Hi all! For my 25 minute talk next week I will be discussing Hyperbolic Geometry and the applications of the Poincare Disk model. In High School we are taught the postulates and theories within Euclidean Geometry. However, the proof for the 5th postulate (the parallel postulate) is not entirely legitimate. This has led to the emergence of alternative geometries, such as Hyperbolic Geometry. For this case, it is accepted that there exists an infinite amount of lines parallel to a given line from a point outside of it. This idea of Hyperbolic Geometry was difficult to conceptualize, but mathematicians soon created different models to represent a plane where Hyperbolic Geometry could exist. One of these was Henri Poincare, who proposed a Disk with certain properties that allowed for multiple parallel lines to be created. To start off, the disk itself is created so that its circumference represents the point at infinity. When a line meets the disk it is understood that it continues on for infinity. Two different lines can be created on this plane: Euclidean lines that pass through the center of the Disk, and Euclidean circles that have two intersections points with disk that are both orthogonal. Given this model, it is found that several objects in nature already utilize Hyperbolic Geometry. In nature some organisms follow this pattern in order to maximize their surface area. I look forward to going into more depth and presenting this topic next week.

# Poincare Disk Model

by solar | Mar 16, 2023 | Uncategorized | 1 comment

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This sounds like an interesting extension to our Geometry class we had last semester. Cannot wait to see how far hyperbolic geometry can be taken since it seemed weird to me at first to learn to concept of parallel lines in a non-Euclidean world.