Hyperbolic geometry

[Vay]

i can't picture it in my head of how it looks like.

I see a front view of it in the book "The Road To Reality" by Dr. Roger Penrose.

The picture is a woodcut by M.C Escher called circle limit I.

 

http://math.slu.edu/escher/upload/thumb/e/e5/Circle-limit-I.jpg/200px-Circle-limit-I.jpg

 

So i was thinking how the universe of hyperbolic plane is round and nothing outside of the circle is existent. But i wondered how it would look like from the side or will it be the same as the way i am looking at it right now? Or will it look like a curve with an opening with two lines consisting of it that extends infinitely.

[the tree]

The disc is just one model of the hyperbolic plane in euclidean space - personally I got taught with the half-plane model.

 

Anyway, there is no point at the edge, each of the creatures in Escher's woodcut has an infinite amount of other creatures between it and the edge.

 

A finite chunk of the hyperbolic plane can be represented in euclidean 3-space quite well with a Saddle shape but there is no correct way to visualise the whole thing, by itself.