There exist numerous conflicting heuristic models to explain Zipf’s law, which is an empirical inverse frequency power law that is ubiquitous in biological, social, and other natural phenomena characterized by complex system dynamics. Given its applicability in diverse systems, it should be possible to derive it as a model-free signature of self-organized criticality. This is what we do in this present paper by proposing a new law f( x ) ∝ x^ - e/3 , which is based on information-theoretic considerations, and it is nearly identical to Zipf’s law. The new law provides a basis for universal criticality as an indication of information optimality, and it also provides a specific mathematical function for Zipf’s law, which is normally stated only as an inverse correlation.

The helical coil is ubiquitous in biological and natural systems, and it is often the basic form in complex structures. This paper considers the question of its dimensionality, D, in biological information as the helical coil goes through recursive coiling as in DNA and RNA molecules in chromatin, in which the D-value is a function of the lengthening of the curve. It is shown that the dimensionality of coiled coils is virtually equal to e. Of the three forms of DNA, the dimensionality of the B-form is nearest to the optimal value, and this might be the reason why it is most common.

It is known that e-dimensionality is optimal for information and by extension for space, and here we ask if this has implications for the dimensionality of time. Given its role as a marker of change, any assumed structure for time may appear arbitrary, but considering the swapping of time and space inside a black hole, it becomes possible to investigate this question. It is proposed that the phase of the normal time, the swapped time, and the transition stage are broadly the types that we can associate with time, and they may exactly equal the number e. The idea that information is stored in the e-dimensions of time swapped into space becomes a solution to the black hole information paradox. (c) 2025 Physics Essays Publication. [http://dx.doi.org/10.4006/0836-1398-38.3.216]

This paper argues that noninteger dimensionality creates a hitherto unexplored source of nonlocal noise, called ND noise or NDN, that is likely to play a role in measurements at very small distances. New analysis on scale invariant probability distributions arising from noninteger dimensionality is presented. The significance of this when considering performance and noise-correction coding in models of quantum computing is indicated. It is argued that NDN noise will lead to decoherence even without interaction with the environment and it is likely to be relevant also in models of cognition.

An optimal representation constitutes an efficient set. It is known that for an aggregating system if the cost of representation increases linearly with the number of bases, ternary coding is superior to binary, and coding in e is optimal. This paper investigates the relative efficiency of bases for the cases when the cost complexity is affine (slope–intercept linear), exponential, and logistic and presents new results. It is shown that for representation of structure in logistic maps, which applies often to biological systems and is true for input–output maps of neurons, the optimal base value is near 1.7632, which is consistent with the unary and space coding of information in songbirds. It is shown that the mathematical basis of this result is the solution to the equation b^b=e.

We review the use of optimality in the investigation of the large-scale geometry of the DNA-protein complex of chromatin using a generalized information principle. Considering systems level optimality requires that not only should all sub-parts of a natural process be optimal, but this also be true for the unfolding of higher recursive levels that include symbolic abstractions. An information-theoretic geometry of the genetic code data, together with the principle of maximum entropy, explains the variation in the codon groupings that map into different amino acids and explains its underlying self-similar structure. This analysis is reviewed for the investigation of the fundamental geometry underlying physical and biological space, which has a dimensionality of e as it gets reflected in aggregates associated with genomic DNA. The analysis is consistent with the fractal dimension of chromatin, but other non-optimal structures with lower dimensionality also exist.

A conscious agent, by virtue of actions that contribute to irreversibility, increases entropy. The agent during the lifespan may be characterized by a passage from the live information state [Formula: see text]1[Formula: see text] to the death state [Formula: see text]0[Formula: see text]. The superposition of such states is well recognized conceptually in psychology and belongs to the field of quantum biology. This information state may be frozen, or even reversed, using the observation-based quantum Zeno effect that, in the case of an individual, would imply self-observation. Since mind and body processes are connected recursively at several levels, a turn back of the information state will have a corresponding effect on aging processes and thus the lifespan of an organism could, in principle, be increased through a system of self-observation.

The paper describes the Hindu philosophers’ understanding of how humans produce speech and draws parallels with our modern scientific understanding of this phenomenon. Hindu philosophical literature divides speech production into four stages. It starts in the most subtle para stage, is followed by paśyanti, the stage of ideation and visualization, progresses to madhyamā, the stage of mind, memory, and word choice, and finally manifests into the outer world as vaikhari. Vaikhari was the only stage that could be heard with our ears, while the others were “concealed” within the body. Modern neuroscientific understanding reveals that there are multiple stepwise processes in the brain that precede the uttering of audible speech, each of which, when absent, is associated with a distinct clinical presentation. We discuss what clinical syndromes might be associated with deficiencies of each of the four Hindu stages of speech. The fourfold classification of speech in Hindu philosophy bears many similarities to findings from modern psycholinguistics and neuroscience. A systematic exploration of these texts might provide useful insights into understanding and managing pathologies of spoken language.

The paper presents analysis showing that the earliest structures in the evolution of noninteger dimensionality in a system will be filaments and spirals. Before the dimensionality has reached 1, the structures are linear, and thereafter these filaments or threads begin to bend and fold to be in accord with fundamental scale-invariance requirements. As the evolution proceeds, threads turn into spirals in a manner that parallels phase changes in a liquid crystal. Evidence is presented that the emergence of threads and spirals is consistent with the findings in bioscience, liquid crystals, galactic center filaments, and early spiral galaxies.

The helical coil is ubiquitous in biological and natural systems and often it is the basic form that leads to complex structures. This paper considers the question of its dimensionality in biological information as the helical coil goes through recursive coiling as happens to DNA and RNA molecules in chromatin. It has been shown that the dimensionality of coiled coils is virtually equal to e. Of the three forms of DNA, the dimensionality of the B-form is nearest to the optimal value and this might be the reason why it is most common.

The chapter will consider the problem of the computability of consciousness and provide evidence from different fields that supports the position that it is non-computable. If consciousness were computable then the present, which includes sentient agents, is completely determined by the past, and so one can emulate it and, therefore, conscious or sentient machines will be built. The arguments in favor of the non-computability of consciousness are: first-person accounts of creativity, the fact that mathematical logic is associated with incompleteness, and that consciousness appears to have the capacity to halt any computation in the brain. The non-computability position is consistent with the Orthodox Copenhagen Interpretation of quantum theory, in which the subject and the object are forever separated by the Heisenberg Cut, which, in turn, implies that materiality and consciousness are like two sides of a coin. The question of how the two can interact, as in controlling the quantum state through observation, will be discussed. Larger philosophical questions related to the problem of consciousness will be considered and assumptions behind the building blocks of reality as in space, time, and matter will be examined together with recent results from logic that are relevant to this discussion.

We investigate the large-scale geometry of the DNA-protein complex of chromatin using a generalized optimality principle, which requires that not only should all sub-parts of a natural process be optimal but also the unfolding of higher recursive levels that include symbolic abstractions. It was shown previously that an information-theoretic geometry of the genetic code data, together with the principle of maximum entropy, explains the variation in the codon groupings that map into different amino acids and explain its underlying self-similar structure. Here we take that analysis forward and investigate the fundamental geometry underling physical and biological space, which has a dimensionality of e, or about 2.718, as it gets reflected in aggregates associated with genomic DNA. The analysis is consistent with the fractal dimension of chromatin that has been measured to be about 2.7 but clearly other non-optimal structures with lower dimensionality also exist.

Motivated by the deep connections that exist between brain activity analyzed through thermodynamics and cognitive processing measured by information, this paper proposes an information principle based on partitions for possible applications to cognition-based judgments with potential applications to artificial intelligence. Looking at information through the lens of variety, which is the set of distinguishable elements of the set, we propose that partitions with only one type of object are counted once, and partitions with k types of objects are counted k times. Put differently, multiple occurrences of an object are considered not to have significance for the observer, or we can say that the objects are indistinguishable unless they are distinct. We explore the implications of this many-to-one logic that has possible applications to cognition centered systems and present a result related to the frequencies of the objects and contrast them with the first digit frequencies as well as the Bose–Einstein distribution.

Abstract This paper considers the matter of representation in Vedānta by examining key claims in the Ṛgveda and the Upaniṣads, which are some of its principal texts. Specifically, we consider the logic behind the paradoxical verses on creation and the conception of consciousness as the ground on which the physical universe exists. This also is the template that explains the logical structure underlying the principal affirmations of the Upaniṣads. The five elements and consciousness are taken to pervade each other, which explains how gross matter is taken to consist of all the four different kinds of atoms that get manifested in different states of the substance. The verses on creation are an example of the use of catuṣkoṭi in Indian philosophy prior to the use of it by Nāgārjuna in the Madhyamaka tradition. It also contrasts central ideas of Vedānta with the corresponding contemporary scientific ideas on consciousness.

It is shown that fractal dimensions may be reasonably estimated in an evolving system when further information is not available by considering the growth in the number of cells in two steps. This may be of use in the estimation of the dimensionality of metamaterials as well as in other natural phenomena. In contrast to algorithm-generated patterns for which a time step is well defined, natural systems evolve in an active space that leads one to the optimal information dimension of e, and this provides a novel perspective on self-similar behavior in natural systems.

Linear fractals associated with weights are investigated. Such fractals are important from a conservation law perspective that is relevant in a variety of physical systems such as materials science, sand dune fractals, barred galaxies, as well as in temporal processes like in the electroencephalogram (EEG). The weight associated with fractals is an additional feature that may be associated with distributions consistent with the ubiquitous power law and the first digit phenomenon. These distributions form a bridge to processes and applications in natural, biological, and engineering systems and, therefore, open up the possibility of the application of linear weighted fractals to these subjects. Two linear fractal algorithms that are near optimal in the information theoretic sense are described. A mechanism for the emergence of these fractals is proposed: it is the indistinguishability amongst the particles in the evolution and transformation of physical systems. Since the fractal approach is an established method of signal processing and coding, the newly proposed weighted fractals have the potential to lead to new useful algorithms.

This paper provides an explanation for why the assignment of codons to amino acids, which range from 1 to 6, is non-uniform. Since mathematical coding theory demands a near uniform assignment, the answer to this question is important to understand deeper aspects of the structure of the genetic code. Our analysis points to 20 different covering regions in an e-dimensional information space, which is equal to the number of amino acids. It is also shown that the assignment of the codons to the amino acids is fractal-like that is well modeled by the Zipf distribution. It is remarkable that the Zipf distribution that holds for the letter frequencies of words in a language also applies to the rank order of triplets the code for amino acids.

Creating machines that are conscious is a long term objective of research in artificial intelligence. This paper look at this idea with new arguments from physics and logic. Observers have no place in classical physics, and although they play a role in measurement in quantum physics there is no explanation for their emergence within the framework. There is suggestion that consciousness, which is implicitly a property of the observer, is a consequence of the complexity of specific brain structures, but this is problematic because one associates free will with consciousness, which goes counter to causal closure of physics. Considering a nested physical system, we argue that even if the system were assumed to have agency, observers cannot exist within it. Since complex systems can be viewed in nested hierarchies, this constitutes a proof against consciousness as a product of complexity, for then we will have nested system of conscious agents. As the existence of consciousness in cognitive agents cannot be denied, the implication is that consciousness belongs to a dimension that is not physical and machine consciousness is unattainable. These ideas are used to take a fresh look at two well-known paradoxes of quantum theory that are important in quantum information theory.

In many natural and engineered systems, the circle and the spiral are the appropriate primitives. As example, rotating motion of fluid around a common centerline constitutes a vortex. In many of these situations, the associated fractal dimension is of analytical interest. The box-counting approach to determining this dimension is not appropriate for many such cases, because unlike squares and boxes that span 2- and 3-dimensional spaces, circles and spheres do not. We propose the computation of the dimension in such cases based on the number of circles or spheres in the mapping process. This is illustrated by a few examples. How this may apply to spiral patterns that are repeated across scale is indicated.