Visual Proof of Triangulating a Sphere

In a 2D plane, uniformly distributed points grow into expanding purple discs, intersecting to form an alpha complex. This method captures the topological features of the dataset, illustrated through persistence diagrams and barcodes. A persistence diagram is a scatter plot that shows the birth and death of topological features, such as connected components, loops, and voids, as the scale changes. A barcode diagram represents these features as horizontal bars, where each bar's length indicates the lifespan of a particular feature. As the discs' radii increase, the process reveals key homological structures, including connected components (0D), loops (1D), and voids (2D). The barcode graph presents the following features: 0D bars show the birth and merging of connected components, 1D bars track the formation and closure of loops or cycles, and 2D bars highlight the emergence and filling of voids, elucidating the data's shape and connectivity in the plane. Of note are the formation of loops (blue), which are homeomorphically equivalent to a circle.

Points uniformly distributed on the surface of a sphere grow into larger spheres, intersecting and forming an alpha complex. This process again captures the topological features of the data on the surface, visualized through persistence diagrams and barcodes. As the radii of these growing spheres increase, the algorithm reveals the underlying homological structures, such as connected components (0D), loops (1D), and voids (2D). Of note is the void created by the accumulation of all the spheres, creating a hollow surface which is homeomorphically equivalent sphere (in green).

An unoptimized reservoir computer was trained on the Lorenz system for 70% of the 20,000 timesteps generated and tested on the remaining 30%. A forecast horizon was defined as the first point where the prediction deviated from the actual system by 25%. This is symbolized above by a red point and as line on each axis subplot to the left. The forecast horizon is highly dependent on definition and hyperparameters used to influence the reservoir. The work here and elsewhere with the Shaheen lab was built upon base code by G. Espitia[^1] at Medium and  M. Francesco at ReservoirComputing.jl[^2].