Topology

What is Topology?

Topology is a mathematical sub-discipline that studies the properties of objects and spaces. Topology is the modern version of geometry; it was used first in 1847 by the German mathematician Johann Benedict Listing. The shapes of objects can change through twisting or stretching them, and topologists ask: what are the object’s properties that remain intact? In these deformations, tearing is not allowed. From a topological point of view, therefore, a circle is equivalent to an ellipse (into which it can be deformed by stretching) and a donut (also called a “torus”, a two-dimensional a surface embedded in three-dimensional space) is equivalent to a coffee cup.

Topology began with the study of curves, surfaces, and other objects in two- and three-dimensional space. One of the central ideas in topology is that spatial objects like circles and spheres can be treated as objects in their own right, and knowledge of objects is independent of how they are embedded in space.

Topology focuses on the inherent connectivity of objects while ignoring their detailed form. Because of this abstraction from the detailed form, it is possible to define the “objects” of topology as “topological spaces”. If two objects have the same topological properties, they are said to be “homeomorphic.”

Moebius Strip

A good example of a topological object is the Moebius strip. It is a one-sided surface that can be constructed by connecting the ends of a rectangular strip after giving one of the ends a one-half twist. This space has interesting properties, such as having only one side and remaining in one piece when splitting down the middle. The properties of the strip were discovered independently and almost simultaneously by two German mathematicians, August Ferdinand Moebius and Johann Benedict Listing, in 1858. The picture above is an example for it.

Here are some more examples for topological structures:

Tesseracts: from 3 to 4 Dimensions.

The short video clip below shows the rotations of a four-dimensional object (4D) in three-dimensional space (3D). The deeper questions concern the nature of a “dimension”. How do we know what a dimension is? Is time the fourth dimension? We live in a 3D world, but could we simultaneously exist in a higher-dimensional universe?

It is best to start with simple examples in order to train the mind to think about these questions. The clip below shows a tesseract, which is the four-dimensional analog of a cube. (In geometry, it is called a regular octachoron or cubic prism.) The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the surface of the tesseract consists of 8 cubical cells.

A generalization of the cube to dimensions greater than three is called a “hypercube”, “n-cube” or “measure polytope”. The tesseract is the four-dimensional hypercube or 4-cube.

According to the Oxford English Dictionary, the word tesseract was coined and first used in 1888 by Charles Howard Hinton in his book A New Era of Thought.

Klein Bottle

The Klein bottle is the next step up from a Moebius strip.  A Moebius Strip is a two-dimensional object in three-dimensional space, and a Klein Bottle is a one-dimensional object in three-dimensional space. Therefore, it has to intersect itself in order to be representable in three dimensions. It is a non-orientable surface with Euler characteristic equal to 0.

A Klein bottle can be made from a rectangular piece of the plane by identifying the top and bottom edges using the same orientation, but identifying the left and right edges with opposite orientation (as in the formation of a Möbius band). The first step forms a tube, but the second step can not be carried out without causing self-intersection: the tube must pass through itself in order to attach the ends correctly.

Another way to make the Klein bottle is to take two Möbius bands and join them along their boundaries (each band has a single boundary curve). Finally, the Klein bottle is the connected sum of two real projective planes, since the projective plane minus a disk is just a Möbius band.

The Klein bottle can not be embedded in three-space, but it can be represented there.

Most containers have an inside and an outside, a Klein bottle is a closed surface with no interior and only one surface. It is unrealizable in 3 dimensions without intersecting surfaces. It can be realized in 4 dimensions. The classical representation is shown below.

 

Turning a Sphere Inside-Out 

Finally, here is an example of the problems that arise from morphing topological objects: How do you turn a sphere inside out, without punching a hole into it? As you can see in the following video, this kind of transformation is possible, but it stretches our imagination.

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