"[…] under plane transformations, like those encountered in the arbitrary stretching of a rubber sheet, certain properties of the figures involved are preserved. The mathematician has a name for them. They are called continuous transformations. This means that very close lying points pass into close lying points and a line is translated into a line under these transformations. Quite obviously, then, two intersecting lines will continue to intersect under a continuous transformation, and nonintersecting lines will not intersect; also, a figure with a hole cannot translate into a figure without a hole or into one with two holes, for that would require some kind of tearing or gluing - a disruption of the continuity." (Yakov Khurgin, "Did You Say Mathematics?", 1974)
"By gluing a mathematical theory on a piece of physical reality we obtain a physical theory. There exist many such theories, covering a great diversity of phenomena. And for a given phenomenon there are usually several different theories. In the better cases one passes from one theory to another one by an approximation (usually an uncontrolled approximation)." (David Ruelle, "Chance and Chaos", 1991)
"Homology theory introduces a new connection between invariants of manifolds. Continuing the 'physical' analogy, we say that a homology theory studies the intrinsic structure of a manifold by breaking it into a system of portions arranged simply, or, more precisely, in a standard way. Then, given certain rules for glueing the portions together, the theory obtains the whole manifold. The main problem consists in proving the resultant geometric quantities that are independent of the decomposition and glueing (i.e., proving the topological invariance of the characteristics)." (Michael I Monastyrsky, "Topology of Gauge Fields and Condensed Matter", 1993)
"Intrinsic properties have to do with the object itself, in contrast to extrinsic properties which describe how the object is embedded in the surrounding space. The cylinder and the band with two twists are intrinsically the same: There is a homeomorphism between them. The difference between them lies in how they sit in our 3-dimensional universe. Both are assembled from a rectangle by the same gluing instructions but one is given two twists before gluing." (L Christine Kinsey. "Topology of Surfaces", 1993)
"The Euler characteristic can be used to shorten the process, but for some cases a lengthy procedure is still necessary. Neither of these options provide a clear accounting for the ways in which the surfaces vary, e.g., which enclose cavities, which are non-orientable, etc., and neither can be completely generalized to higher-dimensional manifolds. Ideally, one would like some sort of algebraic invariant or computable quantity that would codify a lot of information: how many connected pieces a space has, how the gluing directions work, whether the surface is orientable or not, etc. The Euler characteristic is a first attempt at this and has the advantage of being quite easy to compute, but it fails to distinguish between the torus and the Klein bottle, which both have x = 0." (L Christine Kinsey. "Topology of Surfaces", 1993)
"The Mobius band has the interesting property of having only one side, in contrast to the cylinder. It is easy to imagine a cylinder with the outside painted one color and the inside another. Try painting a Möbius strip. Another peculiarity of the Mobius band occurs when one cuts it along the dotted line (called the meridian), and then follows the gluing instructions of the edges [...]. The end result is a planar diagram for a cylinder. However, if one actually constructs a Mobius band of paper and cuts it as described above, one gets something that looks like a cylinder with two twists." (L Christine Kinsey. "Topology of Surfaces", 1993)
"Determination of transition functions makes it possible to restore the whole manifold if individual charts and coordinate maps are already given. Glueing functions may belong to different functional classes, which makes it possible to specify within a certain class of topological manifolds more narrow classes of smooth, analytic, etc. manifolds." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)
"Geometrical intuition plays an essential role in contemporary algebro-topological and geometric studies. Many profound scientific mathematical papers devoted to multi-dimensional geometry use intensively the 'visual slang' such as, say, 'cut the surface', 'glue together the strips', 'glue the cylinder', 'evert the sphere' , etc., typical of the studies of two and three-dimensional images. Such a terminology is not a caprice of mathematicians, but rather a 'practical necessity' since its employment and the mathematical thinking in these terms appear to be quite necessary for the proof of technically very sophisticated results.(Anatolij Fomenko, "Visual Geometry and Topology", 1994)
"Roughly speaking, manifolds are geometrical objects obtained by glueing open discs (balls) like a papier-mache is glued of small paper scraps. To this end, one first prepares a clay or plastecine figure which is then covered with several sheets of paper scraps glued onto one another. After the plasticine is removed, there remains a two-dimensional surface."
"Geometrical intuition plays an essential role in contemporary algebro-topological and geometric studies. Many profound scientific mathematical papers devoted to multi-dimensional geometry use intensively the 'visual slang' such as, say, 'cut the surface', 'glue together the strips', 'glue the cylinder', 'evert the sphere' , etc., typical of the studies of two and three-dimensional images. Such a terminology is not a caprice of mathematicians, but rather a 'practical necessity' since its employment and the mathematical thinking in these terms appear to be quite necessary for the proof of technically very sophisticated results." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)
"Today the network of relationships linking the human race to itself and to the rest of the biosphere is so complex that all aspects affect all others to an extraordinary degree. Someone should be studying the whole system, however crudely that has to be done, because no gluing together of partial studies of a complex nonlinear system can give a good idea of the behaviour of the whole." (Murray Gell-Mann, 1997)
"Starting with a torus with the graph of a layer embedded in it and a second torus with the the graph of a pipe, imagine gluing the two toruses together so that the four points of a shared tile coincide. Now imagine puncturing the two tiles that are glued together to make a door out of the pipe-torus and into the layer-torus. We now have a figure-eight, a two-holed torus." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"In geometry, shapes like circles and polyhedra are rigid objects; the tools of the trade are lengths, angles and areas. But in topology, shapes are flexible things, as if made from rubber. A topologist is free to stretch and twist a shape. Even cutting and gluing are allowed, as long as the cut is precisely reglued. A sphere and a cube are distinct geometric objects, but to a topologist, they’re indistinguishable." (David E Richeson, "Topology 101: The Hole Truth", 2021)
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