i love physics but this shit is to complex so ELI5
In: Mathematics
Spelling: Poincaré conjecture. ELI5: If you can take a 3-dimensional shape and from a point on the outer surface blow it up to a sphere, then the original shape is basically equivalent to a sphere. A donut is not like a sphere; your smushed silly-putty is like a sphere only not round.
Further: What the conjecture says is that if a 3-dimensional shape without a boundary, i.e. closed, supports an unbroken loop and if the loop’s length can be continuously reduced down to nothing, a point, then the shape is topologically equivalent, homeomorphic, to a sphere.
The Poincare Conjecture is very difficult to ELI5, so this may end up being long because I need to use an analogy. The Poincare conjecture is about 3-dimensional spaces that it’s quite difficult for us to imagine, so I’ll start by explaining something analogous in two dimensions.
Suppose we were living on a planet and we didn’t know what shape the planet was. So we are going to run some experiments just on the surface of the planet. The surface of the planet is 2-dimensional. Here is the first hurdle, most people get confused about what the dimension of the object is. Even though we usually think of spheres living in a 3-dimensional space, that doesn’t make the sphere itself 3-dimensional. When you are living on the sphere, the ground looks flat(ish). This is how we define dimension, so whatever planet we are on is 2-dimensional.
OK. So our experiment will be to travel around the planet until we end up back where we started. You can do this by just walking in a 5-meter circle, or you can walk thousands of miles and get back where you started. If we do this on a globe, then we can slightly tweak the path that we took and through a series of minor tweaks, shrink our path to a single point, as if we never left. This is called the loop shrinking property. If however we are in the video game Chrono Trigger, then we can travel East in a straight line and end up back where we started, or travel North in a straight line and end up back where we started. This may not seem odd yet, because this also works on a globe, but in Chrono Trigger, we can’t shrink the path. The connections are very peculiar and don’t work like a sphere should. I encourage you to play around with it. It’s similar in a lot of other games, too, like most (if not all) of the games in the Final Fantasy and Dragon Quest series. So these games don’t take place on a spherical planet.
So what does this have to do with Poincare. So first is the easy part, if a surface does not have the loop shrinking property then it cannot be a sphere. But what if a surface does have the loop shrinking property, can you guarantee that it will be a sphere? Poincare proved that the answer is yes. Then he raised the whole scenario up a dimension and asked if a 3-dimensional space with the loop shrinking property was guaranteed to be a 3-dimensional analogue of a sphere. He guessed that the answer was yes in 1904 and Grigori Perelman proved that it was true in 2006.
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