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

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SUMMARY

The Poincaré disc is a topological 2-disc characterized by a metric where geodesics are circular arcs that intersect the boundary at right angles, resulting in infinitely long paths. As an observer moves along these geodesics towards the boundary, their steps appear to shrink, necessitating infinitely many steps to reach the edge. This phenomenon illustrates the unique properties of non-Euclidean geometry, where the parallel postulate does not hold. Understanding these concepts provides insight into the nature of infinity within this geometric framework.

PREREQUISITES
  • Basic understanding of topology
  • Familiarity with non-Euclidean geometry
  • Knowledge of geodesics and their properties
  • Ability to interpret geometric representations
NEXT STEPS
  • Explore the properties of non-Euclidean geometries
  • Learn about the metric used in the Poincaré disc model
  • Study the concept of geodesics in various geometric contexts
  • Investigate applications of the Poincaré disc in modern mathematics and art
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Mathematicians, geometry enthusiasts, and students of topology seeking to deepen their understanding of non-Euclidean spaces and their implications in both theoretical and practical 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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