Biological objects demonstrate remarkably repeatable patterns, governed by simple mathematical laws and regularities. Biological systems frequently exhibit symmetry and regularity on various spatial scales, starting from the genomic level and biomolecules and extending to the entire organism ^{[1][2][3][4][5][6]}[1,2,3,4,5,6]. A nearly universal observation, which was reported recently, states the subunits in protein assemblies are arranged in symmetric ways (the subunits themselves may be not symmetrical) [7]. It was hypothesized that the beautiful symmetries of biomolecules may reflect basic principles about the energy landscape in biology, just as symmetry relations do in particle physics [8]. Symmetry is inherent to the bodies of practically all animals (with rare exceptions). Animals are characterized by some kind of overall body symmetry, and these are of only a few types: translational, radial, biradial and bilateral symmetry [9]. Symmetry, in turn, represents a kind of ordering in physical and biological systems (“ordering” is understood as the process of putting elements of a biological pattern in a particular order) ^[10][11][10,11]. When reswesarchers address the symmetry of ordered patterns, resesarchers we usually restrict ourselves to mainly considering periodic order [12]. At the same time, order without periodicity has emerged to properly describe an increasing number of complex systems, and in particular biological ones [12]. This kind of ordering was referred to as aperiodic ordering [12]. An outstanding example of aperiodic ordering is supplied by the Fibonacci numbers, or Fibonacci series ^{[13][14][15][16]}[13,14,15,16]. Fibonacci and Lucas series appear in biological patterns [17]. Perhaps the most striking example of such samples is supplied by phyllotaxis, which is the arrangement of leaves on a plant stem ^{[13][14][15][16][17]}[13,14,15,16,17]. These patterns are surprisingly regular, so regular in fact that a physicist can compare their order to that of crystals. However, ordering in biological systems is usually not perfect, and quantitative measures of the deviation from the perfect ordering, which were introduced recently, will be discussed below in detail. It should be mentioned that non-ordered, asymmetrical biological systems exist. It is generally agreed that sponges are completely asymmetrical (see Figure 1); however, this thesis may be debated and will be addressed below in detail.

Moreover, it was suggested that breaking symmetry is a prevalent process in biology ^[3][18][3,18]. However, to be broken, symmetry in biological patterns must first appear, and it appears on different levels of organization of biological systems ^{[1][2][4][18]}[1,2,4,18]. First of all, let us define rigorously the notion of "symmetry". Symmetry in the text is understood as invariance of the biological object under some mathematical transformations, which may be: translation, reflection, rotation or scaling (including fractal scaling).

A reasonable question is as follows: what is the biological reasoning for periodic and aperiodic ordering in biological systems? In other words, why does nature prefer ordered patterns? The possible answers to this fundamental question may be classified as follows: (i) The appearance of symmetry and other sample patterns is due to the external physical constraints implied on the biological system ^{[5][19][20][21]}[5,19,20,21]. This hypothesis accepts that just physical effects, which in many cases act as proximate, direct, tissue-shaping factors during ontogenesis, are also the ultimate causes, (in other words) the indirect factors that provide a selective advantage, of animal or plant symmetry, from organs to body plan level patterns ^[19][20][21][19,20,21]. These physical constraints, in turn, may be responsible for the time evolution of biological species. In particular, it was suggested that the last common ancestor of metazoans (multicellular animals) was probably cylindrically symmetrical, with a concomitant axial polarity generated by early embryonic signaling. Changes in symmetry type occurred repeatedly with time, with respect to environmental parameters as the main constraining forces, providing an evolutionary change in the macroscopic symmetry of metazoans [21]. (ii) The second idea explaining the abundance of the symmetric patterns in biology implies that the symmetry of biological systems stems from the symmetry of molecules themselves and potentials describing interactions between molecules ^[22][23][22,23]. It was demonstrated that the symmetry of these potentials governs the symmetry of biological systems, such as actin, tubulin and the ubiquitous icosahedral shell structures of viral capsids ^[22][23][22,23]. (iii) The third approach relates the appearance of mathematical ordering in biological systems to pure survival reasons. For example, periodic cicadas emerge from their underground homes to mate every 13 or 17 years, and 13 and 17 are primes (this kind of temporal ordering also represents aperiodic ordering) ^{[24][25][26][27][28][29]}[24,25,26,27,28,29]. The philosophy of the evolutionary-based explanation of the mathematical ordering is that if cicadas have 12-year cycles, all the predators with 2-, 3-, 4-, and 6-year cycles will eat them; in other words, the cicadas with prime number cycles will have a higher probability of survival. Rigorously speaking, cicadas will leave more offspring if their cycles are described by primes ^[28][29][28,29]. Let reuseasrchers quote [29]: “a prey with a 12-year cycle will meet-every time it appears-properly synchronized predators appearing every 1, 2, 3, 4, 6 or 12 years, whereas a mutant with a 13-year period has the advantage of being subject to fewer predators”. A second explanation, proposed by Cox, Carlton and Yoshimura, concerns the avoidance not of predators but of hybridization with similar subspecies (consider that these explanations are not mutually exclusive) ^[25][26][27][25,26,27]. Genuine reasoning for the prime-shaped life cycle of cicadas remains debatable, and the discussion of this reasoning gave rise to the deep philosophical discussion of the nature and roots of the notion of “explanation” of natural/biological phenomena in [30]. Survival/reproductive reasons were also involved in the explanation of symmetry appearing in the color of zebra finches, shown in Figure 2.

It was demonstrated that symmetrically banded males produced more offspring that survived past the period of parental care than males in either of the asymmetric treatments. This appeared to be the effect of female choice processes and female-based parental investment and not male intra-sexual dominance. Thus, it was shown that symmetrically manipulated males gain reproductive advantages in controlled laboratory conditions, which further supports recent theories indicating the evolutionary importance of symmetry in signaling-trait design [31]. Moreover, it was demonstrated that symmetric patterns are attractive not only to females, and it was found that males associated more with symmetrical than asymmetrical females, indicating a preference for symmetry [32]. (iv) Finally, rwesesarchers  discuss the more hypothetical relation of the appearance of symmetrical patterns in biological systems to informational reasoning. It was suggested that symmetric biological structures and patterns preferentially arise not only due to natural selection but also because they require less specific information to encode and are therefore much more likely to appear as phenotypic variation through random mutations [33]. This novel concept, which is well tailored to the general informational paradigm of exact sciences and was criticized recently in [34], will be discussed below in relation to the Curie–Minnigerode–Neumann and Landauer principles ^[35][36][37][35,36,37].

Alternative reasoning for the abundance of symmetrical patterns in biological systems was suggested in [33]. The authors of [33] noted that it is plausible to assume (by a certain analogy to engineering design) that symmetry may stem from natural selection, as discussed in Section 2.1, in which it was demonstrated that bilateral symmetry of sea inhabitants is favorable for their maneuverable locomotion in water [19]. However, evolution, unlike engineers, cannot plan ahead, and so these symmetrical features must also afford some immediate selective advantage which is hard to reconcile with the breadth of systems where symmetry is observed. It was suggested in [33] that the symmetric structures preferentially arise not only due to natural selection but also because they require less specific information (and consequently less energy according to the Landauer principle ^[35][36][37][35,36,37]) to encode and are therefore much more likely to appear as phenotypic variation through random mutations. Arguments from algorithmic information theory enabled the formalization of this hypothesis, leading to the prediction that many genotype–phenotype maps are exponentially biased toward phenotypes with low descriptional complexity (preference for symmetry is a special case of the bias toward compressible descriptions ^[38][86]). The authors of the aforementioned hypothesis validated the predictions of this idea with biological data, showing that protein complexes, RNA secondary structures and a model gene regulatory network all exhibit the expected exponential bias toward simpler (and more symmetric) phenotypes [33]. The authors of [33] supplied arguments supporting their concept and rooted in the algorithmic information theory, in which it is well accepted that when the space of algorithms is considered, outputs that can be generated by short programs are exponentially more likely to be produced than outputs that can only be generated by long programs. It was demonstrated in [33] that formalism developed in the algorithmic information theory may be successfully applied for the analysis of genotype–phenotype maps. Thus, symmetry appearing in biological systems emerges from the “informational arguments”, providing an economy of biological information necessary for the description of a biological entity ^[33][38][33,86].

The “informational biological paradigm” introduced and developed in [33] was recently criticized in [34], in which the role of symmetry breaking in biological systems was stressed. It was noted that while symmetry may arise more commonly in biological structures with low complexity, there is evolutionary pressure to develop asymmetry in many biological structures with high complexity. The emergence of symmetry cannot be fully understood without considering the emergence of asymmetry as well [34]. Consider, for example, the human brain, one of the most complex and mysterious biological structures ^[34][39][34,87]. While the two halves of the brain look roughly symmetric at first glance, a recent large-scale neuroimaging demonstrated that structural left–right asymmetries are the rule, rather than the exception, for cortical brain areas ^[39][87]. Importantly, the human central nervous system is not the only one that shows such striking asymmetries. Breaking symmetry is therefore a crucial step in the development of all nervous systems [34]. This statement is in striking correspondence with the Curie–Minnigerode–Neumann principle, formulated by Curie as follows: “asymmetry is what creates a phenomenon” ^[40][41][82,83], which may reshaped as follows: “asymmetry is what creates a biological phenomenon”.

The explanation of symmetrical patterns abundant in biological systems with arguments rooted in the algorithmic information theory seems deep and promising. RWesearchers  propose stretching this approach to the grounding of other kinds of periodic and aperiodic ordering appearing in biological systems, such as Fibonacci series and Archimedean and Lucas spirals ^{[13][14][15][16][17][42]}[13,14,15,16,17,88]. Indeed, the Fibonacci series found in phenotypic structures of plants and animals, defined by Equation (16).

The idea that symmetry in biological patterns is deeply rooted in the informational basic structure of reality fits with ideas introduced by John Archibald Wheeler, who suggested that fundamentals of physics should be re-built on the informational groundings and assumed that the main notions of physics are deeply rooted in the “bit-based” scientific paradigm ^[43][89]. This approach may be very briefly and aphoristically summarized as follows: “all physical things are information-theoretic in origin”, aphoristically reduced to “it from bit” ^[43][89]. The idea was developed recently within the highly debated and controversial Landauer principle, suggesting the thermodynamic equivalent of information, establishing the lower theoretical limit of energy consumption of computation ^[35][36][35,36]. It holds that “any logically irreversible manipulation of information, such as the erasure of a bit or the merging of two computation paths, must be accompanied by a corresponding entropy increase in non-information-bearing degrees of freedom of the information-processing apparatus or its environment” ^[35][36][35,36]. In other words, there is a minimum possible amount of energy E  required to erase one bit of information, known as the Landauer limit and supplied by Equation (38): where kB=1.38×10−23JK is the Boltzmann constant and T is the absolute temperature of the heat sink ^{[44][45][46][47]}[90,91,92,93]. The Landauer principle was experimentally tested in ^[48][49][94,95]. Extensions of the Landauer principle to the realms of quantum mechanics ^[50][96] and general relativity ^[51][97] were reported. The Landauer principle applied to mechanical motion demonstrates that dissipation of energy is the key process through which mechanical motion becomes observable ^[52][98]. The analysis of the performance of photon detectors (such as eyes) leads to the conclusion that only efficiency is limited by the Landauer energy bounds on information gain and information erasure ^[53][99]. Estimation of information contained in molecular motion based on the Landauer principle was performed in ^[54][100]. The Landauer principle restricts the informational capacity of biological systems; thus, it is closely related to the abundance of ordering in biological systems; indeed, periodic and aperiodic ordering enable the saving of memory/energy available for the organism. The Landauer principle bridges the informational and thermodynamic paradigms of life, which explains the ability of organisms to maintain low levels of entropy that explain order ^[55][101]. The informational paradigm of life enabled the analysis of the SARS-CoV-2 virus using Shannon’s information theory ^[56][102]. A relationship between the information entropy of genomes and their mutation dynamics was established. In particular, it was revealed that genomes undergo genetic mutations over time driven by a tendency to reduce their overall information entropy ^[56][102]. Let reusesarchers roughly estimate the informational capacity of living cells with the Landauer principle. Consider that the characteristic spatial range of living cells, namely, l≅1−100 μm, spans the dimensions of a majority of prokaryotic and eukaryotic cells ^[56][102]. Thus, the maximal informational capacity of a living cell may be estimated, according to the Landauer principle, with Equation (49); if researcherswe speculate that information exchange occurs only via the surface of a living cell researcherswe estimate the following: where ES and γint the total and specific interfacial energies of a cell. Assuming γint≅1.0×10−3Jm2 yields ξ≅3.5×105−3.5×109  ^[57][103]. (This value should not be confused with the genomic capacity of a cell ^[58][59][104,105].) Thus, rwesearchers  may estimate the informational interfacial capacity of a small micro-scaled cell as  ξ≅3.5×105 bits and compare this with the DNA-based code, which enables the storage of 5.2×106 bits of information ^[60][106]. Thus, the informational capacity of DNA and cells is restricted; hence, assessing the thermodynamic efficiency of the computations performed by organisms becomes crucial. The authors of ^[61][107] posed and addressed the following fundamental question: how close has life come to maximally efficient computation (presumably under the pressure of natural selection)? The suggested answer is summarized as follows: despite inevitable shifts across the architectures of life, the authors revealed a surprising consistency in the efficiency of translation, one of the most universal types of computation carried out in biological systems ^[61][107]. The analyses demonstrated that as bacteria become larger, their overall translational efficiency converges on that of a single ribosome ^[61][107]. In addition, this efficiency is maintained for unicellular eukaryote and mammalian cells ^[61][107]. Astonishingly, this efficiency is only about an order of magnitude larger than the Landauer bound, supplied by Equation (8) (see ^[61][107]). It is instructive to compare the efficiency of biological computations to that of the best supercomputers. Actually, the cost of computation in supercomputers is about eight orders of magnitude worse than the Landauer bound given by Equation (8), which is about six orders of magnitude less efficient than biological translation when both are compared to the appropriate Landauer bound ^[61][107]. Biology is beating theour current engineered computational thermodynamic efficiencies by an astonishing degree ^[61][107].

It should be emphasized that symmetry and order (periodic and aperiodic) inherent in biological systems improve the efficiency of biological computation; indeed, when an n-fold symmetry is present, the single computation act governs the location of a number of n “spots” in the biological pattern. An interface between artificially created digital information and information produced by organisms was addressed in 108. It was demonstrated that human-related digital information has reached a similar magnitude to information in the biosphere ^[62][108].

Information aspects of order in DNA were discussed in [12]. When the structure of DNA was discovered, it was first assumed that the basic repeating unit of DNA polymer was a tetranucleotide, in which the four different kinds of nucleotides recurred in regular sequence [12]. Thus, DNA was originally viewed as a trivially periodic macromolecule, characterized by translational symmetry, which is unable to store the amount of information required for the governance of cell function [12]. It was Schrödinger who suggested that genetic material should consist of a long sequence of a few repeating elements exhibiting a well-defined order without the recourse of periodic repetition ^[63][109]. In this way, the notion of a 1D aperiodic crystal was introduced and the deep connection between symmetry and information was established ^[12][63][12,109]. Again, the correspondence: “more symmetry–less information” takes place.