"Because of its foundation in topology, catastrophe theory is qualitative, not quantitative. Just as geometry treated the properties of a triangle without regard to its size, so topology deals with properties that have no magnitude, for example, the property of a given point being inside or outside a closed curve or surface. This property is what topologists call 'invariant' -it does not change even when the curve is distorted. A topologist may work with seven-dimensional space, but he does not and cannot measure (in the ordinary sense) along any of those dimensions. The ability to classify and manipulate all types of form is achieved only by giving up concepts such as size, distance, and rate. So while catastrophe theory is well suited to describe and even to predict the shape of processes, its descriptions and predictions are not quantitative like those of theories built upon calculus. Instead, they are rather like maps without a scale: they tell us that there are mountains to the left, a river to the right, and a cliff somewhere ahead, but not how far away each is, or how large." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"Topology has been called 'rubber-sheet geometry', and it is helpful to think of the surfaces as made of rubber. They can be deformed in any way at an, as long as they are not torn and as long as no new qualitative features appear on them. That is, the actual behavior surface for a process with two control factors may be any distortion of the canonical cusp surface, but it can have no local singularity more complex than a pleat." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"Every branch of geometry can be defined as the study of properties that are unaltered when a specified figure is given specified symmetry transformations. Euclidian plane geometry, for instance, concerns the study of properties that are 'invariant' when a figure is moved about on the plane, rotated, mirror reflected, or uniformly expanded and contracted. Affine geometry studies properties that are invariant when a figure is stretched" in a certain way. Projective geometry studies properties invariant under projection. Topology deals with properties that remain unchanged even when a figure is radically distorted in a manner similar to the deformation of a figure made of rubber." (Martin Gardner, "Aha! Insight", 1978)
"There are two topological reasons for adopting normal surfaces as the basic forms for drawings. A sufficiently small distortion of a stable mapping of a surface can be returned to its original shape by an isotopy of the ambient space. In other words, there is a one parameter family of coordinate changes which removes the distortion. Moreover, arbitrarily near any smooth mapping there is a stable approximation to it. A practical way to check that a certain surface feature is unstable is to remove it from the surface by means of a small perturbartions of its parametrization." (George K Francis, "A Topological Picturebook", 1987)
"[...] there is no area of mathematics where thinking abstractly has paid more handsome dividends than in topology, the study of those properties of geometrical objects that remain unchanged when we deform or distort them in a continuous fashion without tearing, cutting, or breaking them." (John L Casti, "Five Golden Rules", 1995)
"Differential geometry is a fine, quantitative geometry, in which relationships between lengths and angles are important. Topology, by contrast, is of a much coarser and more qualitative nature. Here only those quantities that are preserved under distortions are studied." (Christian Bär, "Elementary Differential Geometry", 2010)
"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)
No comments:
Post a Comment