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Friday, March 25, 2011


In 1956, John Milnor found a 7-dimensional mathematical object (7-D), identical to a sphere of 7-D, under the rules of topology (shapes with mathematical properties unchanged by stretching or torsion), though different in structure (exotic spheres). It was the first time in which was identified a way to share topological properties, but not differentiable structures of their smaller dimensional counterparts. Although it is difficult to imagine how form will have such a sphere, it is possible to form larger dimensional spheres through ways other than the usual two-dimensional.

Split a sphere into 2 halves at its middle, so that each half has a copy of each point of its Ecuador. Then, assemble the 2 halves so that one point in its southern part is not in line with another of its northern counterpart. At 2-D scale, there is only one way to do that: twisting the sphere. At 7-D, the points are mixed together in multiple ways (up to 28 exotic spheres, and more at 7-D). Until recently topologists knew that inflating a cube this could be turned into a sphere (2 topologically identical shapes). At 3-D or less, the sphere and the cube have the same differentiable Justificar a ambos ladosstructures and topological form. However, you can not turn a sphere into a doughnut without drilling holes. Although topologically different high dimensional spheres areas can be built making them smoothed differentiated. It is sensed that the human brain has very small exotic topological forms. Pioneering studies in topology, geometry and algebra that have earned John Milnor (Stony Brook University, NY), to receive the Abel Prize for Mathematics 2011, from the Norwegian Academy of Science and Letters.


En 1956 John Milnor descubrió un objeto matemático de 7 dimensiones (7-D), idéntico a una esfera de 7-D, según las reglas de la topología (formas con propiedades matemáticas que no cambian por estiramiento o torsión), aunque diferentes en estructura (esferas exóticas). Era la primera véz que se identificaba una forma compartiendo propiedades topológicas, aunque no las estructuras diferenciables de sus contrapartes de menor dimensión. Aunque es dificil imaginar la forma de tal esfera, es posible conformar esferas de mayores dimensiones por vias diferentes a las usuales bidimensionales.

Parta una esfera en 2 mitades por su parte media, de modo tal que cada mitad tenga una copia de cada punto de su ecuador. Luego, reuna las 2 mitades de modo que un punto de su parte sur no se corresponda con otra de su contraparte norte. A escala de 2-D, solo hay un modo de hacerlo :torciendo la esfera. En 7-D, los puntos se mezclan entre sí en múltiples diferentes formas (hasta 28 esferas exóticas –y mas- en 7-D). Hasta hace poco los topólogos sabian que un cubo se convertía en una esfera inflándolo (2 formas topologicamente idénticas). Para formas 3-D o menos, la esfera y el cubo tienen las mismas estructuras diferenciables y topológicas. No obstante, no se puede convertir una esfera en un doughnut sin realizarle agujeros. Aunque topológicamente diferentes, se pueden construir esferas haciendolas ligeramente diferenciables. Se intuye que el cerebro humano tiene formas topológicas exóticas muy pequeñas. Estudios pioneros en topologia, geometría y algebra que le han valido a John Milnor (Stony Brook University, NY), recibir el Premio Abel de Matematicas, 2011 de parte de la Academia Noruega de Ciencias y Letras.



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