"At first glance the theory of numbers is deprived of any
geometricity. But this is actually not the case. At the contemporary stage of development
of computers it has become possible to explain to a wide range of readers that
visual geometry helps not only to illustrate some abstract situations from the
number theory, but sometimes also to solve new problems."
"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)
"Every mathematician knows and can give many examples from his scientific work when it appears much more difficult to feel or 'see' a correct hypothesis than later to prove it. Visual images are particularlyo ften used in geometry and topology where one has to work with multidimensional objects which, in principle, do not always admit picturing in a three-dimensional space." (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."
"Geometry and topology most often deal with geometrical
figures, objects realized as a set of points in a Euclidean space (maybe of
many dimensions). It is useful to view these objects not as rigid (solid)
bodies, but as figures that admit continuous deformation preserving some qualitative
properties of the object. Recall that the mapping of one object onto another is
called continuous if it can be determined by means of continuous functions in a
Cartesian coordinate system in space. The mapping of one figure onto another is
called homeomorphism if it is continuous and one-to-one, i.e. establishes a
one-to-one correspondence between points of both figures."
"Homeomorphism is one of the basic concepts in topology. Homeomorphism,
along with the whole topology, is in a sense the basis of spatial perception.
When we look at an object, we see, say, a telephone receiver or a ring-shaped
roll and first of all pay attention to the geometrical shape (although we do not concentrate on it specially) - an
oblong figure thickened at the ends or a round rim with a large hole in the
middle. Even if we deliberately concentrate on the shape of the object and
forget about its practical application, we do not yet 'see' the
essence of the shape. The point is that oblongness, roundness, etc. are metric
properties of the object. The topology of the form lies 'beyond
them'."
"Modem geometry and topology take a special place in mathematics because many of the objects they deal with are treated using visual methods. […] Each mathematician has his own system of concepts of the intrinsic geometry of his (specific) mathematical world and visual images which he associated with some or other abstract concepts of mathematics (including algebra, number theory, analysis, etc.). It is noteworthy that sometimes one and the same abstraction brings about the same visual picture in different mathematicians, but these pictures born by imagination are in most cases very difficult to represent graphically, so to say, to draw." (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."
"The concept of homeomorphism appears to be convenient for establishing those important properties of figures which remain unchanged under such deformations. These properties are sometimes referred to as topological, as distinguished from metrical, which are customarily associated with distances between points, angles between lines, edges of a figure, etc." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)
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