"However, it turns out that a one-to-one mapping of the points in a square into the points on a line cannot be continuous. As we move smoothly along a curve through the square, the points on the line which represent the successive points on the square necessarily jump around erratically, not only for the mapping described above but for any one-to-one mapping whatever. Any one-to-one mapping of the square onto the line is discontinuous."
"The whole fabric of the space-time continuum is not merely curved, it is in fact totally bent." (Douglas N Adams, "The Restaurant at the End of the Universe", 1980)
"The ‘eyes of the mind’ must be able to see in the phase space of mechanics, in the space of elementary events of probability theory, in the curved four-dimensional space-time of general relativity, in the complex infinite dimensional projective space of quantum theory. To comprehend what is visible to the ‘actual eyes’, we must understand that it is only the projection of an infinite dimensional world on the retina." (Yuri I Manin, "Mathematics and Physics", 1981)
"Mathematics is more than doing calculations, more than solving equations, more than proving theorems, more than doing algebra, geometry or calculus, more than a way of thinking. Mathematics is the design of a snowflake, the curve of a palm frond, the shape of a building, the joy of a game, the frustration of a puzzle, the crest of a wave, the spiral of a spider's web. It is ancient and yet new. Mathematics is linked to so many ideas and aspects of the universe." (Theoni Pappas, "More Joy of Mathematics: Exploring Mathematics All Around You", 1986)
"A surface which can be regarded as the set of successive position of a curve moving in space is said to be generated by the curve. The utility of this notion in constructing a surface geometrically, in a picture or as a model is increased as the complexity of the generator and its motion is decreased. When the generator is a straight line, it is called a ruled surface. Since you can exchange X and Y in the above analysis, the hyperbolic paraboloid is generated by a line in two ways. It is a doubly ruled surface." (George K Francis, "A Topological Picturebook", 1987)
"Linking topology and dynamical systems is the possibility of using a shape to help visualize the whole range of behaviors of a system. For a simple system, the shape might be some kind of curved surface; for a complicated system, a manifold of many dimensions. A single point on such a surface represents the state of a system at an instant frozen in time. As a system progresses through time, the point moves, tracing an orbit across this surface. Bending the shape a little corresponds to changing the system's parameters, making a fluid more visous or driving a pendulum a little harder. Shapes that look roughly the same give roughly the same kinds of behavior. If you can visualize the shape, you can understand the system." (James Gleick, "Chaos: Making a New Science", 1987)
"[...] the image of a stable map of a surface into space looks like in the neighborhood of each point. If no neighborhood, no matter how small, of a given point looks like a mildly bent disc, then it is a singular point. A stable map can have three kinds of singular points. In a neighborhood of a double point a surface looks like two sheets of some fabric crossing along a so-called double curve. A neighborhood of a triple point looks like three surface sheets crossing transversely. Thus triple points are isolated. You can see why a quadruple point is unstable. A slight perturbation of one of the sheets would make four sheets cross each other so as to produce a little tetrahedral cell. Double curves are either closed, extend to infinity, terminate on the border or simply end at very special points, called pinch points." (George K Francis, "A Topological Picturebook", 1987)
"These then are the elements of descriptive topology: normal surfaces with their border curves and double curves, triple points and pinch points, and their pictures with cusps and contours." (George K Francis, "A Topological Picturebook", 1987)
"Thus the unit normal N/|N is independent of the parametrization and defines a mapping from the surface to the unit sphere, called the Gauss map. The Jacobian of the Gauss map is called the Gauss curvature function on the surface. Here is a less complicated way of thinking about this function. For each plane normal to a surface at a point, the curvature of the curve of intersection is a sectional curvature. The Gauss curvature is the product of the maximum and the minimum sectional curvatures at the point. (The average sectional curvature is the mean curvature and this vanishes on surfaces minimizing area locally; soap films, for example.)" (George K Francis, "A Topological Picturebook", 1987)
"Bodies like the earth are not made to move on curved orbits by a force called gravity; instead, they follow the nearest thing to a straight path in a curved space, which is called a geodesic. A geodesic is the shortest" (or longest) path between two nearby points." (Stephen Hawking, "A Brief History of Time", 1988)
"Nonlinear systems" (the graph of at least one relationship displays some curved feature) are notoriously more difficult to comprehend than linear systems, that is, they are more complex. Consequently they are also more difficult to control. This is exemplified by the volumes of elegant mathematics that have been developed in the search for optimal control of linear systems." (Robert L Flood & Ewart R Carson, "Dealing with Complexity: An introduction to the theory and application of systems", 1988)
"The metaphor never goes very far, anymore than a curve can long be confused with its tangent." (Henri Bergson,"A World of Ideas", 1989)
"It is now known to science that there are many more dimensions than the classical four. Scientists say that these don’t normally impinge on the world because the extra dimensions are very small and curve in on themselves, and that since reality is fractal most of it is tucked inside itself. This means either that the universe is more full of wonders than we can hope to understand or, more probably, that scientists make things up as they go along." (Terry Pratchett, Pyramids, 1989)
"The metaphor never goes very far, anymore than a curve can long be confused with its tangent." (Henri Bergson, "A World of Ideas", 1989)
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