"All forms of complex causation, and especially nonlinear transformations, admittedly stack the deck against prediction. Linear describes an outcome produced by one or more variables where the effect is additive. Any other interaction is nonlinear. This would include outcomes that involve step functions or phase transitions. The hard sciences routinely describe nonlinear phenomena. Making predictions about them becomes increasingly problematic when multiple variables are involved that have complex interactions. Some simple nonlinear systems can quickly become unpredictable when small variations in their inputs are introduced." (Richard N Lebow, "Forbidden Fruit: Counterfactuals and International Relations", 2010)
"Strange attractors, unlike regular ones, are geometrically very complicated, as revealed by the evolution of a small phase-space volume. For instance, if the attractor is a limit cycle, a small two-dimensional volume does not change too much its shape: in a direction it maintains its size, while in the other it shrinks till becoming a 'very thin strand' with an almost constant length. In chaotic systems, instead, the dynamics continuously stretches and folds an initial small volume transforming it into a thinner and thinner 'ribbon' with an exponentially increasing length." (Massimo Cencini et al, "Chaos: From Simple Models to Complex Systems", 2010)
"Symmetry may have its appeal but it is inherently stale. Some kind of imbalance is behind every transformation." (Marcelo Gleiser, "A Tear at the Edge of Creation: A Radical New Vision for Life in an Imperfect Universe", 2010)
"[…] topology is the study of those properties of geometric objects which remain unchanged under bi-uniform and bi-continuous transformations. Such transformations can be thought of as bending, stretching, twisting or compressing or any combination of these." (Lokenath Debnath, "The Legacy of Leonhard Euler - A Tricentennial Tribute", 2010)
"A crucial difference between topology and geometry lies in the set of allowable transformations. In topology, the set of allowable transformations is much larger and conceptually much richer than is the set of Euclidean transformations. All Euclidean transformations are topological transformations, but most topological transformations are not Euclidean. Similarly, the sets of transformations that define other geometries are also topological transformations, but many topological transformations have no counterpart in these geometries. It is in this sense that topology is a generalization of geometry."
"What distinguishes topological transformations from geometric ones is that topological transformations are more 'primitive'. They retain only the most basic properties of the sets of points on which they act." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)
"The potential freedom in the choice of a particular mathematical representation of physical objects is loosely called symmetry. In mathematical terms, physical symmetries are intimately related to groups in the sense that symmetry transformations form a group." (Teiko Heinosaari & Mario Ziman, "The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement", 2012)
"Mathematics does not merely describe the problem in an abstract way, it allows us to find a previously unknown 'solution' from the abstract description. It is surprising that the unknown can be transformed into the well known when we succeed in describing the problem mathematically." (Waro Iwane, "Mathematics in Our Company: What Does It Describe?", [in "What Mathematics Can Do for You"] 2013)
"[…] the symmetry group of the infinite logarithmic spiral is an infinite group, with one element for each real number . Two such transformations compose by adding the corresponding angles, so this group is isomorphic to the real numbers under addition." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)
"Intersections of lines, for example, remain intersections, and the hole in a torus (doughnut) cannot be transformed away. Thus a doughnut may be transformed topologically into a coffee cup (the hole turning into a handle) but never into a pancake. Topology, then, is really a mathematics of relationships, of unchangeable, or 'invariant', patterns." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)
"It is evident that chaotic behavior, in the new scientific sense of the term, is very different from random, erratic motion. With the help of strange attractors a distinction can be made between mere randomness, or 'noise', and chaos. Chaotic behavior is deterministic and patterned, and strange attractors allow us to transform the seemingly random data into distinct visible shapes."
"Topology is a geometry in which all lengths, angles, and areas can be distorted at will. Thus a triangle can be continuously transformed into a rectangle, the rectangle into a square, the square into a circle, and so on. Similarly, a cube can be transformed into a cylinder, the cylinder into a cone, the cone into a sphere. Because of these continuous transformations, topology is known popularly as 'rubber sheet geometry'. All figures that can be transformed into each other by continuous bending, stretching, and twisting are called 'topologically equivalent'." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)
"[…] topology is concerned precisely with those properties of geometric figures that do not change when the figures are transformed. Intersections of lines, for example, remain intersections, and the hole in a torus (doughnut) cannot be transformed away. Thus a doughnut may be transformed topologically into a coffee cup (the hole turning into a handle) but never into a pancake. Topology, then, is really a mathematics of relationships, of unchangeable, or 'invariant', patterns."
"[…] the role that symmetry plays is not confined to material objects. Symmetries can also refer to theories and, in particular, to quantum theory. For if the laws of physics are to be invariant under changes of reference frames, the set of all such transformations will form a group. Which transformations and which groups depends on the systems under consideration." (William H Klink & Sujeev Wickramasekara, "Relativity, Symmetry and the Structure of Quantum Theory I: Galilean quantum theory", 2015)
"Symmetries are transformations that keep certain parameters (properties, equations, and so on) invariant, that is, the parameters they refer to are conserved under these transformations. It is to be expected, therefore, that the identification of conserved quantities is inseparable from the identification of fundamental symmetries in the laws of nature. Symmetries single out 'privileged' operations, conservation laws single out 'privileged' quantities or properties that correspond to these operations. Yet the specific connections between a particular symmetry and the invariance it entails are far from obvious. For instance, the isotropy of space (the indistinguishability of its directions) is intuitive enough, but the conservation of angular momentum based on that symmetry, and indeed, the concept of angular momentum, are far less intuitive." (Yemima Ben-Menahem, "Causation in Science", 2018)
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