10 citations · 27 across the 14 of their papers we have counts for
14 papers
Illustrating Hyperbolic Surfaces with Mesh Embeddings
Fabian Lander, Erik Löffelholtz, Diaaeldin Taha +2
Hyperbolic geometry exhibits geometric phenomena, such as fast area growth, that are difficult to visualize faithfully in Euclidean space, and which standard models like the Poinca…
Resolving Thinness of Degree-Six Symplectic Hypergeometric Monodromy Groups
J. Maxwell Riestenberg, Diaaeldin Taha, Steve Trettel +1
We prove arithmeticity for two degree-six symplectic hypergeometric monodromy groups and thinness for a third case. These are the three open cases called C-32, C-47, and C-55 in th…
Elliptic Curves and the Hopf Fibration
Nadir Hajouji, Steve Trettel
By combining tools from different areas of mathematics, we obtain 3D visualizations of elliptic curves over different fields that faithfully capture the underlying algebra and geom…
Classical Optics for Charged Black Holes
Steve Trettel
We describe a method for rendering multiple extremally charged black holes using an analogous system in classical optics, simplifying the prerequisite mathematics for generating ac…
Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matrices
Wei Zhao, Federico Lopez, J. Maxwell Riestenberg +3
Recent research has shown that alignment between the structure of graph data and the geometry of an embedding space is crucial for learning high-quality representations of the data…
Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices
Federico López, Beatrice Pozzetti, Steve Trettel +2
We propose the use of the vector-valued distance to compute distances and extract geometric information from the manifold of symmetric positive definite matrices (SPD), and develop…
Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach
Federico López, Beatrice Pozzetti, Steve Trettel +2
Learning faithful graph representations as sets of vertex embeddings has become a fundamental intermediary step in a wide range of machine learning applications. We propose the sys…
Hermitian Symmetric Spaces for Graph Embeddings
Federico López, Beatrice Pozzetti, Steve Trettel +1
Learning faithful graph representations as sets of vertex embeddings has become a fundamental intermediary step in a wide range of machine learning applications. The quality of the…
Ray-marching Thurston geometries
Rémi Coulon, Elisabetta A. Matsumoto, Henry Segerman +1
We describe algorithms that produce accurate real-time interactive in-space views of the eight Thurston geometries using ray-marching. We give a theoretical framework for our algor…
Algebraic Number Starscapes
Edmund Harriss, Katherine E. Stange, Steve Trettel
We study the geometry of algebraic numbers in the complex plane, and their Diophantine approximation, aided by extensive computer visualization. Motivated by these images, called a…
Non-Euclidean Virtual Reality III: Nil
Rémi Coulon, Elisabetta A. Matsumoto, Henry Segerman +1
We describe a method of rendering real-time scenes in Nil geometry, and use this to give an expository account of some interesting geometric phenomena. You can play around with the…
Non-Euclidean Virtual Reality IV: Sol
Rémi Coulon, Elisabetta A. Matsumoto, Henry Segerman +1
This article presents virtual reality software designed to explore the Sol geometry. The simulation is available on 3-dimensional.space/sol.html
Families of Geometries, Real Algebras, and Transitions
Steve Trettel
This thesis details the results of four interrelated projects. The first of these presents a new proof of the theorem of Cooper, Danciger and Wienhard classifying the limits under…
The Heisenberg Plane
Steve J. Trettel
The geometry of the Heisenberg group acting on the plane arises naturally in geometric topology as a degeneration of the familiar spaces and $\mathbb{E}…