What Is the Poincaré Disc and How Do Its Edges Represent Infinity?

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Discussion Overview

The discussion revolves around the Poincaré disc, focusing on its definition, the nature of its geodesics, and the concept of infinity as represented by the edges of the disc. The conversation touches on topology, geometry, and the implications of the parallel postulate.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants describe the Poincaré disc as a topological 2 disc with a specific metric where geodesics are circles intersecting the boundary at right angles, which are infinitely long.
  • One participant expresses confusion about the concept of infinitely long geodesics and seeks a clearer understanding.
  • Another participant explains that from the perspective of a person walking along a geodesic, their steps would appear to shrink as they approach the boundary, requiring infinitely many steps to reach it, thus making the geodesic infinitely long.
  • It is noted that the same geometry can arise from a plane geometry where the parallel postulate does not hold, with geodesics being straight lines that are also infinitely long.
  • Some participants express appreciation for the discussion and suggest engaging with the metric to compute geodesics on the Poincaré disc.
  • Links to artistic renderings of the Poincaré disc are shared, indicating interest in visual representations.

Areas of Agreement / Disagreement

Participants generally agree on the nature of the Poincaré disc and the concept of infinitely long geodesics, but there is no consensus on the understanding of these concepts, particularly for those unfamiliar with topology.

Contextual Notes

Some participants indicate a lack of familiarity with topology, which may limit their understanding of the discussed concepts. The discussion also reflects varying levels of mathematical background among participants.

Who May Find This Useful

This discussion may be useful for individuals interested in topology, geometry, and the philosophical implications of infinity in mathematical contexts.

htetaung
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hi there
What is a Poincare' disc and why is the edges of disc represent infinity?
thanks
 
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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.
 
Thank you.
I think I got it.
 
htetaung said:
Thank you.
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...
http://www.hnorthrop.com/escher.html
 
Last edited by a moderator:
g_edgar said:
Here is the most famous artist's rendering of the Poincare disk...
http://www.hnorthrop.com/escher.html

very cool
 
Last edited by a moderator:

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