"A hemispherical bowl has a single, circular edge. So does a disc. If you sew the disc to the hemisphere you obtain a closed, two-sided surface; it has an inside and an outside. Topologists still call this a sphere, and it could be inflated back to a geometrical sphere if you like. Now take a long, narrow strip of paper, give it a half twist and glue the ends together. The resulting surface is not only one sided, it has but a single, closed edge. What happens if this edge is sewn to the rim of a disc?" (George K Francis, "A Topological Picturebook", 1987)
"A regular homotopy of an immersion is a deformation through immersions for which the matrix of first partials itself varies smoothly." (George K Francis, "A Topological Picturebook", 1987)
"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)
"A twist about a circle that separates a surface is isotopic to the identity. You can convince yourself of this by representing the surface as a sphere with holes and handles (even cross caps) and twisting along the equator. Smoothly turning the northern component once clockwise against the southern component undoes the cut-turn-paste operation of the twist." (George K Francis, "A Topological Picturebook", 1987)
"Good design of a topological picture involves imagining something in 3-space that embodies the mathematical idea to be illustrated. Then you must draw it in such a way that the viewer has no difficulty in recognizing the idea. The picture should cause him to imagine the same object without his having to consult a long verbal description." (George K Francis, "A Topological Picturebook", 1987)
"Since the days of Descartes, expressing geometrical information in that universal language of mathematics, algebra, has been immensely useful in the service of precision and economy of thought. Nevertheless, something is inevitably lost in this transcription. The task of descriptive topology is to unfold the visual secrets so often compressed into algebraic shorthand." (George K Francis, "A Topological Picturebook", 1987)
"For complicated objects it is often impossible to find a view which does not hide some important structure behind a surface sheet. One remedy is to remove a regular patch from the object, creating a transparent window through which this structure can be seen in the picture." (George K Francis, "A Topological Picturebook", 1987)
"Perspective is the simplest and most direct way of creating the illusion of depth in a picture of spatially extended objects. The more or less correctly placed vanishing points of parallel lines, the estimated regression of evenly spaced points on a line, the elliptically compressed circles: all these tricks of perspective do more than merely please the eye. They help the viewer guess correctly where the artist meant to place things relative to each other. For example, even a modest amount of perspective convergence prevents you from mistaking a three-dimensional picture for a two-dimensional diagram." (George K Francis, "A Topological Picturebook", 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)
"The mode in which analytical expressions and coordinate equations are formulated has considerable influence on the speed and precision with which the reader glimpses the same thing the writer means to describe. At times, efficiency requires a departure from customary style in analytical geometry. This is especially true for 3-dimensional objects and phenomena." (George K Francis, "A Topological Picturebook", 1987)
"The story of everting spheres in 3-space by regular homotopies is the case history of a nontrivial visualization problem of remarkable complexity and compelling beauty. The task is to show the motion of a spherical surface through itself in space so that, without tearing or creasing, the surface is turned inside out. That so many different graphical methods were applied to the same problem, when it is more in the nature of mathematics to display the versatility of one method by applying it to a variety of different problems, makes it a paradigm for descriptive topology." (George K Francis, "A Topological Picturebook", 1987)
"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 are two important groups associated with a closed surface: the fundamental group and the mapping class group. [...] For the fundamental group, choose a base point on the surface and consider the set of all continuous paths that begin and end at this point. The possibility of deforming one such loop into another establishes an equivalence relation which respects the binary operation on loops. [...] For the mapping class group, consider the set of homeomorphisms of the surface to itself (the self-maps). In this context, a self-map is always a one-to-one transformation of a closed, orientable surface onto itself which, together with its inverse, is continuous. The possibility of deforming one self-map into another through a continuous family of self-maps, called an isotopy, establishes an equivalence relation which respects composition of mappings. The product of two self-maps is just the self-map obtained by performing the second immediately after the first." (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)
"[...] twists about isotopic circles are in the same mapping class. This allows me to shape the pictures conveniently. In describing the effect of a homeomorphism on a surface, it is a good idea to imagine a transparent, flexible copy of the surface lying on top of the fixed copy. The target circle x is on the fixed copy, the working circle y is on the flexible copy. A small isotopy will move y into general position with respect to x, so that y crosses x from one side to the other at a finite number of points. There are global isotopies that reduce such crossings to a minimum, even among an entire generating set of circles [...]"(George K Francis, "A Topological Picturebook", 1987)
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