<|im_start|>user
Could you explain why in a hyperbolic space two lines will ever intersect ?<|im_end|>
<|im_start|>assistant
Could you be a little more specific? Two lines in hyperbolic space can be such that they intersect, but they may also be such that they never intersect. It depends on the lines. Parallelity does work differently in hyperbolic space, compared to euclidean space.<|im_end|>
<|im_start|>user
Yes, I mean parallel lines<|im_end|>
<|im_start|>assistant
In a Euclidean space, parallel lines will never intersect either.

The key difference is that in Euclidean space, these parallel lines must be exactly the same direction, i.e. if you draw a line between them the sum of the angles on each side of the line is 180°; but in Hyperbolic space, lines can be parallel and not in the same direction. In fact, two lines in Hyperbolic geometry can be parallel and have any direction except perpendicular, provided that they are far enough away from each other.

Furthermore, parallel lines in Euclidean space are always the same distance from each other, i.e. if you pick any point on line *A* it will have the same distance to line *B*; but in Hyperbolic space, there is a single point on both lines where the distance between them is the least. If this point is 0, the lines are called **limiting parallel**; if it's nonzero, the lines are called **diverging parallel** or **ultra-parallel**. Limiting parallel lines in Hyperbolic space look like:

```
\  /
 \/
  |
 /\
/  \
```

While diverging parallel lines in Hyperbolic space look like:

```
\    /
 \  /
 |  |
 |--|
 |  |
 /  \
/    \
```

Parallel lines in Euclidean space look like:

```
 |  |
 |  |
 |  |
 |--|
 |  |
 |  |
 |  |
```

For more info, see https://en.wikipedia.org/wiki/Hyperbolic_geometry<|im_end|>
