Exploring the Poincaré Disc: Understanding Infinity

In summary, the Poincare disc is a topological 2-disc with infinitely long geodesics represented by circles that intersect the boundary at right angles. This is due to the fact that the parallel postulate is false in this axiomatic version, making the geodesics straight lines that are infinitely long. This concept can be better understood by imagining a person walking along a geodesic towards the boundary from a bird's eye view, where they would appear to shrink and take infinitely many steps to reach the boundary. The Poincare disc can be further explored by computing its geodesics using the given metric. The famous artist, M.C. Escher, has created a rendering of the Poincare disc which can
  • #1
htetaung
15
0
hi there
What is a Poincare' disc and why is the edges of disc represent infinity?
thanks
 
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  • #2
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.
 
  • #3
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?
 
  • #4
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.
 
  • #5
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.
 
  • #6
Thank you.
I think I got it.
 
  • #7
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.
 
  • #8
Here is the most famous artist's rendering of the Poincare disk...
http://www.hnorthrop.com/escher.html
 
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  • #9
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:

Related to Exploring the Poincaré Disc: Understanding Infinity

1. What is the Poincaré Disc?

The Poincaré Disc is a geometric model of a hyperbolic plane, created by mathematician Henri Poincaré in the late 19th century. It is a two-dimensional representation of the infinite hyperbolic space, which is a non-Euclidean geometry.

2. What is the purpose of exploring the Poincaré Disc?

The Poincaré Disc allows us to understand and visualize the concept of infinity in a finite space. It also helps us to study and solve problems related to hyperbolic geometry and its applications in various fields such as physics, computer science, and art.

3. How is the Poincaré Disc different from a Euclidean plane?

Unlike a Euclidean plane, the Poincaré Disc has a constant negative curvature, which means that the distance between two points on its surface is not the same as their Euclidean distance. It also follows different rules for parallel lines and angles, and has infinitely many parallel lines through a given point.

4. Can the Poincaré Disc be extended to three dimensions?

Yes, the Poincaré Disc can be extended to three dimensions to create the Poincaré Ball model. This model is useful in visualizing the hyperbolic space, but it is not as convenient for calculations as the Poincaré Disc.

5. What are some real-life applications of the Poincaré Disc?

The Poincaré Disc has various applications, such as in computer graphics, where it is used to create realistic-looking 3D objects with curved surfaces. It is also used in physics to study the behavior of particles in curved space-time, and in art to create visually interesting designs with hyperbolic patterns.

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