Document Type
Thesis
Date of Award
2026
Degree Name
Master of Science (MS)
Department
Mathematics
First Advisor
Catalin Georgescu
Abstract
Barycentric subdivision of a triangle is the geometrical process of repeatedly subdividing a triangle by connecting the midpoints of the sides to the opposite vertices. The transformations which determine this subdivision form a group acting on the hyperbolic plane, action which we will show is topologically transitive. We find specific cases when the barycentric subdivision process leads to flat triangles (all vertices on the x-axis) and, on the contrary, situations when shapes are positioned on an orbit that is a circle, hence never becoming flat. We will also analyse this process when the starting triangle is already flat and we will show that this could lead to a chaotic dynamic of shapes. Finally, we will explore the process of barycentric subdivision as a random process to show that the distance between the shape of a triangle after n iterations and a given point converges in probability to a strictly positive number which is called the Lyapunov exponent.
Subject Categories
Mathematics
Keywords
barycentric, geometry, hyperbolic, Lyapunov, triangle
Number of Pages
78
Publisher
University of South Dakota
Recommended Citation
Steger, Hannah Elisabeth, "Barycentric Subdivision and Hyperbolic Geometry" (2026). Dissertations and Theses. 426.
https://red.library.usd.edu/diss-thesis/426