Topology is a branch of mathematics. Its purpose is to study the structure of objects regardless of their size and initial shape, just as geometry does. Geometry mathematically describes a figure and topology analyzes the possibilities of the figures. Let’s think of a circle. On the one hand, it is a figure in which all the points are at the same distance from the center. If the circumference were in three dimensions and were a ball, it could be turned into a cube.
Topology understands objects as if they were made of rubber and could be transformed. In fact, the properties of objects remain unchanged even if their shape is alterable. If we think of a circle, it is a geometric figure, but if we can manipulate it, it becomes another figure: a triangle or an ellipse. This concrete example gives a guideline of a basic principle of topology: the equivalence between the figures. Two figures are equivalent if one is convertible into the other.
If we start from the idea that the surfaces of objects are modifiable (think of a sheet of paper that can be cut or folded), it is easy to realize that the concrete applications of topology are immense. In computing, programs are used to modify images. In optics, the structure of the lenses is altered. In industry, objects are subject to variations in their shapes.
These examples demonstrate the versatility of the topology.
From a theoretical point of view, topology is related to other mathematical operations (statistics, differential equations…). However, what is striking about topology is its ability to solve practical problems: analyzing the best path for the delivery of goods or how to modify an object without breaking it. At the same time, topology has provided a very useful model and basic structure for biology, specifically for the explanation of DNA. The genetic material is distributed in two complementary chains, the double helix, which wind through the same axis. And the curvature of the axis is a topological form.
In conclusion, topology is based on a series of theoretical and abstract principles and from these it is possible to apply them to a multitude of areas of knowledge. In fact, despite the complexity of this branch of mathematics, according to psychology, children intuitively handle the principles of topology in their games and in the manipulation of objects.
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