How mathematics can help us explore the notion that there may be more than 3 spatial dimensions. If you liked this post you might also like: Everybody understood that if the proof is correct, then no other recognition is needed.вЂќ In explaining why he turned down $1 million he said that the prize, вЂњwas completely irrelevant for me. Grigori Perelman however was not interested in either the acclaim or the money on offer for solving one of the worldвЂ™s most difficult mathematics problems. Poincare conjectured that the same would be true in higher dimensions вЂ“ i.e 4 dimensional cubes (a tesseract, as shown above) could be squashed and squeezed to make a 4 dimensional sphere. Answer: The Poincar conjecture is currently the only millennium prize problem that currently has a verifiable solution, making it the easiest one out of the seven problems. In regular 3 dimensions, conventional 3 dimensional shapes without a hole in them (cubes, cuboids etc) can all be squashed and squeezed to create a sphere. So, with those terms defined we can simplify the Poincare conjecture. In the above animation, the coffee mug and the donut (torus) are shown to be homeomorphic.ДЈ-Sphere means a sphere in 4 dimensions (i.e with a 3 dimensional surface area). Basically, if 2 shapes can be sqeezed or stretched to form another shape then they are homeomorphic. Homeomorphic means it is mathematically equivalent in terms of the relationship between points. A 4 dimensional sphere is one which has a fixed radius in 4 dimensions (unlike in 3 dimensions for a sphere and 2 dimensions for a circle). So a 3 dimensional surface on a sphere would require a 4 dimensional sphere. Imagine the surface of a sphere вЂ“ that is a 2 dimensional surface. In 3 dimensions, a sphere and cube are simply connected, but a donut shape (torus) is not.ДЈD manifold means a 3 dimensional surface. The two shapes on the left above are simply connected, the two on the right are not. Simply connected means a shape without holes. The Poincare Conjecture is that, вЂњ Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.вЂќ At first glance that may look quite complicated вЂ“ so looking at the definitions in turn: What had Perelman done to achieve such (apparently unwanted) acclaim? He had solved a puzzle that had frustrated mathematicians for over 100 years вЂ“ the Poincare conjecture. In 2010 he was the first mathematician to be awarded $1 million вЂ“ he turned it down. In 2006 the Russian mathematician Grigori Perelman was awarded the mathematical equivalent of the mathematical Nobel prize (the Fields Medal). The Poincare Conjecture and Grigori Perelman

Pointcarre conjecture