Tagged: Topology

Poincaré Conjecture

This explanation is quoted from the  The Clay Mathematics Institute, Poincaré Conjecture. (solved by: Grigoriy Perelman, 2002-3) “If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface...

Knot theory

Knot theory is a very fast-growing field of mathematics.  Knots are not natural phenomena, and there exists only a finite number of distinct knots in three-dimensional space. Knots define spaces because we can think of a knot as a way in which different dimensions can be connected. Mathematicians are working on notation systems for knots, which leads to a form of arithmetic for knots. This has  fascinating consequences for other disciplines, and for our understanding of reality in general.  What is a knot? Complex knots can oftentimes be simplified with a few moves, which the German mathematician Reidemeister organized into...

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...