What is a Poincare' disc and why is the edges of disc represent infinity?
The Poincare disc is the topological 2 disc given a metric whose geodesics are circles that intersect the boundary in right angles. These geodesics are infinitely long.
Thanks for your reply.
But I don't know about topology. So is there anyway to understand its infinitely long geodesics?
Why are those things infinitely long?
If you look down from space on a man walking along one of these geodesics towards the boundary, he would keep shrinking and his steps would look increasingly smaller. For him it would take infinitely many steps to get to the boundary. This is true even if he walks at what he considers to be constant speed. So for him the geodesic is infinitely long.
the same geometry comes from a plane geometry in which the parallel postulate is false.
The geodesics are just straight lines in this axiomatic version and like any line in a plane geometry they are infinitely long.
I think I got it.
I think it would be enjoyable for you to compute the geodesics on the Poincare disc starting with the metric. It is not hard.
Here is the most famous artist's rendering of the Poincare disk...
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