Book: the multidimensional world

Tozzi A, Peters JF.  2017.  The multidimensional world.  Lambert Academic Publishing, Saarbrücken, Germany.  ISBN-13: 978-3-330-03530-0.

Recently introduced versions of the Borsuk-Ulam theorem (BUT) reveal that a feature vector on a n-manifold projects two feature vectors (matching descriptions of a single object) onto an n+1 manifold.  Starting from this rather simple, yet far-reaching, computational topology observation, we build a fruitful general framework, able to elucidate disparate “real” physical and biological phenomena, from quantum entanglement to gauge theories.  Summarizing this novel topological approach, we take into account projections among functional or real dimensions.  We achieve a system of mappings that fit very well with experimental results, making it possible to assess countless systems in far-flung scientific branches. This book highlights the computational character of matching descriptions (arising from descriptively proximal objects) that display a widest range of possible uses.  Such observations point to BUT not just from the standpoint of a novel interpretation of almost all the biological and physical phenomena, but also as suitable tools in evaluating the slight (objective and subjective) differences that make our world an astonishing realm of rich heterogeneity.  


A novel theorem

Gravitational lensing

Pauli exclusion principle

Small world networks


Group theory

Thermodynamic entropy: the arrow of time

Shannon, Rényi entropy

The dimensions of living beings

Natural projections and evolution

How to detect hidden dimensions?



Chaotic 4D paths & Feigenbaum constant in a multidimensional world

Tozzi A, Peters JF, James III C, Deli E.  2016.  Multidimensional chaotic dynamics and entropies.  Archives of Information Science and Technology, 1(1): 10-19. 

The possible presence of further dimensions hidden in our three-dimensional-plus time world might help to elucidate countless physical and biological systems’ behaviors, from quantum entanglement to brain function. Nevertheless, suggestions concerning multidimensional arrangement of physical and biological systems do not deserve the role of scientific claims, unless the suggested additional dimensions can be verified via empirically testable hypotheses and experimental apparatus. Here we suggest that the widespread nonlinear dynamics and chaotic behavior of physical and biological collective systems might mirror further dimensions hidden in our world. Indeed, bringing together disparate knowledge from seemingly unrelated fields (brane cosmology, fluid dynamics, algebraic topology, computational topology, dynamic systems theory, logic and statistical mechanics), we show how, in logistic maps derived from nonlinear dynamical equations, the typical bifurcation diagrams might arise from linear flow paths, that intersect largesized hidden dimensions at the canonical phase parameter’s values between three and four. Therefore, chaotic dynamics suggests the existence of a further hidden dimension in our Universe. We also provide a thermodynamic framework which suggests that the cosmic entropy is encompassed in a multidimensional manifold.  PDF



Dynamic systems theory and evolution (my comment published by Nature)

Tozzi A. 2014. Evolution: Networks and Energy Count.  Nature 515: 343. doi:10.1038/515343c.

Lysenko,suggested the heritability of acquired characteristics.  His heretic ideas were dismissed with disgust in favor of the “post-Darwinist” standard evolution theory (SET), one of the most pervasive paradigms of the modern science.  However, after half a century of oblivion, the debate is once again an hot topic of current research.  In particular, the possible epigenetic inheritance within organisms have been suggested as neo-Lamarckian in nature and talks about a picture different from SET, despite Wray’s skeptical claims.  If we examine the problem from the novel perspective of the supramolecular chemistry, we notice that the epigenetic information involves the storage of information at the molecular level and its retrieval, transfer and processing at the supramolecular level, via transitory processes that are self-organized, self-assembled and dynamic.  SET does not keep into account that the complexity of adaptive evolving systems (including species, niches and environment) is best understood as dynamic networks of relationships, aiming to decrease their free energy via entropy transfer.  The DNA is just one of the countless functional tasks of interest in the study of evolution: changes propagate through interlinked levels of organization, inducing connectivity and interaction on all scales of the multilevel system, with no preferred level of granularity.  Models of fitness attractors intended to capture the process of natural selection are starting to be developed, taking into account power laws, non-equilibrium steady-state at the edge of the chaos and energetic landscapes made of basins, valleys, floors, ridges and saddle points.  In conclusion, it would be useful to investigate SET in the framework of dynamical system theories.  PDF



Building fractals from noise  

Tozzi A. 2015.   How to Turn an Oscillation in a Pink One.  Journal of Theoretical Biology 377, 117–18. doi:10.1016/j.jtbi.2015.04.018.

Scale-free dynamics are an intrinsic feature of a large class of natural models, from earthquakes to brain activity.  Assessing a geometrical/mathematical model of synthetic power law oscillations, we noticed that a wave containing a fractal-like structure can be produced by summing a random oscillation to a carefully chosen one.  This observation gives rise to countless applications: a “hidden” oscillation may cause a scale-free behavior in a random noise; a fractal system can be produced by simply choosing the appropriate oscillation to bring in; if power laws are involved in random walks, phase transitions and self-organized criticality, then the superimposition of a carefully chosen oscillation may lead to systems of increased complexity; “nested” waves from the central nervous system’s spontaneous networks  may be the source of the scale-free dynamics seen in EEG and fMRI; in the event of brain 1/f scaling disruption caused by illnesses such as Alzheimer’s disease, an external wave - for instance, via transcranial stimulation - could restore the broken symmetry.   PDF


A Gauge theory for living cells

Tozzi A, Peters JF, Navarro J, Kun W, Lin B, Marijuán PC.  2017.   Cellular Gauge Symmetry and the Li Organization Principle.  Progress in Biophysics and Molecular Biology.

Based on novel topological considerations, we postulate a gauge symmetry for living cells and proceed to interpret it from a consistent Eastern perspective: the li organization principle. Gauge theories had a tremendous impact in particle physics and have been recently proposed in order to assess nervous activity too. Herein, taking into account novel claims from topology, the mathematical branch that allows the investigation of the most general systems activity, we aim to sketch a gauge theory addressed to the fundamentals of cellular organization. In our framework, the reference system is the living cell, equipped with general symmetries and energetic constraints standing for the intertwined biochemical, metabolic and signaling pathways that allow the global homeostasis of the system. Abstractly, these functional movements would follow donut-like trajectories. Environmental stimuli stand for forces able to locally break the symmetry of metabolic and signaling pathways, while the species-specific DNA is the gauge field that restores the global homeostasis after external perturbations. We show how the Borsuk-Ulam Theorem (BUT), which states that a single point on a circumference maps two points on a sphere, allows an inquiry on the evolution from inorganic to organic structures as well as the comparison between prokaryotic and eukaryotic metabolisms and modes of organization. Furthermore, using recently developed BUT variants, we operationalize a methodology for the description of cellular activity in terms of topology/gauge fields and discuss about the experimental implications and feasible applications. We converge on the strategic role that second messengers have played regarding the emergence of such a unitary gauge field for the cell, and the subsequent evolutionary implications for multicellulars. A new avenue for a deeper investigation of biological complexity looms. Philosophically, along this overall exploration of cellular dynamics and biological complexity, we might be reminded of the duality between two essential concepts proposed by the great Chinese synthesizer Zhu Xi (in the XIII Century). His explanatory scheme epitomizes a feasible philosophical interpretation of the present proposal: on the one side, the li organization principle, which may be taken as equivalent to the dynamic interplay between symmetry and information; and on the other side, the qi principle, which can be interpreted as the energy participating in the process, and which always appears as interlinked with the former. In contemporary terms, it would mean the required interconnection between information and energy, and at the same time it would be pointing at essential interpretive principles of information philosophy.   PDF


Geometric curves underlying physical and biological dynamics

Yurkin A, Tozzi A, Peters JF, Marijuan PC.  2017.  Quantifying Energetic Dynamics in Physical and Biological Systems Through a Simple Geometric Tool and Geodetic Curves.  Addendum to: Cellular Gauge Symmetry and the Li Organization Principle.  Progress in Biophysics and Molecular Biology. 

The present Addendum complements the accompanying paper “Cellular Gauge Symmetry and the Li Organization Principle”; it illustrates a recently-developed geometrical physical model able to assess electronic movements and energetic paths in atomic shells. The model describes a multi-level system of circular, wavy and zigzag paths which can be projected onto a horizontal tape. This model ushers in a visual interpretation of the distribution of atomic electrons’ energy levels and the corresponding quantum numbers through rather simple tools, such as compasses, rulers and straightforward calculations. Here we show how this geometrical model, with the due corrections, among them the use of geodetic curves, might be able to describe and quantify the structure and the temporal development of countless physical and biological systems, from Langevin equations for random paths, to symmetry breaks occurring ubiquitously in physical and biological phenomena, to the relationships among different frequencies of EEG electric spikes. Therefore, in our work we explore the possible association of binomial distribution and geodetic curves configuring a uniform approach for the research of natural phenomena, in biology, medicine or the neurosciences.  PDF



When holes impair the cellular homogeneity and the surface/volume ratio.

Tozzi, A.  2022. Living Cell’s Feeling of Holes: The Mathematics of Cavities in Biophysical Structures. Preprints. doi: 10.20944/preprints202204.0156.v1


Mechanical properties such as shape, volume and size affect the dynamics of biological systems. Most of the current methodological approaches are inclined to remove the existence of holes and impurities from systems’ description, regarding them as routes toward mechanical failure. On the contrary, we suggest that the occurrence of holes might be of utmost functional importance, allowing reversible transformations of cellular structures. The focus here is on the widespread occurrence of intracytoplasmic holes, that deeply modifies the topology of living cells and provides researchers with novel operational tools to investigate intracellular dynamics. We take as example the prokaryotic gas vesicles, i.e., intracellular cavities filled with gases spreading from the nearby medium. The mechanical and topological cellular properties dictated by intracytoplasmic holes are investigated, focusing on the physical constraints imposed by their very existence. For instance, the presence of gas vesicles breaks the cytoplasmic homogeneity, leading to inhomogeneity in functional activities and modifications in intracellular flows. Also, a topological approach to cytoplasmic holes suggests novel physiological roles for gas vesicles. For example, the gas vesicles’ ability to increase/decrease cellular volumes provides a mechanism that counteracts the detrimental effects of the surface/volume ratio. In conclusion, a structural/methodological approach based on the occurrence of holes testifies once again how the simple biophysical structure alone can dictate the function.



A Timeless biology; when time does not count

Tozzi A, Peters JF, Chafin C, De Falco D, Today J.  2018.  A timeless biology.  Progr Biophys Mol Biol. 134,  38-43.  doi: 10.1016/j.pbiomolbio.2017.12.002. 

Contrary to claims that physics is timeless while biology is time-dependent, we take the opposite standpoint: physical systems’ dynamics are constrained by the arrow of time, while living assemblies are time-independent.  Indeed, the concepts of “constraints” and “displacements” shed new light on the role of continuous time flow in life evolution, allowing us to sketch a physical gauge theory for biological systems in long timescales.  In the very short timescales of biological systems’ individual lives, time looks like “frozen” and “fixed”, so that the second law of thermodynamics is momentarily wrecked. The global symmetries (standing for biological constrained trajectories, i.e. the energetic gradient flows dictated by the second law of thermodynamics in long timescales) are broken by local “displacements” where time is held constant, i.e., modifications occurring in living systems.  Such displacements stand for brief local forces, able to temporarily “break” the cosmic increase in entropy.  The force able to restore the symmetries (called “gauge field”) stands for the very long timescales of biological evolution.  Therefore, at the very low speeds of life evolution, time is no longer one of the four phase space coordinates of a spacetime Universe: but it becomes just a gauge field superimposed to three-dimensional biological systems.  We discuss the implications in biology: when assessing living beings, the underrated role of isolated “spatial” modifications needs to be emphasized, living apart the evolutionary role of time.   PDF



Ramsey’s economic theory of savings may help to quantify the energetic requests of the living cells.

Tozzi A.  2022.  An economic approach to energy budgets: how many resources should living organisms spare? Biosystems.  Volume 211, 104584.      

Ramsey's economic theory of saving (RTS) estimates how much of its commodities a nation should save to safeguard the well-being of future generations. Since RTS retains many attractive qualities such as simplicity, strength, breadth and generality, here we ask if it would be useful to investigate biophysical issues. Specifically, we focus on a biological topic that lends itself as a backdrop for the study of the imbalance between intake and expenditure, i.e., the evaluation of the multicellular living organisms' energetic requirements and constraints. Our problem is to find at each time the optimum distribution and the right balance of the cellular energy budget between consumption and storage: how much must a living organism spare to increase its chances of survival over long periods? To give an operational example, we discuss the ATP requirements in the central nervous system during the spontaneous and the evoked activity of the brain, showing that the experimentally detected values of energetic expenditure during neural computations match well with the estimations provided by RTS. Suggesting how to find the optimum allocation of the available energy between expenditure and saving at each time, RTS approaches to biological energy budgets may have a wide range of experimental applications, such as: a) optimization of the long-term survival chances of either immortalized cell cultures, or beneficial bacterial colonies and exogenous probiotic mixtures; b) eradication of detrimental biofilms, such as, e.g., heart valves' Streptococcus colonies; c) novel anti-stress and anti-ageing strategies.




Time-reversal entropy: an underrated actor

Tozzi A. Peters JF.  2017.  Critique of pure free energy principle: Comment on “Answering Schrödinger's question: A free-energy formulation” by Maxwell James DésormeauRamstead et al. Physics of Life Reviews.DOI: 10.1016/j.plrev.2017.10.003. 

The paper by Ramstead et al. reminds us the efforts of eminent scientists such as Whitehead and Godel. After having produced influential manuscripts, they turned to more philosophical issues, understanding the need for a larger formalization of their bounteous scientific results. In a similar way, the successful free-energy principle has been generalized, in order to encompass not only the brain activity of the original formulation, but also the whole spectrum of life. Here we go through philosophical (the principle of identity) and physical (temperature, Pandemonium architecture, time reversal entropy) issues that might be correlated with the free energy principle.  PDF



Curvatures in biology: when the misused "analogy" is still helpful in scientific affairs

Tozzi A.  2018.  Curvature Changes in Biological Dynamics. Preprints, 2018100299 (doi: 10.20944/preprints201810.0299.v1).

Geometry is correlated with both analogical thinking and physical/biological observables.  Indeed, naïve, common-sense descriptions of objects’ shapes and biological trajectories in geometric phase spaces may help experimental investigation.  For example, different biological dynamics, such as the developmental growth patterns of the oldest known animal (the extinct Dickinsonia) and the human brain electric oscillations, display striking analogies: when encompassed in abstract spaces, their paths describe the same changes in curvature, from convex, to flat, to concave and vice versa.  Such dynamical behavior, anticipated by Nicholas de Cusa in his 1440 analogic account of “coincidentia oppositorum”, can be used to describe widespread biological paths in terms of concave, flat and convex curves on a donut-like structure.  Every one of the achieved trajectories on the torus can be located, through a topological technique called Hopf fibration, into a four-dimensional space with positive curvature.  We discuss how the correlation between Hopf fibration and Navier-Stokes equation allows us to treat biological and neuroscientific issues in terms of flows taking place in a viscous fluid medium which can be experimentally assessed and quantified. 


Alexander horned sphere: a toroidal phase space for both linear and nonlinear dynamics

Linear dynamics of physical and biological systems can be described in terms of  trajectories taking place inside manageable donut-like manifolds equipped with genus one.  In turn, widespread nonlinear dynamics, such as paths at the edge of chaos, power law and free scale phenomena, systems’ bifurcations, iterative maps, ordinary differential equations, Bayesian issues, are difficult to map and locate inside toroidal phase spaces.  Here we ask: is it feasible to build a single donut-like phase space, where both linear and non-linear dynamics might be mapped?  The answer is positive, if we consider a torus equipped with an Alexander horn.  Each path is arranged inside a phase space consisting of an horned ring, i.e., a modification of the pathological object termed Alexander horned sphere.  This relatively simple description of both linear and nonlinear trajectories taking place inside an horned torus-shaped manifold allows to standardize the assessment of a wide range of natural and artificial paths, from time series analysis to iterative maps and population dynamics.  


Horizontal transmission modifies our concept of species: another case of decision limit problems

Tozzi A.  2019.  Does horizontal transmission amend our concept of living species?  (electronic response to: Chen J, Quiles-Puchalt N, Chiang YN, Bacigalupe R, Fillol-Salom A, et al. 2018. Genome hypermobility by lateral transduction. Science, Vol. 362, Issue 6411, pp. 207-212. DOI: 10.1126/science.aat5867). 

Demarcation among objects and things is somewhat arbitrary, because our mind tends to exclude the continuity among hidden or unknown structures of the world. We are used to draw lines of separation among things that we judge different, arbitrarily excluding or including issues in our description, to achieve positive demarcations that allow a pragmatic treatment of the world based on regularity and uniformity. In touch with set theory, observers tend to spatially and temporally split the set of the entire world in different, arbitrary, fictious subsets that could not really stand for different objects or events. There’s a plenty of boundaries in math, physics and biology. In these disciplines, the concept of boundary is grounded on the presence of internal and external surfaces. This concept seems straightforward in math, and in particular in topology, where we are in front of planes split into an “interior” region bounded by curves and an “exterior” one, containing all of the nearby and far away exterior points.

Concerning the biophysical realm, many accounts of our real world are used to think at living being as individual, self-preserving, unique subjects equipped with borders, such as cellular membranes, envelopments, Markov blankets, and so on. The canonical illustration of biological organisms depicts “something” equipped with some peculiar activity, confined into itself, which struggles against (and cooperates with) the external world, in order to keep its entropy as low as possible and devoted to self-preservation. However, the demarcation among objects, things and living beings could be somewhat arbitrary. The same limitation holds for the very concept of evolution, which is based on species. Our concept of species is very arbitrary, because it throws (not experimentally demonstrated) borders among living beings. Thinking, e.g., to different populations of Homo, despite we are used to consider Sapiens, Neandertals, Denisovians, Floresians as different species, genetic studies undoubtedly point towards their ancient ibridation, which occurred more than once in different prehistorical contexts (Slon et al., 2018). Concerning our DNA, the delimitation of a species from another is sometimes difficult, due, e.g., to the presence in animal genome of viral and bacterial sequences. To make an example, the horizontal transmission and lateral transduction of genetic material described by Chen et al. (2018) is able to overtake the so-called canonical “species-specific” barriers, making sometimes difficult to clearly encompass individual living beings in a given species.


Facial muscles of extinct hominids

Tozzi A. 2016.  Muscles of facial expression in extinct species of the genus Homo.  bioRxiv 072884; doi:

We present a detailed description of mimetic muscles in extinct human species, framed in comparative and phylogenetic contexts.  Using known facial landmarks, we assessed the arrangement of muscles of facial expression in Homo sapiens, neanderthalensis, erectus, heidelbergensis and ergaster.  In modern humans, several perioral muscles are proportionally smaller in size (levator labii superioris, zygomaticus minor, zygomaticus major and triangularis) and/or located more medially (levator labii superioris, zygomaticus minor and quadratus labii inferioris) than in other human species.  As mimetic musculature is examined in the most ancient specimens up to the most recent, there is a general trend towards an increase in size of corrugator supercillii and triangularis.  Homo ergaster’s mimetic musculature closely resembles modern Homo, both in size and in location; furthermore, Homo erectus and Homo neanderthalensis share many muscular features.  The extinct human species had an elaborate and highly graded facial communication system, but it remained qualitatively different from that reported in modern Homo.  Compared with other human species, Homo sapiens exhibits a lower degree of facial expression, possibly correlated with more sophisticated social behaviours and with enhanced speech capabilities.  The presence of anatomical variation among species of the genus Homo raises important questions about the possible taxonomic value of mimetic muscles.   PDF



When Neanderthal and ancient humans converge, here you are the modern humans

Tozzi A 2021.  Novel Insights from Non-ultrametric Phylogenetic Trees: The Case of Neanderthal Introgression.  viXra:2112.0069.

 Ultrametric spaces are widely used to depict evolutionary times in phylogenetic trees, since they assume that every population/species is located at the tips of bifurcating branches of the same length.  The discrete branching of ultrametric trees permits the measurement of distances between pairs of individuals that are proportional to their divergence time.  Here the traditional ultrametric concept of bifurcating, divergent phylogenetic tree is overturned and a new type of non-ultrametric diagram is introduced to describe gene flows in terms of convergent branches.  To provide an operational example, the paleoanthropological issue of Neanderthal genome’s introgression in non-African humans is examined.  Neanderthals and ancient humans are not anymore two species that exchange chunks of DNA, rather become a single, novel cluster ox extant hominins that must be considered by itself.  The novel converging, non-ultrametric phylogenetic trees permit the calibration of molecular clocks with a twofold benefit. When the date of the branching of two population/species from a common ancestor is known, the novel approach allows to calculate the time of subsequent introgressions.  On the contrary, when the date of the introgression between two population/species is known, the novel approach allows to detect the time of their previous branching from a common ancestor.