standpoint. Why didn't he? In a logical progression, Galileo generalized and extended ideas about size and scaling in his book \textit{Two New Sciences} to animal size. Scaling provided a conce...

Sir Lawrence Bragg wrote: ``The fun in science lies not in discovering facts, but in discovering new ways of thinking of them.'' This article gives a short account of the provenance of his remark.

A mean path length in a number of steps equal to the natural logarithm optimally scales an isotropic network. One possible implication for discretely named subsystems of an organism is that they must overlap, if the subsystems cannot or do not connect by a mean path length equal to the natural logarithm. The July 2019 issue of Scientific American m...

Network entropy as a concept derives from thermodynamics and statistical mechanics arising out of it emerges that a network's mean path length can scale the network. Network entropy connects the rate of English lexical growth and glottochronology. Does network entropy also connect lexical growth with phonemic diversity? It seems not. But the rate u...

Different approaches imply that two contemporaneous differently dimensioned reference frames exist in a variety of contexts. The existence of two cosmological reference frames in turn implies that three dimensional space expands, an astronomical phenomenon called dark energy. Astronomical observations are consistent with a two reference frame solut...

The two reference frame solution, if applicable, implies that the cosmic scale factor $a(t)$ is a constant that does not vary with time. Instead $a(t)$ is a consequence of a ratio of dimensions of two differently dimensioned homologous systems. The physics is related to the principle of dimensional capacity and an invariant ratio that applies to th...

This is an introduction to articles planned to discuss the two reference frame solution.

The Hart-Fuller Debate consists of two articles in the Harvard Law Review in 1958. Hart supported legal positivism: laws are no more than what societies formally promulgate. Fuller argued to the contrary that laws must conform to fundamental principles embodied by natural law. Relocating the Debate to a conceptual landscape that adds collective pro...

This article attempts to situate ideas and phenomena relating energy and dimension as outcomes of general principles and general methods. Instead of a scaling conceptual reference frame, the conceptual reference frame is based on dimension and an invariant ratio of features of related structures and processes. The invariant ratio approach appears t...

A diamond anvil cell is able to create a large pressure P=F/A$ with a moderate force F applied to a small area A. The trick is to increase pressure not by increasing force F but by greatly reducing area A in the denominator. In problem solving, the same problem solving effort, analogous to F applied to a problem smaller in scope, analogous to A, pe...

Using the dimensional standpoint that energy equally distributes among available dimensions in corresponding systems leads to the idea of an invariant cosmological size ratio. An invariant cosmological size ratio is inconsistent with a singularity at the time of the origin of the universe.

Implications that derive from the principle of dimensional capacity lead to a formula for network entropy, the connection of the natural logarithm to isotropy, a basis for Kleiber's Law, and, possibly, a theoretical basis for expanding space. These implications are not yet, it seems, part of accepted physics. If the implications are sound, then the...

A ratio of dimensions plays a role in Galileo's 1638 discussion of the strength of materials, and in the 1838 work of Sarrus and Rameaux about why breathing is slower in larger animals. The concepts developed by Galileo and by Sarrus and Rameaux suggest existence of a principle of dimensional capacity, that a system that receives heat, energy or in...

Distribution of energy in an animal by a circulatory system and in the universe by radiation (which expands the universe) have analogous properties. Both may be modeled by a $4:3$ ratio of entropies or more simply as a ratio of the dimensions of a transmitting system and a corresponding receiving system. A circulatory system distributes and supplie...

This article attempts to formally word a question arising from a dimensional standpoint about the scaling of corresponding systems-when heat, energy or information transmits from an x + 1 dimensional system to an x dimensional system.

West Brown and Enquist's 1997 paper (WBE 1997) describing a general model of allometric scaling is important. Kozlowski and Konarzewski's 2004 critique (KK 2004) raises objections. Each set of authors wrote a follow up article. The four articles are a scientific debate. Resolution may require a mathematical derivation different than WBE 1997's.

The dimensional standpoint giving a plausible explanation of 3/4 metabolic scaling, the theory (the principle of dimensional capacity) and the application (deriving 3/4 metabolic scaling) mutually reinforce the validity of each.
The 3/4 scaling of mass is a 3/4 scaling of the 4/3 scaling of energy supply capacity arising from the ratio of a 4 dimen...

Developing a notation for use in understanding the principle of dimensional capacity may assist in characterizing and understanding it and working out its implications. Not only does dimensional capacity relate to the paradigm example Galileo gave relating to weight and bones, it relates in a similar way to the astronomical observation of expanding...

Adapting the dimensional approach used in the 1838 work of Sarrus and Rameaux yields a succinct theoretical basis for Kleiber's Law that basal metabolism $Y \propto M^{3/4}$ for animal mass $M$.
The point implicit in Sarrus and Rameaux 1838 is that the scaling they described scaled the higher capacity of supply, heat supply in their article, energy...

Languages are resistant to abrupt and material changes because the enormous utility of languages depends on their stability during a person's lifetime. The skepticism that is part of the implicit infrastructure of language likely begets the skepticism of science when science is confronted with revisions to existing ideas and with new ideas. What is...

Two perspectives on analyzing the effect of growth on physical and biological systems are (1) scaling and (2) dimension. The scaling perspective seems mathematically unexceptional. The dimension perspective for the same growth, yielding the same results, implies that the capacity of a system is proportional to its dimensions. But the dimension pers...

The Science section of the January 3, 2019 issue of The Economist discusses plans for astronomical observations intended to shed light on dark energy. The article below suggests that a possible explanation for dark energy may have roots in Galileo's Two New Sciences (1638) and may be less exotic than supposed.

Inverting a conceptual framework can sometimes help solve a problem in theoretical physics. In particular, inverting conceptual reference frames may help account for dark energy. Redshift and luminosity distances for type 1A supernovae differ. The relationship of the two distances implies a force causing space to grow --- dark energy. Instead of su...

## Questions

Questions (436)

Can you give possible examples or instances? Are there any journal eeferences?

This is a generalization and extension of a question posed a couple of months ago:

In Two New Sciences, 1638, Galileo applies his analysis of the strength of a beam to an animal’s weight bearing bone. For a larger animal, weight scales by 3 when cross sectional area of bone scales by 2.

The relative dimensional capacity to contain or impress W varies by exponents 3:2, or (D+1)/D.

Question. Does (D+1)/D extend from D+2, Galileo’s case, to D+3?

Minkowski’s space time can be considered a D=3 example.

If extension to D=3 were valid, then the ratio of dimensions may explain 3/4 metabolic scaling, Stefan’s Law, Peto’s Paradox, brain weight scaling, the fractal envelope of Brownian motion, and expansion of cosmological space, among other problems.

An exploration of this question is in:

What supports (D+1)/D working for D+3? What undermines it?

Is the extension to D= 3 valid?

Can this discussion point be upheld?

A paper that is related is:

Preprint Thermodynamic advantages of morality

Or a tool used in various different kinds of arguments?

Or an equivalence class of arguments that use comparisons and similarities?

I refer to :

Bartha, P. (2013). Analogy and Analogical Reasoning. Stanford Encyclopedia of Philosophy. E. N. Zalta. Stanford, Stanford University: 1 - 69. URL = http://plato.stanford.edu/archives/fall2013/entries/reasoning-analogy

This question relates to the characterization of analogy generally.

For example, in English pronunciation, b is a vocalized variation and analogy to p.

Formation of analogous phonemes would facilitate teaching and learning them.

Are there articles which specifically deal with these kinds of analogies?

A review of analogies used in high energy physics is a 2020 Ph.D dissertation by Gunnar Kreisel, Analogies in Physics — Analysis of an Unplanned Epistemic Strategy, Gottfried Wilhelm Leibniz Universitat Hannover.

Are there any books or articles for analogies in classical mechanics?

For example, how long did it take for the market to adjust the price of automobiles relative to horse drawn carriages, so that their price/benefit ratios were the same.

Similar questions apply to steam engines, electricity, digital computers, the internet and so on.

This question relates to the question of how long it takes for stocks in a broad based ETF fund to reflect market conditions?

What are books and articles that discuss this?

My suggestion is: trying to solve it in a conceptual reference frame that is not optimal.

Here is my example. Kleiber’s Law is Max Klieber’s empirical inference that metabolism scales by a 3/4 power of mass. Accordingly, much effort has been invested in trying to deduce a 3/4 exponent from a mathematically based reasoning. An example is the geometric, fracctally based reasoning in A General Model for the Origin of Allometric Scaling Laws in Biology , 1997, Science , Vol. 276. The 3/4 power relates to energy use. Energy use is the conceptual reference frame. Instead, it appears that a better conceptual reference frame focuses on how much energy distribution capacity increases with increased animal size. In that case, the 3/4 scaling of the rate of metabolism is how evolution responded to the 4/3 scaling of energy supply, to render energy per cell invariant. This is discussed in:

Preprint Size, scaling, and invariant ratios

Other examples:

The laws of motion without the concept of inertia (Galileo’s marbles experiments).

The nature of heat without connecting energy, motion and heat.

Equating redshift and luminosity distances for SN 1A. I suspect this is a conceptual reference frame problem.

Do you have other examples?

The question above arises out of a recently posted preceding question.

Stam Nicolis replied to the question, posted recently: If space-time validly joins two fundamental aspects of the universe, shouldn't there be abundant four dimensional analogues?, in part: "No-there's a distinction between spacetime symmetries and other symmetries."

Is the question here different that the original question? If the question is different, are there systems that are analogues to space-time symmetries and invariances?

The article,

seems to imply that there are 4D analogues.

Combining space and time into four dimensions demonstrates invariance under Lorentz transformation.

Space-time being so fundamental an attribute of our universe, there must be numerous analogues. What invariances apply to them, if any?

This is a corollary to the question: https://www.researchgate.net/post/Apart_from_space-time_are_there_any_other_mathematical_models_of_physical_phenomena_using_a_fourth_dimension

And:

Preprint Size, scaling, and invariant ratios

Space and time being so fundamental, shouldn’t four dimensional analogues all be in plain sight?

Where are they? Has physics overlooked or missed them?

This is a follow up to the question:

And see:

I searched yesterday and could not find any references, apart from hypercubes etc, to mathematical modeling using 4 dimensions other than my articles on arXiv and RG. That may explain why the role of 4/3 scaling has been unnoticed by physics.

I think a fourth dimension does play a role in modeling:

3/4 metabolic scaling.

Peto’s paradox

Brain weight scaling

4/3 fractal envelope of Brownian motion.

Clausius 1860 article on gas molecular mean path lengths.

Waterston on the energy to maintain a levitating elastic plane in a gravitational field (Roy Soc 1892 publication of 1845 submission).

Dark energy.

Are there any others?

Several articles on RG discuss 4/3 scaling, which involves the 4th dimension, including:

Preprint Dark energy modeled by scaling

Preprint Flow as a fourth dimension

and several other RG articles back to .

This question arises out of proceeding question: Are information exchange and collective appraisal of ideas necessary for the emergence of collective consciousness?

If networked computers acquired collective consciousness, then their faster collective computing speed might enable emergence of a collective electronic consciousness more powerful than that of human society?

Is that a risk?

Would regulations about networking computers mitigate risk?

The 1992 movie The Lawnmower Man, and the Skynet network in the movie Terminator, have plots that raises similar issues.

This question arises out of the essay

Preprint Collective consciousness first

That essay observes thatlanguage is invented and improved by society collectively. The collection of ideas thus enabled creates a collective consciousness. Individuals then can acquire individual consciousness by acquiring language and ideas from the collective consciousness.

The essay further observes that collective appraisal is more indicative of collective problem solving than is invention that adds to a society’s lexicon and store of ideas.

Perhaps networking that enables collective problem solving is not just a means of collective problem solving but is necessary for consciousness itself. Perhaps appraisal of new words and ideas proposed by an individual or specialized group can enable the development of collective consciousness only if the words are repeatedly and continuously tested by the entire network. In other words, does deciding whether a new idea is added to the network’s collective information resources require appraisal by the rest of the network acing as an audience for consciousness to emerge?

And if networking combined collective appraisal leads to collective consciousness for human societies, should we be wary of networking computers?