/sci/ - Science & Math

>>11635429
Less than 15 minutes!

>>8767669
This depends on what you mean by infinity. Consider two models, the "upper halfplane" and the "Poincaré disk". The first is [math]\{ z\in\mathbb{C}\ |\ \text{Im}(z)>0\}[/math], and the other [math]\{ z\in\mathbb{C}\ |\ |z|<1\}[/math]. As you may see, the upper halfplane goes infinitely far from the origin in all directions except down, whereas the Poincaré disk is bounded. Therefore, in the Euclidean sense, they can be both infinite and finite.

On the other hand, the axis [math]\{ z \in\mathbb{C}\ |\ \text{Im}(z)=0\}[/math], and the perimeter line of the Poincaré disk [math]\{ z\in\mathbb{C}\ |\ |z|=1\}[/math] are hyperbolically infinitely far from any point on the plane. This is due to the nature of the hyperbolic metric (of the considered model), and the homeomorphism of the upper halfplane and the disk taking the real axis together with the infinity point to be the boundary of the disk. This also preserves distance, and, since the real axis is infinitely far from any point on the upper halfplane, so is the perimeter from any point on the disk. If you want to interpret this somehow, consider the thing how gravity is usually visualized, a bowling ball on a rubber sheet bending it. Similarly, the center of the disk is where the ball is, and the closer you get to the boundary, the steeper the the climb up (or down) is eventually becoming essentially vertical. This follows from the hyperbolic metric, and the hyperbolic metric makes it so that the Euclidean distance may look small, but the hyperbolic distance can be long, making the disk hyperbolically infinite.

The same knid of idea applies for other planar models, too. Unless someone comes to counter this with a hyperbolically finite hyperbolic plane, they are all infinite in their own sense, regardless of their Euclidean boundedness. Does this satisfy you?

>>8455011
That would be Ika Musume.

Here's another one:
"I don't know how to prove this diagram's commutativity, so I'll just draw devils around it and leave it to the reader"
- Grothendieck in a letter of his

>>8448283
It only intersects itself in that pic because the pic has too few dimensions. In 4D it doesn't.

>>8385062
This. Science is restricted to the boring physical world. Ascend beyond it and start studying mathematics.