.topology

.homework#1: topological operations

topology (greek topos = place, logos = word) is a mathematical science that studies spatial properties of objects and their continuous transformations - which can be understood as deformations without tearing or gluing. word topology can also refer to an object that is being studied.

topology is sometimes called rubber-sheet geometry - this idea perfectly describes the logic of continuous transformations and equivalency of objects in topology. two objects are equivalent if you can continuously transform (by a homeomorphism) one to the other (as if it was made out of rubber) - or, alternatively: two spaces are topologically equivalent if one can be deformed into the other without cutting it apart or gluing pieces of it together. this is illustrated by this simple animation (it is not absolutely correct:-):

the animation shows how a circle can be continuously transformed to a squere, or, you can imagine that it shows the profile of transformation of a sphere to a cube.

formal definition of a homeomorphism: homeomorphism is defined as a continuous bijection with a continuous inverse.
classic joke: the topologist can't tell the coffee mug she is drinking out of from the doughnut she is eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle. the shape of a doughnut is called a torus.

planar models of 3d objects: for better representation we can "cut" the 3d objects and then unwrap/place them into a plane. it is something like paper models that used to be in technical magazines for kids such as "abc", etc. - by connecting planar polygonal objects, connetcting them along their edges with keeping some rules you can get a 3d object and reverse. these pictures represent how the torus can be "unwrapped" into a plane after cutting it in a dispayed way:

the colors and arrows emphasize that the planar model must be glued in along the complementary edges with respect to the orientation of arrows. since the model of torus was analyzed during the class on monday, 9. oct, i would like to continue with an aplication of the this planar model.

problem of a granary, a well and a shed:
the task is to connect the three houses with the granary, the well and the shed in such way, that the paths do not intersect each other:

quite soon you can see, that this goal is not accomplishable in planar euklides-geometry. with the tools of graph theory you can even proove it. the next picture demonstrates how the problem can be solved on a torus (instead of plane):

the red and green lines represent the planar model of a torus that was described in this text.

summary: topology is a powerful branch of mathematics, which brings joy and pleasure to those who understand it and use it in a proper way.