The paper presents analysis showing that the earliest structures in the evolution of noninteger dimensionality in a system will be threads and spirals. Before the dimensionality has reached 1, the structures are linear, and thereafter these threads or strings 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.

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. 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 as it gets reflected in aggregates associated with genomic DNA. The analysis is consistent with the measured fractal dimension of chromatin.

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.

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.

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.

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 such cases, because unlike squares and boxes that span 2- and 3-dimensional spaces, circles and spheres don’t. Since self-similarity or scale invariance is not an exact property, we propose the computation of the dimension is 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.

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 the EEG.  Weighted fractals lead to distributions that are consistent with the ubiquitous power law and the first digit phenomenon. Two linear fractal algorithms that are near optimal in the information theoretic sense are described. It is proposed that one mechanism for the emergence of these fractals is the indistinguishability amongst the particles in the evolution and transformation of physical systems.

The question of coding of information is connected with that of dimensionality and efficient processing in both engineered and natural systems. When the cost of coding of data or that of classes 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) or exponential. The affine cases for which binary and ternary bases are optimal are presented.

The paper proposes an information principle based on partitions for possible applications to cognition-based judgments and applications in natural phenomena. 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 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.

The genetic code has two puzzling anomalies: first, when viewed as 64 sub-cubes of a 4x4x4 cube, the codons for serine (S) are not contiguous; and there are amino acid codons with zero redundancy, which goes counter to the objective of error-correction. To make sense of this, the paper shows that the genetic code must be viewed not only on stereochemical, coevolution, and error-correction considerations, but also on two additional factors of significance to natural systems, that of an information-theoretic dimensionality of the code data, which requires that the optimum natural code, as opposed to an engineered one, be characterized by e-dimensionality, and the principle of maximum entropy. One implication of noninteger dimensionality is self-similarity to different scales, and it is shown that the genetic code does satisfy this property, and it is further shown that the maximum entropy principle operates through the scrambling of the elements in the sense of maximal algorithmic information complexity. These two additional considerations explain several aspects of its organization.

This paper considers several aspects of the relationship between size, structure, speed of propagation and the number of autonomous cognitive agents in a neural network. Whereas, memory and function generation capacities of neural networks with scale invariant structure have been investigated extensively, the number of autonomous agents has not received prior attention. We propose the emergence of the dichotomy of causal and noncausal regions that is related to speed of propagation, in which the autonomous cognitive agents are not bound in a causal relationship with other agents. Arguments are presented for why the count of autonomous agents is best estimated with respect to the dimensionality of the underlying space. The number of autonomous agents obtained for the human brain equals twenty-five, and it is significant that the number in the sub-system modules also turns out to be close to the same value. It is possible that this near equality across layers provides a special uniqueness to the human brain. We argue that the findings of this study will be useful in the design of neural-network based AI systems that are designed to emulate human cognitive capacity.

The Newcomb-Benford (NB) distribution of first digits has been applied widely in many areas ranging from engineering to natural and biological sciences for the investigation of self-similarity and randomness. In this article, we consider systems for which the data is not enough to obtain proper first digit statistics, and we propose the use of an iterated version of the distribution where the statistics are aggregated over different scales on grounds that the first digit distribution is approximately scale invariant across a wide range of phenomena and also because scaling and recomputing first digits is not a linear process and so this process generates new data. We provide examples of the use of the iterated test for data in two different biological applications, viz. that of the secretome and the genetic code in both of which the raw data does not include all the nine different first digits. The paper includes proposals for further research on the idea of data amplification using scaling transformations.