On Homogeneity

"The power of differential calculus is that it linearizes all problems by going back to the 'infinitesimally small', but this process can be used only on smooth manifolds. Thus our distinction between the two senses of rotation on a smooth manifold rests on the fact that a continuously differentiable coordinate transformation leaving the origin fixed can be approximated by a linear transformation at О and one separates the (nondegenerate) homogeneous linear transformations into positive and negative according to the sign of their determinants. Also the invariance of the dimension for a smooth manifold follows simply from the fact that a linear substitution which has an inverse preserves the number of variables." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"An 'empty world', i. e., a homogeneous manifold at all points at which equations (1) are satisfied, has, according to the theory, a constant Riemann curvature, and any deviation from this fundamental solution is to be directly attributed to the influence of matter or energy." (Howard P Robertson, "On Relativistic Cosmology", 1928)

"When the statistician looks at the outside world, he cannot, for example, rely on finding errors that are independently and identically distributed in approximately normal distributions. In particular, most economic and business data are collected serially and can be expected, therefore, to be heavily serially dependent. So is much of the data collected from the automatic instruments which are becoming so common in laboratories these days. Analysis of such data, using procedures such as standard regression analysis which assume independence, can lead to gross error. Furthermore, the possibility of contamination of the error distribution by outliers is always present and has recently received much attention. More generally, real data sets, especially if they are long, usually show inhomogeneity in the mean, the variance, or both, and it is not always possible to randomize." (George E P Box, "Some Problems of Statistics and Everyday Life", Journal of the American Statistical Association, Vol. 74 (365), 1979)

"[…] homogeneous functions have an interesting scaling property: they reproduce themselves upon rescaling. This scaling invariance can shed light into some of the darker corners of physics, biology, and other sciences, and even illuminate our appreciation of music." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"In contrast to gravitation, interatomic forces are typically modeled as inhomogeneous power laws with at least two different exponents. Such laws (and exponential laws, too) are not scale-free; they necessarily introduce a characteristic length, related to the size of the atoms. Power laws also govern the power spectra of all kinds of noises, most intriguing among them the ubiquitous (but sometimes difficult to explain)." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"Scaling invariance results from the fact that homogeneous power laws lack natural scales; they do not harbor a characteristic unit (such as a unit length, a unit time, or a unit mass). Such laws are therefore also said to be scale-free or, somewhat paradoxically, 'true on all scales'. Of course, this is strictly true only for our mathematical models. A real spring will not expand linearly on all scales; it will eventually break, at some characteristic dilation length. And even Newton's law of gravitation, once properly quantized, will no doubt sprout a characteristic length." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"Fitting data means finding mathematical descriptions of structure in the data. An additive shift is a structural property of univariate data in which distributions differ only in location and not in spread or shape. […] The process of identifying a structure in data and then fitting the structure to produce residuals that have the same distribution lies at the heart of statistical analysis. Such homogeneous residuals can be pooled, which increases the power of the description of the variation in the data." (William S Cleveland, "Visualizing Data", 1993)

"When the distributions of two or more groups of univariate data are skewed, it is common to have the spread increase monotonically with location. This behavior is monotone spread. Strictly speaking, monotone spread includes the case where the spread decreases monotonically with location, but such a decrease is much less common for raw data. Monotone spread, as with skewness, adds to the difficulty of data analysis. For example, it means that we cannot fit just location estimates to produce homogeneous residuals; we must fit spread estimates as well. Furthermore, the distributions cannot be compared by a number of standard methods of probabilistic inference that are based on an assumption of equal spreads; the standard t-test is one example. Fortunately, remedies for skewness can cure monotone spread as well." (William S Cleveland, "Visualizing Data", 1993)

"Descriptive statistics are built on the assumption that we can use a single value to characterize a single property for a single universe. […] Probability theory is focused on what happens to samples drawn from a known universe. If the data happen to come from different sources, then there are multiple universes with different probability models. If you cannot answer the homogeneity question, then you will not know if you have one probability model or many. [...] Statistical inference assumes that you have a sample that is known to have come from one universe." (Donald J Wheeler, "Myths About Data Analysis", International Lean & Six Sigma Conference, 2012)

"The four questions of data analysis are the questions of description, probability, inference, and homogeneity. [...] Descriptive statistics are built on the assumption that we can use a single value to characterize a single property for a single universe. […] Probability theory is focused on what happens to samples drawn from a known universe. If the data happen to come from different sources, then there are multiple universes with different probability models.  [...] Statistical inference assumes that you have a sample that is known to have come from one universe." (Donald J Wheeler," Myths About Data Analysis", International Lean & Six Sigma Conference, 2012)

"The Second Law of Thermodynamics states that in an isolated system (one that is not taking in energy), entropy never decreases. (The First Law is that energy is conserved; the Third, that a temperature of absolute zero is unreachable.) Closed systems inexorably become less structured, less organized, less able to accomplish interesting and useful outcomes, until they slide into an equilibrium of gray, tepid, homogeneous monotony and stay there." (Steven Pinker, "The Second Law of Thermodynamics", 2017) [source] 

On Homogeneity: Trivia

"For thought raised on specialization the most potent objection to the possibility of a universal organizational science is precisely its universality. Is it ever possible that the same laws be applicable to the combination of astronomic worlds and those of biological cells, of living people and the waves of the ether, of scientific ideas and quanta of energy? [...] Mathematics provide a resolute and irrefutable answer: yes, it is undoubtedly possible, for such is indeed the case. Two and two homogenous separate elements amount to four such elements, be they astronomic systems or mental images, electrons or workers; numerical structures are indifferent to any element, there is no place here for specificity." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"Economics is a science of thinking in terms of models joined to the art of choosing models which are relevant to the contemporary world. It is compelled to be this, because, unlike the typical natural science, the material to which it is applied is, in too many respects, not homogeneous through time. The object of a model is to segregate the semi-permanent or relatively constant factors from those which are transitory or fluctuating so as to develop a logical way of thinking about the latter, and of understanding the time sequences to which they give rise in particular cases." (John M Keynes, [letter to Roy Harrod] 1938)

"On the most usual assumption, the universe is homogeneous on the large scale, i. e. down to regions containing each an appreciable number of nebulae. The homogeneity assumption may then be put in the form: An observer situated in a nebula and moving with the nebula will observe the same properties of the universe as any other similarly situated observer at any time." (Sir Hermann Bondi, "Review of Cosmology," Monthly Notices of the Royal Astronomical Society, 1948)

"The plane is the mainstay of all graphic representation. It is so familiar that its properties seem self-evident, but the most familiar things are often the most poorly understood. The plane is homogeneous and has two dimensions. The visual consequences of these properties must be fully explored." (Jacques Bertin, Semiology of graphics [Semiologie Graphique], 1967)

"The sciences have started to swell. Their philosophical basis has never been very strong. Starting as modest probing operations to unravel the works of God in the world, to follow its traces in nature, they were driven gradually to ever more gigantic generalizations. Since the pieces of the giant puzzle never seemed to fit together perfectly, subsets of smaller, more homogeneous puzzles had to be constructed, in each of which the fit was better." (Erwin Chargaff, "Voices in the Labyrinth", 1975)

"Cybernetics is a homogenous and coherent scientific complex, a science resulting from the blending of at least two sciences - psychology and technology; it is a general and integrative science, a crossroads of sciences, involving both animal and car psychology. It is not just a discipline, circumscribed in a narrow and strictly defined field, but a complex of disciplines born of psychology and centered on it, branched out as branches of a tree in its stem. It is a stepwise synthesis, a suite of multiple, often reciprocal, modeling; syntheses and modeling in which, as a priority, and as a great importance, the modeling of psychology on the technique and then the modeling of the technique on psychology. Cybernetics is an intellectual symphony, a symphony of ideas and sciences." (Stefan Odobleja, "Psihologia consonantista ?i cibernetica" ["Consonatist and Cybernetic Psychology"], 1978)

"The standard process of organizing knowledge into departments, and subderpartments, and further breaking it up into separate courses, tends to conceal the homogeneity of knowledge, and at the same time to omit much which falls between the courses." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"Cellular automata (henceforth: CA) are discrete, abstract computational systems that have proved useful both as general models of complexity and as more specific representations of non-linear dynamics in a variety of scientific fields. Firstly, CA are (typically) spatially and temporally discrete: they are composed of a finite or denumerable set of homogenous, simple units, the atoms or cells. [...] Secondly, CA are abstract: they can be specified in purely mathematical terms and physical structures can implement them. Thirdly, CA are computational systems: they can compute functions and solve algorithmic problems." (Francesco Berto & Jacopo Tagliabue, "Cellular Automata", Stanford Encyclopedia of Philosophy, 2012) [source]

"A significant factor missing from any form of artificial intelligence is the inability of machines to learn based on real life experience. Diversity of life experience is the single most powerful characteristic of being human and enhances how we think, how we learn, our ideas and our ability to innovate. Machines exist in a homogeneous ecosystem, which is ok for solving known challenges, however even Artificial General Intelligence will never challenge humanity in being able to acquire the knowledge, creativity and foresight needed to meet the challenges of the unknown." (Tom Golway, 2021)

On Scale-free

"In contrast to gravitation, interatomic forces are typically modeled as inhomogeneous power laws with at least two different exponents. Such laws (and exponential laws, too) are not scale-free; they necessarily introduce a characteristic length, related to the size of the atoms. Power laws also govern the power spectra of all kinds of noises, most intriguing among them the ubiquitous (but sometimes difficult to explain)." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"In physics, there are numerous phenomena that are said to be 'true on all scales', such as the Heisenberg uncertainty relation, to which no exception has been found over vast ranges of the variables involved (such as energy versus time, or momentum versus position). But even when the size ranges are limited, as in galaxy clusters (by the size of the universe) or the magnetic domains in a piece of iron near the transition point to ferromagnetism (by the size of the magnet), the concept true on all scales is an important postulate in analyzing otherwise often obscure observations." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"Scaling invariance results from the fact that homogeneous power laws lack natural scales; they do not harbor a characteristic unit (such as a unit length, a unit time, or a unit mass). Such laws are therefore also said to be scale-free or, somewhat paradoxically, 'true on all scales'. Of course, this is strictly true only for our mathematical models. A real spring will not expand linearly on all scales; it will eventually break, at some characteristic dilation length. And even Newton's law of gravitation, once properly quantized, will no doubt sprout a characteristic length." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"In networks belonging to the second category, the winner takes all, meaning that the fittest node grabs all links, leaving very little for the rest of the nodes. Such networks develop a star topology, in which all nodes are connected to a central hub. In such a hub-and-spokes network there is a huge gap between the lonely hub and everybody else in the system. Thus a winner-takes-all network is very different from the scale-free networks we encountered earlier, where there is a hierarchy of hubs whose size distribution follows a power law. A winner-takes-all network is not scale-free. Instead there is a single hub and many tiny nodes. This is a very important distinction." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"Networks are not en route from a random to an ordered state. Neither are they at the edge of randomness and chaos. Rather, the scale-free topology is evidence of organizing principles acting at each stage of the network formation process." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"[…] networks are the prerequisite for describing any complex system, indicating that complexity theory must inevitably stand on the shoulders of network theory. It is tempting to step in the footsteps of some of my predecessors and predict whether and when we will tame complexity. If nothing else, such a prediction could serve as a benchmark to be disproven. Looking back at the speed with which we disentangled the networks around us after the discovery of scale-free networks, one thing is sure: Once we stumble across the right vision of complexity, it will take little to bring it to fruition. When that will happen is one of the mysteries that keeps many of us going." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"The first category includes all networks in which, despite the fierce competition for links, the scale-free topology survives. These networks display a fit-get-rich behavior, meaning that the fittest node will inevitably grow to become the biggest hub. The winner's lead is never significant, however. The largest hub is closely followed by a smaller one, which acquires almost as many links as the fittest node. At any moment we have a hierarchy of nodes whose degree distribution follows a power law. In most complex networks, the power law and the fight for links thus are not antagonistic but can coexist peacefully."(Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"At an anatomical level - the level of pure, abstract connectivity - we seem to have stumbled upon a universal pattern of complexity. Disparate networks show the same three tendencies: short chains, high clustering, and scale-free link distributions. The coincidences are eerie, and baffling to interpret." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"In a random network the loss of a small number of nodes can cause the overall network to become incoherent - that is, to break into disconnected subnetworks. In a scale-free network, such an event usually won’t disrupt the overall network because most nodes don’t have many links. But there’s a big caveat to this general principle: if a scale-free network loses a hub, it can be disastrous, because many other nodes depend on that hub." (Thomas Homer-Dixon, "The Upside of Down: Catastrophe, Creativity, and the Renewal of Civilization", 2006)

"Scale-free networks are particularly vulnerable to intentional attack: if someone wants to wreck the whole network, he simply needs to identify and destroy some of its hubs. And here we see how our world’s increasing connectivity really matters. Scientists have found that as a scale-free network like the Internet or our food-distribution system grows- as it adds more nodes - the new nodes tend to hook up with already highly connected hubs." (Thomas Homer-Dixon, "The Upside of Down: Catastrophe, Creativity, and the Renewal of Civilization", 2006)

On Perodicity III

"Since the ellipse is a closed curve it has a total length, λ say, and therefore f(l + λ) = f(l). The elliptic function f is periodic, with 'period' λ, just as the sine function is periodic with period 2π. However, as Gauss discovered in 1797, elliptic functions are even more interesting than this: they have a second, complex period. This discovery completely changed the face of calculus, by showing that some functions should be viewed as functions on the plane of complex numbers. And just as periodic functions on the line can be regarded as functions on a periodic line - that is, on the circle - elliptic functions can be regarded as functions on a doubly periodic plane - that is, on a 2-torus." (John Stillwell, "Yearning for the impossible: the surpnsing truths of mathematics", 2006)

"A typical control goal when controlling chaotic systems is to transform a chaotic trajectory into a periodic one. In terms of control theory it means stabilization of an unstable periodic orbit or equilibrium. A specific feature of this problem is the possibility of achieving the goal by means of an arbitrarily small control action. Other control goals like synchronization and chaotization can also be achieved by small control in many cases." (Alexander L Fradkov, "Cybernetical Physics: From Control of Chaos to Quantum Control", 2007)

"In parametrized dynamical systems a bifurcation occurs when a qualitative change is invoked by a change of parameters. In models such a qualitative change corresponds to transition between dynamical regimes. In the generic theory a finite list of cases is obtained, containing elements like ‘saddle-node’, ‘period doubling’, ‘Hopf bifurcation’ and many others." (Henk W Broer & Heinz Hanssmann, "Hamiltonian Perturbation Theory (and Transition to Chaos)", 2009)

"In fact, contrary to intuition, some of the most complicated dynamics arise from the simplest equations, while complicated equations often produce very simple and uninteresting dynamics. It is nearly impossible to look at a nonlinear equation and predict whether the solution will be chaotic or otherwise complicated. Small variations of a parameter can change a chaotic system into a periodic one, and vice versa." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)

"The main defining feature of chaos is the sensitive dependence on initial conditions. Two nearby initial conditions on the attractor or in the chaotic sea separate by a distance that grows exponentially in time when averaged along the trajectory, leading to long-term unpredictability. The Lyapunov exponent is the average rate of growth of this distance, with a positive value signifying sensitive dependence (chaos), a zero value signifying periodicity (or quasiperiodicity), and a negative value signifying a stable equilibrium." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)

"In dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behaviour. Generally, at a bifurcation, the local stability properties of equilibria, periodic orbits or other invariant sets changes." (Gregory Faye, "An introduction to bifurcation theory", 2011)

"Chaos is just one phenomenon out of many that are encountered in the study of dynamical systems. In addition to behaving chaotically, systems may show fixed equilibria, simple periodic cycles, and more complicated behaviors that defy easy categorization. The study of dynamical systems holds many surprises and shows that the relationships between order and disorder, simplicity and complexity, can be subtle, and counterintuitive." (David P Feldman, "Chaos and Fractals: An Elementary Introduction", 2012)

"A limit cycle is an isolated closed trajectory. Isolated means that neighboring trajectories are not closed; they spiral either toward or away from the limit cycle. If all neighboring trajectories approach the limit cycle, we say the limit cycle is stable or attracting. Otherwise the limit cycle is unstable, or in exceptional cases, half-stable. Stable limit cycles are very important scientifically - they model systems that exhibit self-sustained oscillations. In other words, these systems oscillate even in the absence of external periodic forcing." (Steven H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", 2015)

"The significance of Fourier’s theorem to music cannot be overstated: since every periodic vibration produces a musical sound (provided, of course, that it lies within the audible frequency range), it can be broken down into its harmonic components, and this decomposition is unique; that is, every tone has one, and only one, acoustic spectrum, its harmonic fingerprint. The overtones comprising a musical tone thus play a role somewhat similar to that of the prime numbers in number theory: they are the elementary building blocks from which all sound is made." (Eli Maor, "Music by the Numbers: From Pythagoras to Schoenberg", 2018)

"It is particularly helpful to use complex numbers to model periodic phenomena, especially to operate with phase differences. Mathematically, one can treat a physical quantity as being complex, but address physical meaning only to its real part. Another possibility is to treat the real and imaginary parts of a complex number as two related (real) physical quantities. In both cases, the structure of complex numbers is useful to make calculations more easily, but no physical meaning is actually attached to complex variables." (Ricardo Karam, "Why are complex numbers needed in quantum mechanics? Some answers for the introductory level", American Journal of Physics Vol. 88 (1), 2020)

On Perodicity II

"Engineers have sought to minimize the effects of noise in electronic circuits and communication systems. But recent research has established that noise can play a constructive role in the detection of weak periodic signals." (Kurt Wiesenfeld & Frank Moss, "Stochastic Resonance and the Benefits of Noise: From Ice Ages to Crayfish and SQUIDs", Nature vol. 373, 1995)

"In addition to dimensionality requirements, chaos can occur only in nonlinear situations. In multidimensional settings, this means that at least one term in one equation must be nonlinear while also involving several of the variables. With all linear models, solutions can be expressed as combinations of regular and linear periodic processes, but nonlinearities in a model allow for instabilities in such periodic solutions within certain value ranges for some of the parameters." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"Chaos appears in both dissipative and conservative systems, but there is a difference in its structure in the two types of systems. Conservative systems have no attractors. Initial conditions can give rise to periodic, quasiperiodic, or chaotic motion, but the chaotic motion, unlike that associated with dissipative systems, is not self-similar. In other words, if you magnify it, it does not give smaller copies of itself. A system that does exhibit self-similarity is called fractal. [...] The chaotic orbits in conservative systems are not fractal; they visit all regions of certain small sections of the phase space, and completely avoid other regions. If you magnify a region of the space, it is not self-similar." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"The double periodicity of the torus is fairly obvious: the circle that goes around the torus in the 'long' direction around the rim, together with the circle that goes around it through the hole in the center. And just as periodic functions can be defined on a circle, doubly periodic functions can be defined on a torus." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001

"In the nonmathematical sense, symmetry is associated with regularity in form, pleasing proportions, periodicity, or a harmonious arrangement; thus it is frequently associated with a sense of beauty. In the geometric sense, symmetry may be more precisely analyzed. We may have, for example, an axis of symmetry, a center of symmetry, or a plane of symmetry, which define respectively the line, point, or plane about which a figure or body is symmetrical. The presence of these symmetry elements, usually in combinations, is responsible for giving form to many compositions; the reproduction of a motif by application of symmetry operations can produce a pattern that is pleasing to the senses." (Hans H Jaffé & Milton Orchin, "Symmetry in Chemistry", 2002)

"In colloquial usage, chaos means a state of total disorder. In its technical sense, however, chaos refers to a state that only appears random, but is actually generated by nonrandom laws. As such, it occupies an unfamiliar middle ground between order and disorder. It looks erratic superficially, yet it contains cryptic patterns and is governed by rigid rules. It's predictable in the short run but unpredictable in the long run. And it never repeats itself: Its behavior is nonperiodic." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"Just as a circle is the shape of periodicity, a strange attractor is the shape of chaos. It lives in an abstract mathematical space called state space, whose axes represent all the different variables in a physical system." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"The existence of equilibria or steady periodic solutions is not sufficient to determine if a system will actually behave that way. The stability of these solutions must also be checked. As parameters are changed, a stable motion can become unstable and new solutions may appear. The study of the changes in the dynamic behavior of systems as parameters are varied is the subject of bifurcation theory. Values of the parameters at which the qualitative or topological nature of the motion changes are known as critical or bifurcation values." (Francis C Moona, "Nonlinear Dynamics", 2003)

"A moderate amount of noise leads to enhanced order in excitable systems, manifesting itself in a nearly periodic spiking of single excitable systems, enhancement of synchronized oscillations in coupled systems, and noise-induced stability of spatial pattens in reaction-diffusion systems." (Benjamin Lindner et al, "Effects of Noise in Excitable Systems", Physical Reports. vol. 392, 2004)

"Double periodicity is more interesting than single periodicity, because it is more varied. There is really only one periodic line, since all circles are the same up to a scale factor. However, there are infinitely many doubly periodic planes, even if we ignore scale. This is because the angle between the two periodic axes can vary, and so can the ratio of period lengths. The general picture of a doubly periodic plane is given by a lattice in the plane of complex numbers: a set of points of the form mA + nB, where A and B are nonzero complex numbers in different directions from O, and m and n run through all the integers. A and B are said to generate the lattice because it consists of all their sums and differences. […] The shape of the lattice of points mA + nB can therefore be represented by the complex number A/B. It is not hard to see that any nonzero complex number represents a lattice shape, so in some sense there is whole plane of lattice shapes. Even more interesting: the plane of lattice shapes is a periodic plane, because different numbers represent the same lattice." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

On Perodicity I

"Since a given system can never of its own accord go over into another equally probable state but into a more probable one, it is likewise impossible to construct a system of bodies that after traversing various states returns periodically to its original state, that is a perpetual motion machine." (Ludwig E Boltzmann, "The Second Law of Thermodynamics", [Address to a Formal meeting of the Imperial Academy of Science], 1886)

"Science works by the slow method of the classification of data, arranging the detail patiently in a periodic system into groups of facts, in series like the strata of the rocks. For each series there must be a vocabulary of special words which do not always make good sense when used in another series. But the laws of periodicity seem to hold throughout, among the elements and in every sphere of thought, and we must learn to co-ordinate the whole through our new conception of the reign of relativity." (William H Pallister, "Poems of Science", 1931)

"Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions. (Edward N Lorenz, "Deterministic Nonperiodic Flow", Journal of the Atmospheric Science 20, 1963)

"Now, the main problem with a quasiperiodic theory of turbulence (putting several oscillators together) is the following: when there is a nonlinear coupling between the oscillators, it very often happens that the time evolution does not remain quasiperiodic. As a matter of fact, in this latter situation, one can observe the appearance of a feature which makes the motion completely different from a quasiperiodic one. This feature is called sensitive dependence on initial conditions and turns out to be the conceptual key to reformulating the problem of turbulence." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)

"All physical objects that are 'self-similar' have limited self-similarity - just as there are no perfectly periodic functions, in the mathematical sense, in the real world: most oscillations have a beginning and an end (with the possible exception of our universe, if it is closed and begins a new life cycle after every 'big crunch' […]. Nevertheless, self-similarity is a useful  abstraction, just as periodicity is one of the most useful concepts in the sciences, any finite extent notwithstanding." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"The digits of pi march to infinity in a predestined yet unfathomable code: they do not repeat periodically, seeming to pop up by blind chance, lacking any perceivable order, rule, reason, or design - ‘random’ integers, ad infinitum." (Richard Preston, "The Mountains of Pi", The New Yorker, March 2, 1992)

"Clearly, however, a zero probability is not the same thing as an impossibility; […] In systems that are now called chaotic, most initial states are followed by nonperiodic behavior, and only a special few lead to periodicity. […] In limited chaos, encountering nonperiodic behavior is analogous to striking a point on the diagonal of the square; although it is possible, its probability is zero. In full chaos, the probability of encountering periodic behavior is zero." (Edward N Lorenz, "The Essence of Chaos", 1993)

"The description of the evolutionary trajectory of dynamical systems as irreversible, periodically chaotic, and strongly nonlinear fits certain features of the historical development of human societies. But the description of evolutionary processes, whether in nature or in history, has additional elements. These elements include such factors as the convergence of existing systems on progressively higher organizational levels, the increasingly efficient exploitation by systems of the sources of free energy in their environment, and the complexification of systems structure in states progressively further removed from thermodynamic equilibrium." (Ervin László et al, "The Evolution of Cognitive Maps: New Paradigms for the Twenty-first Century", 1993) 

"There is no question but that the chains of events through which chaos can develop out of regularity, or regularity out of chaos, are essential aspects of families of dynamical systems [...]  Sometimes [...] a nearly imperceptible change in a constant will produce a qualitative change in the system’s behaviour: from steady to periodic, from steady or periodic to almost periodic, or from steady, periodic, or almost periodic to chaotic. Even chaos can change abruptly to more complicated chaos, and, of course, each of these changes can proceed in the opposite direction. Such changes are called bifurcations." (Edward Lorenz, "The Essence of Chaos", 1993)

"As with subtle bifurcations, catastrophes also involve a control parameter. When the value of that parameter is below a bifurcation point, the system is dominated by one attractor. When the value of that parameter is above the bifurcation point, another attractor dominates. Thus the fundamental characteristic of a catastrophe is the sudden disappearance of one attractor and its basin, combined with the dominant emergence of another attractor. Any type of attractor static, periodic, or chaotic can be involved in this. Elementary catastrophe theory involves static attractors, such as points. Because multidimensional surfaces can also attract (together with attracting points on these surfaces), we refer to them more generally as attracting hypersurfaces, limit sets, or simply attractors." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

On Power Laws

"In contrast to gravitation, interatomic forces are typically modeled as inhomogeneous power laws with at least two different exponents. Such laws (and exponential laws, too) are not scale-free; they necessarily introduce a characteristic length, related to the size of the atoms. Power laws also govern the power spectra of all kinds of noises, most intriguing among them the ubiquitous (but sometimes difficult to explain)." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"[…] power laws, with integer or fractional exponents, are one of the most fertile fields and abundant sources of self-similarity." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"Scaling invariance results from the fact that homogeneous power laws lack natural scales; they do not harbor a characteristic unit (such as a unit length, a unit time, or a unit mass). Such laws are therefore also said to be scale-free or, somewhat paradoxically, "true on all scales." Of course, this is strictly true only for our mathematical models. A real spring will not expand linearly on all scales; it will eventually break, at some characteristic dilation length. And even Newton's law of gravitation, once properly quantized, will no doubt sprout a characteristic length." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"Nature normally hates power laws. In ordinary systems all quantities follow bell curves, and correlations decay rapidly, obeying exponential laws. But all that changes if the system is forced to undergo a phase transition. Then power laws emerge-nature's unmistakable sign that chaos is departing in favor of order. The theory of phase transitions told us loud and clear that the road from disorder to order is maintained by the powerful forces of self-organization and is paved by power laws. It told us that power laws are not just another way of characterizing a system's behavior. They are the patent signatures of self-organization in complex systems." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"From a purely mathematical perspective, a power law signifies nothing in particular - it's just one of many possible kinds of algebraic relationship. But when a physicist sees a power law, his eyes light up. For power laws hint that a system may be organizing itself. They arise at phase transitions, when a system is poised at the brink, teetering between order and chaos. They arise in fractals, when an arbitrarily small piece of a complex shape is a microcosm of the whole. They arise in the statistics of natural hazards - avalanches and earthquakes, floods and forest fires - whose sizes fluctuate so erratically from one event to the next that the average cannot adequately stand in for the distribution as a whole." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"An event occurring at one node will cause a cascade of events: often this cascade or avalanche propagates to affect only one or two further elements, occasionally it affects more, and more rarely it affects many. The mathematical theory of this - which is very much part of complexity theory - shows that propagations of events causing further events show characteristic properties such as power laws (caused by many and frequent small propagations, few and infrequent large ones), heavy tailed probability distributions (lengthy propagations though rare appear more frequently than normal distributions would predict), and long correlations (events can and do propagate for long distances and times)." (W Brian Arthur, "Complexity and the Economy", 2015) 

"But note that any heavy tailed process, even a power law, can be described in sample (that is finite number of observations necessarily discretized) by a simple Gaussian process with changing variance, a regime switching process, or a combination of Gaussian plus a series of variable jumps (though not one where jumps are of equal size […])." (Nassim N Taleb, "Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)

Manfred Schroeder - Collected Quotes

"A Markov process is a stochastic process in which present events depend on the past only through some finite number of generations. In a first-order Markov process, the influential past is limited to a single earlier generation: the present can be fully accounted for by the immediate past." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"[…] a pink (or white, or brown) noise is the very paradigm of a statistically self-similar process. Phenomena whose power spectra are homogeneous power functions lack inherent time and frequency scales; they are scale-free. There is no characteristic time or frequency -whatever happens in one time or frequency range happens on all time or frequency scales. If such noises are recorded on magnetic tape and played back at various speeds, they sound the same […]" (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"All physical objects that are 'self-similar' have limited self-similarity - just as there are no perfectly periodic functions, in the mathematical sense, in the real world: most oscillations have a beginning and an end (with the possible exception of our universe, if it is closed and begins a new life cycle after every 'big crunch' […]. Nevertheless, self-similarity is a useful  abstraction, just as periodicity is one of the most useful concepts in the sciences, any finite extent notwithstanding." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"[…] an epidemic does not always percolate through an entire population. There is a percolation threshold below which the epidemic has died out before most of the people have." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"Apart from power laws, iteration is one of the prime sources of self-similarity. Iteration here means the repeated application of some rule or operation - doing the same thing over and over again. […] A concept closely related to iteration is recursion. In an age of increasing automation and computation, many processes and calculations are recursive, and if a recursive algorithm is in fact repetitious, self-similarity is waiting in the wings."(Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"Formally, a Cantor set is defined as a set that is totally disconnected, closed, and perfect. A totally disconnected set is a set that contains no intervals and therefore has no interior points. A closed set is one that contains all its boundary elements. (A boundary element is an element that contains elements both inside and outside the set in arbitrarily small neighborhoods.) A perfect set is a nonempty set that is equal to the set of its accumulation points. All three conditions are met by our middle-third—erasing construction, the original Cantor set." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"[…] homogeneous functions have an interesting scaling property: they reproduce themselves upon rescaling. This scaling invariance can shed light into some of the darker corners of physics, biology, and other sciences, and even illuminate our appreciation of music." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"In a white-noise process, every value of the process (e.g., the successive frequencies of a melody) is completely independent of its past - it is a total surprise. By contrast, in 'brown music' (a term derived from Brownian motion), only the increments are independent of the past, giving rise to a rather boring tune." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"In contrast to gravitation, interatomic forces are typically modeled as inhomogeneous power laws with at least two different exponents. Such laws (and exponential laws, too) are not scale-free; they necessarily introduce a characteristic length, related to the size of the atoms. Power laws also govern the power spectra of all kinds of noises, most intriguing among them the ubiquitous (but sometimes difficult to explain)." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"In physics, there are numerous phenomena that are said to be 'true on all scales', such as the Heisenberg uncertainty relation, to which no exception has been found over vast ranges of the variables involved (such as energy versus time, or momentum versus position). But even when the size ranges are limited, as in galaxy clusters (by the size of the universe) or the magnetic domains in a piece of iron near the transition point to ferromagnetism (by the size of the magnet), the concept true on all scales is an important postulate in analyzing otherwise often obscure observations." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"Nature abounds with periodic phenomena: from the motion of a swing to the oscillations of atoms, from the chirping of a grasshopper to the orbits of the heavenly bodies. […] Of course, nothing in nature is exactly periodic. All motion has a beginning and an end, so that, in the mathematical sense, strict periodicity does not exist in the real world. Nevertheless, periodicity has proved to be a supremely useful concept in elucidating underlying laws and mechanisms in many fields." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"Percolation is a widespread paradigm. Percolation theory can therefore illuminate a great many seemingly diverse situations. Because of its basically geometric character, it facilitates the analysis of intricate patterns and textures without needless physical complications. And the self-similarity that prevails at critical points permits profitably mining the connection with scaling and fractals." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"[…] physicists have come to appreciate a fourth kind of temporal behavior: deterministic chaos, which is aperiodic, just like random noise, but distinct from the latter because it is the result of deterministic equations. In dynamic systems such chaos is often characterized by small fractal dimensions because a chaotic process in phase space typically fills only a small part of the entire, energetically available space." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"[…] power laws, with integer or fractional exponents, are one of the most fertile fields and abundant sources of self-similarity." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"Scaling invariance results from the fact that homogeneous power laws lack natural scales; they do not harbor a characteristic unit (such as a unit length, a unit time, or a unit mass). Such laws are therefore also said to be scale-free or, somewhat paradoxically, 'true on all scales'. Of course, this is strictly true only for our mathematical models. A real spring will not expand linearly on all scales; it will eventually break, at some characteristic dilation length. And even Newton's law of gravitation, once properly quantized, will no doubt sprout a characteristic length." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"The only prerequisite for a self-similar law to prevail in a given size range is the absence of an inherent size scale." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"The unifying concept underlying fractals, chaos, and power laws is self-similarity. Self-similarity, or invariance against changes in scale or size, is an attribute of many laws of nature and innumerable phenomena in the world around us. Self-similarity is, in fact, one of the decisive symmetries that shape our universe and our efforts to comprehend it." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"[…] the world is not complete chaos. Strange attractors often do have structure: like the Sierpinski gasket, they are self-similar or approximately so. And they have fractal dimensions that hold important clues for our attempts to understand chaotic systems such as the weather." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"Understandably, invariant sets (and their complements) play a crucial role in dynamic systems in general because they tell the most important fact about any initial condition, namely, its eventual fate: will the iterates be bounded, or will they be unstable and diverge? Or will the orbit be periodic or aperiodic?" (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

On the Monte Carlo Method

"The Monte Carlo method is a numerical method of solving mathematical problems by the simulation of random variables. [...] One advantageous feature of the Monte Carlo method is the simple structure of the computation algorithm. As a rule, a program is written to carry out one random trial [...]. This trial is repeated N times, each trial being independent of the rest, and then the results of all trials are averaged. Therefore, the Monte Carlo method is sometimes called the method of statistical trials." (Ilya M Sobol, "A Primer for the Monte Carlo Method", 1994)

"To understand what kinds of problems are solvable by the Monte Carlo method, it is important to note that the method enables simulation of any process whose development is influenced by random factors. Second, for many mathematical problems involving no chance, the method enables us to artificially construct a probabilistic model (or several such models), making possible the solution of the problems." (Ilya M Sobol, "A Primer for the Monte Carlo Method", 1994)

"Indeed, the frequency of crashes in the Monte Carlo simulations was much smaller than the frequency of crashes in the real data: if one of the most frequently used benchmarks of the industry is incapable of reproducing the observed frequency of crashes, this indeed means that there is something to explain that may require new concepts and methods." (Didier Sornette, "Why Stock Markets Crash: Critical Events in Complex Systems", 2003)

"Monte Carlo simulations handle uncertainty by using a computer’s random number generator to determine outcomes. Done over and over again, the simulations show the distribution of the possible outcomes. [...] The beauty of these Monte Carlo simulations is that they allow users to see the probabilistic consequences of their decisions, so that they can make informed choices. [...] Monte Carlo simulations are one of the most valuable applications of data science because they can be used to analyze virtually any uncertain situation where we are able to specify the nature of the uncertainty [...]" (Gary Smith & Jay Cordes, "The 9 Pitfalls of Data Science", 2019)

"The Monte Carlo tree search method is naturally suited to non-deterministic settings such as card games or backgammon. Minimax trees are not well suited to non-deterministic settings because of the inability to predict the opponent’s moves while building the tree. On the other hand, Monte Carlo tree search is naturally suited to handling such settings, since the desirability of moves is always evaluated in an expected sense. The randomness in the game can be naturally combined with the randomness in move sampling in order to learn the expected outcomes from each choice of move." (Charu C Aggarwal, "Artificial Intelligence: A Textbook", 2021)

"The nice thing with Monte Carlo is that you play a game of let’s pretend, like this: first of all there are ten scenarios with different probabilities, so let’s first pick a probability. The dice in this case is a random number generator in the computer. You roll the dice and pick a scenario to work with. Then you roll the dice for a certain speed, and you roll the dice again to see what direction it took. The last thing is that it collided with the bottom at an unknown time so you roll dice for the unknown time. So now you have speed, direction, starting point, time. Given them all, I know precisely where it [could have] hit the bottom. You have the computer put a point there. Rolling dice, I come up with different factors for each scenario. If I had enough patience, I could do it with pencil and paper. We calculated ten thousand points. So you have ten thousand points on the bottom of the ocean that represent equally likely positions of the sub. Then you draw a grid, count the points in each cell of the grid, saying that 10% of the points fall in this cell, 1% in that cell, and those percentages are what you use for probabilities for the prior for the individual distributions." (Henry R Richardson) [in (Sharon B McGrayne, "The Theory That Would Not Die", 2011)]

On Prediction XV: Systems III

"The state of a system at a given moment depends on two things - its initial state, and the law according to which that state varies. If we know both this law and this initial state, we have a simple mathematical problem to solve, and we fall back upon our first degree of ignorance. Then it often happens that we know the law and do not know the initial state. It may be asked, for instance, what is the present distribution of the minor planets? We know that from all time they have obeyed the laws of Kepler, but we do not know what was their initial distribution. In the kinetic theory of gases we assume that the gaseous molecules follow rectilinear paths and obey the laws of impact and elastic bodies; yet as we know nothing of their initial velocities, we know nothing of their present velocities. The calculus of probabilities alone enables us to predict the mean phenomena which will result from a combination of these velocities. This is the second degree of ignorance. Finally it is possible, that not only the initial conditions but the laws themselves are unknown. We then reach the third degree of ignorance, and in general we can no longer affirm anything at all as to the probability of a phenomenon. It often happens that instead of trying to discover an event by means of a more or less imperfect knowledge of the law, the events may be known, and we want to find the law; or that, instead of deducing effects from causes, we wish to deduce the causes." (Henri Poincaré, "Science and Hypothesis", 1902)

"Although a system may exhibit sensitive dependence on initial conditions, this does not mean that everything is unpredictable about it. In fact, finding what is predictable in a background of chaos is a deep and important problem. (Which means that, regrettably, it is unsolved.) In dealing with this deep and important problem, and for want of a better approach, we shall use common sense." (David Ruelle, "Chance and Chaos", 1991)

"Although the detailed moment-to-moment behavior of a chaotic system cannot be predicted, the overall pattern of its 'random' fluctuations may be similar from scale to scale. Likewise, while the fine details of a chaotic system cannot be predicted one can know a little bit about the range of its 'random' fluctuation." (F David Peat, "From Certainty to Uncertainty", 2002)

"[…] while chaos theory deals in regions of randomness and chance, its equations are entirely deterministic. Plug in the relevant numbers and out comes the answer. In principle at least, dealing with a chaotic system is no different from predicting the fall of an apple or sending a rocket to the moon. In each case deterministic laws govern the system. This is where the chance of chaos differs from the chance that is inherent in quantum theory." (F David Peat, "From Certainty to Uncertainty", 2002)

"Chaos is a phenomenon encountered in science and mathematics wherein a deterministic (rule-based) system behaves unpredictably. That is, a system which is governed by fixed, precise rules, nevertheless behaves in a way which is, for all practical purposes, unpredictable in the long run. The mathematical use of the word 'chaos' does not align well with its more common usage to indicate lawlessness or the complete absence of order. On the contrary, mathematically chaotic systems are, in a sense, perfectly ordered, despite their apparent randomness. This seems like nonsense, but it is not." (David P Feldman, "Chaos and Fractals: An Elementary Introduction", 2012)

David P Feldman - Collected Quotes

"Chaos is a phenomenon encountered in science and mathematics wherein a deterministic (rule-based) system behaves unpredictably. That is, a system which is governed by fixed, precise rules, nevertheless behaves in a way which is, for all practical purposes, unpredictable in the long run. The mathematical use of the word 'chaos' does not align well with its more common usage to indicate lawlessness or the complete absence of order. On the contrary, mathematically chaotic systems are, in a sense, perfectly ordered, despite their apparent randomness. This seems like nonsense, but it is not." (David P Feldman, "Chaos and Fractals: An Elementary Introduction", 2012)

"Chaos is just one phenomenon out of many that are encountered in the study of dynamical systems. In addition to behaving chaotically, systems may show fixed equilibria, simple periodic cycles, and more complicated behaviors that defy easy categorization. The study of dynamical systems holds many surprises and shows that the relationships between order and disorder, simplicity and complexity, can be subtle, and counterintuitive." (David P Feldman, "Chaos and Fractals: An Elementary Introduction", 2012)

"Understanding chaos requires much less advanced mathematics than other current areas of physics research such as general relativity or particle physics. Observing chaos and fractals requires no specialized equipment; chaos is seen in scores of everyday phenomena - a boiling pot of water, a dripping faucet, shifting weather patterns. And fractals are almost ubiquitous in the natural world. Thus, it is possible to teach the central ideas and insights of chaos in a rigorous, genuine, and relevant way to students with relatively little mathematics background." (David P Feldman, "Chaos and Fractals: An Elementary Introduction", 2012)

"A dynamical system is any mathematical system that changes in time according to a well specified rule." (David P Feldman, "Chaos and Dynamical Systems", 2019)

"Typically, in a mathematical model or the real world, one only expects to observe stable fixed points. An unstable fixed point is susceptible to a small perturbation; a tiny external influence will move the system away from the unstable fixed point." (David P Feldman, "Chaos and Dynamical Systems", 2019)

Richard Bandler - Collected Quotes

"Deletion is a process by which we selectively pay attention to certain dimensions of our experience and exclude others. Take, for example, the ability that people have to filter out or exclude all other sound in a room full of people talking in order to listen to one particular person's voice. Using the same process, people are able to block themselves from hearing messages of caring from other people who are important to them. [...] Deletion reduces the world to proportions which we feel capable of handling. The reduction may be useful in some contexts and yet be the source of pain for us in others." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"Distortion is the process which allows us to make shifts in our experience of sensory data. Fantasy, for example, allows us to prepare for experiences which we may have before they occur. People will distort present reality when rehearsing a speech which they will later present. It is this process which has made possible all the artistic creations which we as humans have produced." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"Generalization is the process by which elements or pieces of a person's model become detached from their original experience and come to represent the entire category of which the experience is an example. bur ability to generalize is essential to coping with the world." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"Generalization may impoverish the client's model by causing loss of the detail and richness of their original experiences. Thus, generalization prevents them from making distinctions which would give them a fuller set of choices in coping with any particular situation. At the same time, the generalization expands the specific painful experience to the level of being persecuted by the universe (an insurmountable obstacle to coping)." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"Models are not intended to either reflect or construct a single objective reality. Rather, their purpose is to simulate some aspect of a possible reality. In NLP, for instance, it is not important whether or not a model is 'true' , but rather that it is 'useful' . In fact, all models can be perceived as symbolic or metaphoric, as opposed to reflective of reality. Whether the description being used is metaphorical or literal, the usefulness of a model depends on the degree to which it allows us to move effectively to the next step in the sequence of transformations connecting deeper structures and surface structures. Instead of 'constructing' reality, models establish a set of functions that serve as a tool or a bridge between deep structures and surface structures. It is this bridge that forms our 'understanding' of reality and allows us to generate new experiences and expressions of reality." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"The most pervasive paradox of the human condition which we see is that the processes which allow us to survive, grow, change, and experience joy are the same processes which allow us to maintain an impoverished model of the world - our ability to manipulate symbols, that is, to create models. So the processes which allow us to accomplish the most extraordinary and unique human activities are the same processes which block our further growth if we commit the error of mistaking the model of the world for reality." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"[…] there is an irreducible difference between the world and our experience of it. We as human beings do not operate directly on the world. Each of us creates a representation of the world in which we live - that is, we create a map or model which we use to generate our behavior. Our representation of the world determines to a large degree what our experience of the world will be, how we will perceive the world, what choices we will see available to us as we live in the world." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"To say that our communication, our language, is a system is to say that it has structure, that there is some set of rules which identify. I which sequences of words will make sense, will represent a model of our experience. In other words, our behavior when creating a I representation or when communicating is rule-governed behavior. Even though we are not normally aware of the structure in the process of representation and communication, that structure, the structure of language, can be understood in terms of regular patterns." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"We are almost never conscious of the way in which we order and structure the words we select. Language so fills our world that we move through it as a fish swims through water. Although we have little or no consciousness of the way in which we form our communication, our activity - the process of using language - is highly structured." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

Bilash K Bala - Collected Quotes

"A model may be defined as a substitute of any object or system. […] A mental image used in thinking is a model, and it is not the real system. A written description of a system is a model that presents one aspect of reality. The simulation model is logically complete and describes the dynamic behaviour of the system. Models can be broadly classified as (a) physical models and (b) abstract models [..] Mental models and mathematical models are examples of abstract models." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)

"Feedback systems are closed loop systems, and the inputs are changed on the basis of output. A feedback system has a closed loop structure that brings back the results of the past action to control the future action. In a closed system, the problem is perceived, action is taken and the result influences the further action. Thus, the distinguishing feature of a closed loop system is a feedback path of information, decision and action connecting the output to input." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)

"For a scientist, a model is useful if it generates insight into the structure of [the] real system, makes correct prediction and stimulates meaningful questions for future research. For the public and political leaders, a model is useful if it explains the causes of important problems and provides a basis for designing policy to improve the behaviour of the system. Validity meaning confidence in a model’s usefulness is inherently relative concepts. One must choose between competing models." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)

"The goal of a system dynamics approach is to understand how a dynamic pattern of behaviour is generated by a system and to find leverage points within the system structure that have the potential to change the problematic trend to a more desirable one. The key steps in a system dynamics approach are identifying one or more trends that characterise the problem, describing the structure of the system generating the behaviour and finding and testing leverage points in the system to change the problematic behaviour. System dynamics is an appropriate modelling approach for sustainability questions because of the long-term perspective and feedback dynamics inherent in such questions." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)

"The model should be robust under extreme conditions. There is an important direct structure test to the robustness of the model under direct extreme conditions, and it evaluates the validity of the equations under extreme conditions by assessing the plausibility of the resulting values against knowledge/anticipation of what would happen under similar conditions in real life." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)

"[…] the system boundary should encompass that portion of the whole system which includes all the important and relevant variables to address the problem and the purpose of policy analysis and design. The scope of the study should be clearly stated in order to identify the causes of the problem for clear understanding of the problem and policies for solving the problem in the short run and long run." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)

"There is nothing in either physical or social science about which we have perfect knowledge and information. We can never say that a model is a perfect representation of the reality. On the other hand, we can say that there is nothing of which we know absolutely nothing. So, models should not be judged on an absolute scale but on relative scale if the models clarify our knowledge and provide insights into systems." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)

On Feedback XII

"[…] feedback is not necessarily transmitted and returned through the same system component - or even through the same system. It may travel through several intervening components within the system first, or return from an external system, before finally arriving again at the component where it started." (Virginia Anderson & Lauren Johnson, "Systems Thinking Basics: From Concepts to Causal Loops", 1997)

"Feedback is the transmission and return of information. […] A system has feedback within itself. But because all systems are part of larger systems, a system also has feedback between itself and external systems. In some systems, the feedback and adjustment processes happen so quickly that it is relatively easy for an observer to follow. In other systems, it may take a long time before the feedback is returned, so an observer would have trouble identifying the action that prompted the feedback." (Virginia Anderson & Lauren Johnson, "Systems Thinking Basics: From Concepts to Causal Loops", 1997)

"In a complex system, it is not uncommon for subsystems to have goals that compete directly with or diverge from the goals of the overall system. […] Feedback gathered from small, local subsystems for use by larger subsystems may be either inaccurately conveyed or inaccurately interpreted. Yet it is this very flexibility and looseness that allow large, complex systems to endure, although it can be hard to predict what these organizations are likely to do next." (Virginia Anderson & Lauren Johnson, "Systems Thinking Basics: From Concepts to Causal Loops", 1997)

"Reinforcing loops can be seen as the engines of growth and collapse. That is, they compound change in one direction with even more change in that direction. Many reinforcing loops have a quality of accelerating movement in a particular direction, a sense that the more one variable changes, the more another changes." (Virginia Anderson & Lauren Johnson, "Systems Thinking Basics: From Concepts to Causal Loops", 1997)

"In negative feedback regulation the organism has set points to which different parameters (temperature, volume, pressure, etc.) have to be adapted to maintain the normal state and stability of the body. The momentary value refers to the values at the time the parameters have been measured. When a parameter changes it has to be turned back to its set point. Oscillations are characteristic to negative feedback regulation […]" (Gaspar Banfalvi, "Homeostasis - Tumor – Metastasis", 2014)

"The work around the complex systems map supported a concentration on causal mechanisms. This enabled poor system responses to be diagnosed as the unanticipated effects of previous policies as well as identification of the drivers of the sector. Understanding the feedback mechanisms in play then allowed experimentation with possible future policies and the creation of a coherent and mutually supporting package of recommendations for change."  (David C Lane et al, "Blending systems thinking approaches for organisational analysis: reviewing child protection", 2015)

"Feedback systems are closed loop systems, and the inputs are changed on the basis of output. A feedback system has a closed loop structure that brings back the results of the past action to control the future action. In a closed system, the problem is perceived, action is taken and the result influences the further action. Thus, the distinguishing feature of a closed loop system is a feedback path of information, decision and action connecting the output to input." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)

Avner Ash - Collected Quotes

"A group is a set along with a rule that tells how to combine any two elements in the set to get another element in the set. We usually use the word composition to describe the act of combining two elements of the group to get a third." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"A symmetry is a function that preserves what we feel is important about an object." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Although it is not difficult to count the holes in a real pretzel in your hand, prior to eating it, when a surface pops out of an abstract mathematical construction it can be very difficult to figure out its properties, such as how many holes it has. The cohomology groups can help us to do so." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Lie groups turn up when we study a geometric object with a lot of symmetry, such as a sphere, a circle, or flat spacetime. Because there is so much symmetry, there are many functions from the object to itself that preserve the geometry, and these functions become the elements of the group." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Many people believe that all of mathematics has already been discovered and codified. Mathematicians (they think) do nothing except rearrange the material in different ways for different types of students. This seems to be the result of the cut-and-dried method of teaching mathematics in many high schools and universities. The facts are laid out in the cleanest logical order. Little attempt is made to show how someone once had to invent it all, at first in a confused way, and that only later was it possible to give it this neat form." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Mathematicians are just lucky that elliptic curves are fairly simple - they only involve two variables and no powers higher than the cube - and yet are so rich." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Mathematicians often get bored by a problem after they have fully understood it and have given proofs of their conjectures. Sometimes they even forget the precise details of what they have done after the lapse of years, having refocused their interest in another area. The common notion of the mathematician contemplating timeless truths, thinking over the same proof again and again - Euclid looking on beauty bare - is rarely true in any static sense." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Mathematics is like a game. It has rules, and to enjoy playing or watching it, you have to know and understand the rules. Mathematicians make up the rules as they go along." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"One thing mathematicians do is connect concepts that occur in different trains of thought." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Some number patterns, like even and odd numbers, lie on the surface. But the more you learn about numbers, both experimentally and theoretically, the more you discover patterns that are not so obvious. […] After a hidden pattern is exposed, it can be used to find more hidden patterns. At the end of a long chain of patterned reasoning, you can get to very difficult theorems, exploring facts about numbers that you otherwise would not know were true." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Still, in the end, we find ourselves drawn to the beauty of the patterns themselves, and the amazing fact that we humans are smart enough to prove even a feeble fraction of all possible theorems about them. Often, greater than the contemplation of this beauty for the active mathematician is the excitement of the chase. Trying to discover first what patterns actually do or do not occur, then finding the correct statement of a conjecture, and finally proving it - these things are exhilarating when accomplished successfully. Like all risk-takers, mathematicians labor months or years for these moments of success." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"The set of complex numbers is another example of a field. It is handy because every polynomial in one variable with integer coefficients can be factored into linear factors if we use complex numbers. Equivalently, every such polynomial has a complex root. This gives us a standard place to keep track of the solutions to polynomial equations." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"The word conjecture means 'guess'. The way it is used in mathematics is 'educated guess'." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"There is a big debate as to whether logic is part of mathematics or mathematics is part of logic. We use logic to think. We notice that our thinking, when it is valid, goes in certain patterns. These patterns can be studied mathematically. Thus, logic is a part of mathematics, called 'mathematical logic'." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006) 

"What is a group? It is just a pattern that certain things can exhibit when you have a composition law for always getting a third thing by combining any two others." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

Avinash K Dixit - Collected Quotes

"A game is a situation of strategic interdependence: the outcome of your choices (strategies) depends upon the choices of one or more other persons acting purposely. The decision makers involved in a game are called players, and their choices are called moves. The interests of the players in a game may be in strict conflict; one person’s gain is always another’s loss. Such games are called zero-sum. More typically, there are zones of commonality of interests as well as of conflict and so, there can be combinations of mutually gainful or mutually harmful strategies. Nevertheless, we usually refer to the other players in a game as one’s rivals." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"Chess experts have been successful at characterizing optimal strategies near the end of the game. Once the chessboard has only a small number of pieces on it, experts are able to look ahead to the end of the game and determine by backward reasoning whether one side has a guaranteed win or whether the other side can obtain a draw. But the middle of the game, when several pieces remain on the board, is far harder. Looking ahead five pairs of moves, which is about as much as can be done by experts in a reasonable amount of time, is not going to simplify the situation to a point where the endgame can be solved completely from there on." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"Chess strategy illustrates another important practical feature of looking forward and reasoning backward: you have to play the game from the perspective of both players. While it is hard to calculate your best move in a complicated tree, it is even harder to predict what the other side will do." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"John Nash’s beautiful equilibrium was designed as a theoretical way to square just such circles of thinking about thinking about other people’s choices in games of strategy. The idea is to look for an outcome where each player in the game chooses the strategy that best serves his or her own interest, in response to the other’s strategy. If such a configuration of strategies arises, neither player has any reason to change his choice unilaterally. Therefore, this is a potentially stable outcome of a game where the players make individual and simultaneous choices of strategies." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"Many mathematical game theorists dislike the dependence of an outcome on historical, cultural, or linguistic aspects of the game or on purely arbitrary devices like round numbers; they would prefer the solution be determined purely by the abstract mathematical facts about the game - the number of players, the strategies available to each, and the payoffs to each in relation to the strategy choices of all. We disagree. We think it entirely appropriate that the outcome of a game played by humans interacting in a society should depend on the social and psychological aspects of the game." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"Science and art, by their very nature, differ in that science can be learned in a systematic and logical way, whereas expertise in art has to be acquired by example, experience, and practice." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"Strategic thinking starts with your basic skills and considers how best to use them. Knowing the law, you must decide the strategy for defending your client. Knowing how well your football team can pass or run and how well the other team can defend against each choice, your decision as the coach is whether to pass or to run. Sometimes, as in the case of nuclear brinkmanship, strategic thinking also means knowing when not to play." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"The essence of a game of strategy is the interdependence of the players’ decisions. These interactions arise in two ways. The first is sequential [...] The players make alternating moves. [...] The second kind of interaction is simultaneous, as in the prisoners’ dilemma [...] The players act at the same time, in ignorance of the others’ current actions. However, each must be aware that there are other active players, who in turn are similarly aware, and so on. Therefore each must figuratively put himself in the shoes of all and try to calculate the outcome. His own best action is an integral part of this overall calculation. When you find yourself playing a strategic game, you must determine whether the interaction is simultaneous or sequential." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"When playing mixed or random strategies, you can’t fool the opposition every time. The best you can hope for is to keep them guessing and fool them some of the time. You can know the likelihood of your success but cannot say in advance whether you will succeed on any particular occasion. In this regard, when you know that you are talking to a person who wants to mislead you, it may be best to ignore any statements he makes rather than accept them at face value or to infer that exactly the opposite must be the truth." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

On Art IX: Metaphysics

"Metaphysics may be, after all, only the art of being sure of something that is not so and logic only the art of going wrong with confidence." (Joseph W Krutch, "The Modern Temper", 1929)

"As to the role of emotions in art and the subconscious mechanism that serves as the integrating factor both in artistic creation and in man's response to art, they involve a psychological phenomenon which we call a sense of life. A sense of life is a pre-conceptual equivalent of metaphysics, an emotional, subconsciously integrated appraisal of man and of existence." (Ayn Rand, "The Romantic Manifesto: A Philosophy of Literature", 1969)

"For some years now the activity of the artist in our society has been trending more toward the function of the ecologist: one who deals with environmental relationships. Ecology is defined as the totality or pattern of relations between organisms and their environment. Thus the act of creation for the new artist is not so much the invention of new objects as the revelation of previously unrecognized relation- ships between existing phenomena, both physical and metaphysical. So we find that ecology is art in the most fundamental and pragmatic sense, expanding our apprehension of reality." (Gene Youngblood, "Expanded Cinema", 1970)

"The mystery of sound is mysticism; the harmony of life is religion. The knowledge of vibrations is metaphysics, the analysis of atoms is science, and their harmonious grouping is art. The rhythm of form is poetry, and the rhythm of sound is music. This shows that music is the art of arts and the science of all sciences; and it contains the fountain of all knowledge within itself." (Inayat Khan, "The Mysticism of Sound and Music", 1996)

 "Science, as usually taught to liberal arts students, emphasizes results rather than method, and tries to teach technique rather than to give insight into and understanding of the scientific habit of thought. What is needed, however, is not a dose of metaphysics but a truly humanistic teaching of science." (Harry D Gideonse)

On Art VIII: Physics

"It is impossible to follow the march of one of the greatest theories of physics, to see it unroll majestically its regular deductions starting from initial hypotheses, to see its consequences represent a multitude of experimental laws down to the smallest detail, without being charmed by the beauty of such a construction, without feeling keenly that such a creation of the human mind is truly a work of art." (Pierre-Maurice-Marie DuhemDuhem, "The Aim and Structure of Physical Theory", 1908)

"What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze." (Paul Klee, "Bauhaus prospectus", 1929)

"Mathematicians who build new spaces and physicists who find them in the universe can profit from the study of pictorial and architectural spaces conceived and built by men of art." (György Kepes, "The New Landscape In Art and Science", 1956)

"A more problematic example is the parallel between the increasingly abstract and insubstantial picture of the physical universe which modern physics has given us and the popularity of abstract and non-representational forms of art and poetry. In each case the representation of reality is increasingly removed from the picture which is immediately presented to us by our senses." (Harvey Brooks, "Scientific Concepts and Cultural Change", 1965)

"Part of the art and skill of the engineer and of the experimental physicist is to create conditions in which certain events are sure to occur." (Eugene P Wigner, "Symmetries and Reflections", 1979)

"There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depends heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood." (Paul Lockhart, "A Mathematician's Lament", 2009)

Subrahmanyan Chandrasekhar - Collected Quotes

"All of us are sensitive to nature's beauty. It is not unreasonable that some aspects of this beauty are shared by the natural sciences." (Subrahmanyan Chandrasekhar, "Beauty and the Quest for Beauty in Science", Physics Today Vol. 32 (7), [lecture] 1979)

"It is, indeed an incredible fact that what the human mind, at its deepest and most profound, perceives as beautiful finds its realization in external nature.[...] What is intelligible is also beautiful." (Subrahmanyan Chandrasekhar, "Beauty and the Quest for Beauty in Science", Physics Today Vol. 32 (7), [lecture] 1979)

"The black holes of nature are the most perfect macroscopic objects there are in the universe: the only elements in their construction are our concepts of space and time." (Subrahmanyan Chandrasekhar, "The Mathematical Theory of Black Holes", 1983)

"A black hole partitions the three-dimensional space into two regions: an inner region which is bounded by a smooth two-dimensional surface called the event horizon; and an outer region, external to the event horizon, which is asymptotically flat; and it is required (as a part of the definition) that no point in the inner region can communicate with any point of the outer region. This incommunicability is guaranteed by the impossibility of any light signal, originating in the inner region, crossing the event horizon. The requirement of asymptotic flatness of the outer region is equivalent to the requirement that the black hole is isolated in space and that far from the event horizon the space-time approaches the customary space-time of terrestrial physics." (Subrahmanyan Chandrasekhar, "On Stars, Their Evolution, and Their Stability",[Nobel lecture] 1983)

"The mathematical theory of black holes is a subject of immense complexity; but its study has convinced me of the basic truth of the ancient mottoes 'The simple is the seal of the true' and 'beauty is the splendor of truth.'" (Subrahmanyan Chandrasekhar, "On Stars, Their Evolution, and Their Stability",[Nobel lecture] 1983)

"Turning to the physical properties of the black holes, we can study them best by examining their reaction to external perturbations such as the incidence of waves of different sorts. Such studies reveal an analytic richness of the Kerr space-time which one could hardly have expected. This is not the occasion to elaborate on these technical matters. Let it suffice to say that contrary to every prior expectation, all the standard equations of mathematical physics can be solved exactly in the Kerr space-time. And the solutions predict a variety and range of physical phenomena which black holes must exhibit in their interaction with the world outside." (Subrahmanyan Chandrasekhar, "On Stars, Their Evolution, and Their Stability",[Nobel lecture] 1983)

"In some strange way, any new fact or insight that I may have found has not seemed to me as a 'discovery' of mine, but rather something that had always been there and that I had chanced to pick up." (Subrahmanyan Chandrasekhar, "Truth and Beauty: Aesthetics and Motivations in Science", 1987)

"The pursuit of science has often been compared to the scaling of mountains, high and not so high. But who amongst us can hope, even in imagination, to scale the Everest and reach its summit when the sky is blue and the air is still, and in the stillness of the air survey the entire Himalayan range in the dazzling white of the snow stretching to infinity? None of us can hope for a comparable vision of nature and of the universe around us. But there is nothing mean or lowly in standing in the valley below and awaiting the sun to rise over Kinchinjunga." (Subrahmanyan Chandrasekhar, "Truth and Beauty: Aesthetics and Motivations in Science, 1987)

"When a supremely great creative mind is kindled, it leaves a blazing trail that remains a beacon for centuries." (Subrahmanyan Chandrasekhar, "Newton and Michelangelo", Current Science Vol. 67 (7), 1994)

"This 'shuddering before the beautiful', this incredible fact that a discovery motivated by a search after the beautiful in mathematics should find its exact replica in Nature, persuades me to say that beauty is that to which the human mind responds at its deepest and most profound." (Subrahmanyan Chandrasekhar)

Figurative Figures IV: On Circumference

"Nature is an infinite sphere of which the center is everywhere and the circumference nowhere." (Blaise Pascal, "Pensées", 1670)

"Many errors, of a truth, consist merely in the application of the wrong names of things. For if a man says that the lines which are drawn from the centre of the circle to the circumference are not equal, he understands by the circle, at all events for the time, something else than mathematicians understand by it." (Baruch Spinoza, "Ethics", Book I, 1677)

"A system of nature proceeding from subjects of the most simple organization to such as are more perfect, or from the circumference to the centre, is called a Mathematical System." (John Lindley, "Some Account of the Spherical and Numerical System of Nature o/M. Elias Fries", ‘Philosophical magazine: a journal of theoretical, experimental and applied physics’ Vol. 68, 1826)

"Philosophical systems do not depend upon individual productions which are subject to continual variation, but upon eternal and unchangeable ideas. These always proceed from the centre to the circumference, or from the most perfect productions to those of a lower order." (John Lindley, "Some Account of the Spherical and Numerical System of Nature o/M. Elias Fries", ‘Philosophical magazine: a journal of theoretical, experimental and applied physics’ Vol. 68, 1826)

"The world can no more have two summits than a circumference can have two centres." (Pierre T de Chardin, "The Divine Milieu", 1960)

"An observer of our biological sciences today sees dark figures moving over a bridge of glass. We are faced with an ever expanding universe of light and darkness. The greater the circle of understanding becomes, the greater is the circumference of surrounding ignorance." (Erwin Chargaff, "Essays on Nucleic Acids", 1963)

"The universe has no circumference, for if it had a center and a circumference there would be some and some thing beyond the world, suppositions which are wholly lacking in truth. Since, therefore, it is impossible that the universe should be enclosed within a corporeal center and corporeal boundary, it is not within our power to understand the universe, whose center and circumference are God. And though the universe." (Nicholas of Cusa[Nicolaus Cusanus])

Geometrical Figures XIX: On Circumference

"Now discourse is necessarily limited by its point of departure and its point of arrival, and since these are in mutual opposition we speak of contradiction. For the discursive reason these terms are opposed and distinct. In the realm of the reason, therefore, there is a necessary disjunction between extremes, as, for example, in the rational definition of the circle where the lines from the center to the circumference are equal and where the center cannot coincide with the circumference." (Nicholas of Cusa, "Apologia Doctae ignorantiae" ["The Defense of Learned Ignorance"], 1449)

"Many errors, of a truth, consist merely in the application of the wrong names of things. For if a man says that the lines which are drawn from the centre of the circle to the circumference are not equal, he understands by the circle, at all events for the time, something else than mathematicians understand by it." (Baruch Spinoza, "Ethics", Book I, 1677)

"The circumference of any circle being given, if that circumference be brought into the form of a square, the area of that square is equal to the area of another circle, the circumscribed square of which is equal to the area of the circle whose circumference is first given." (John A Parker, "The Quadrature of the Circle", 1874)

"Infinity is the land of mathematical hocus pocus. There Zero the magician is king. When Zero divides any number he changes it without regard to its magnitude into the infinitely small [great?], and inversely, when divided by any number he begets the infinitely great [small?]. In this domain the circumference of the circle becomes a straight line, and then the circle can be squared. Here all ranks are abolished, for Zero reduces everything to the same level one way or another. Happy is the kingdom where Zero rules!" (Paul Carus, "The Nature of Logical and Mathematical Thought"; Monist Vol 20, 1910)

"To square a circle means to find a square whose area is equal to the area of a given circle. In its first form this problem asked for a rectangle whose dimensions have the same ratio as that of the circumference of a circle to its radius. The proof of the impossibility of solving this by use of ruler and compasses alone followed immediately from the proof, in very recent times, that π cannot be the root of a polynomial equation with rational coefficients." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"The digits of pi beyond the first few decimal places are of no practical or scientific value. Four decimal places are sufficient for the design of the finest engines; ten decimal places are sufficient to obtain the circumference of the earth within a fraction of an inch if the earth were a smooth sphere" (Petr Beckmann, "A History of Pi", 1976)

"There are a number of diagrams in the literature of Sacred Geometry all related to the single idea known as the 'Squaring of the Circle'. This is a practice which seeks, with only the usual compass and straight-edge, to construct a square which is virtually equal in perimeter to the circumference of a given circle, or which is virtually equal in area to the area of a given circle. Because the circle is an incommensurable figure based on π, it is impossible to draw a square more than approximately equal to it." (Robert Lawlor, "Sacred Geometry", 1982)

"The mathematician's circle, with its infinitely thin circumference and a radius that remains constant to infinitely many decimal places, cannot take physical form. If you draw it in sand, as Archimedes did, its boundary is too thick and its radius too variable." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"In the case of circle squaring, since the problem requires pinpointing the ratio between a circle’s diameter and circumference, the irrational number the investigator bumps into is pi (π). Perhaps because of its extreme (in fact, total) difficulty - similar to the alchemist’s hope of turning lead into gold - circle squaring offered its pursuers the dream of international fame in the discovery of an unknown quantity seemingly woven into the fabric of the universe." (Daniel J Cohen, "Equations from God: Pure Mathematics and Victorian Faith", 2007)

"Is it possible to construct a square, using only a compass and a straightedge, that is exactly equal in area to the area of a given circle? If π could be expressed as a rational fraction or as the root of a first- or second-degree equation, then it would be possible, with compass and straightedge, to construct a straight line exactly equal to the circumference of a circle. The squaring of the circle would quickly follow. We have only to construct a rectangle with one side equal to the circle’s radius and the other equal to half the circumference. This rectangle has an area equal to that of the circle, and there are simple procedures for converting the rectangle to a square of the same area. Conversely, if the circle could be squared, a means would exist for constructing a line segment exactly equal to π. However, there are ironclad proofs that π is transcendental and that no straight line of transcendental length can be constructed with compass and straightedge. " (Martin Gartner, "Sphere Packing, Lewis Carroll, and Reversi", 2009)

"The engineer and the mathematician have a completely different understanding of the number pi. In the eyes of an engineer, pi is simply a value of measurement between three and four, albeit fiddlier than either of these whole numbers. [...] Mathematicians know the number pi differently, more intimately. What is pi to them? It is the length of a circle’s round line (its circumference) divided by the straight length (its diameter) that splits the circle into perfect halves. It is an essential response to the question, ‘What is a circle?’ But this response – when expressed in digits – is infinite: the number has no last digit, and therefore no last-but-one digit, no antepenultimate digit, no third-from-last digit, and so on." (Daniel Tammet, "Thinking in Numbers" , 2012)

"It just so happens that π can be characterised precisely without any reference to decimals, because it is simply the ratio of any circle’s circumference to its diameter. Likewise can be characterised as the positive number which squares to 2. However, most irrational numbers can’t be characterised in this way." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Mathematically, circles embody change without change. A point moving around the circumference of a circle changes direction without ever changing its distance from a center. It’s a minimal form of change, a way to change and curve in the slightest way possible. And, of course, circles are symmetrical. If you rotate a circle about its center, it looks unchanged. That rotational symmetry may be why circles are so ubiquitous. Whenever some aspect of nature doesn’t care about direction, circles are bound to appear. Consider what happens when a raindrop hits a puddle: tiny ripples expand outward from the point of impact. Because they spread equally fast in all directions and because they started at a single point, the ripples have to be circles. Symmetry demands it." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"Pi is fundamentally a child of calculus. It is defined as the unattainable limit of a never-ending process. But unlike a sequence of polygons steadfastly approaching a circle or a hapless walker stepping halfway to a wall, there is no end in sight for pi, no limit we can ever know. And yet pi exists. There it is, defined so crisply as the ratio of two lengths we can see right before us, the circumference of a circle and its diameter. That ratio defines pi, pinpoints it as clearly as can be, and yet the number itself slips through our fingers." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"The incommensurability of the diagonal of a square was initially a problem of measuring length but soon moved to the very theoretical level of introducing irrational numbers. Attempts to compute the length of the circumference of the circle led to the discovery of the mysterious number. Measuring the area enclosed between curves has, to a great extent, inspired the development of calculus." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)

"Imagine a person with a gift of ridicule [He might say] First that a negative quantity has no logarithm [ln(-1)]; secondly that a negative quantity has no square root [√-1]; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter [π]." (Augustus De Morgan) [attributed]

"Ten decimal places of pi are sufficient to give the circumference of the earth to a fraction of an inch, and thirty decimal places would give the circumference of the visible universe to a quantity imperceptible to the most powerful microscope." (Simon Newcomb)

"The attempt to apply rational arithmetic to a problem in geometry resulted in the first crisis in the history of mathematics. The two relatively simple problems - the determination of the diagonal of a square and that of the circumference of a circle - revealed the existence of new mathematical beings for which no place could be found within the rational domain." (Tobias Dantzig)