5/3/2023 0 Comments Hyperbolic tessellationThere are infinite copies of the fundamental domain in a hyperbolic tiling, so we need a rendering approach that is not computationally expensive or inaccurate. Render the tiling in hyperbolic space.We also want to find a suitable symmetry group in hyperbolic space (ie a reflective hyperbolic symmetry group for a reflective euclidean group, a rotational group for a rotational group, and so on). We want to find a way to stretch the fundamental domain of a wallpaper group to a fundamental domain of a group in Hyperbolic space such that the new tiling preserves the identity of the floor pattern. Find a suitable mapping from the fundamental domain and group in Euclidean space to Hyperbolic space. The groups classify the tiling of the plane through operations like reflections, gyrations, glide reflections and translations of an initial fundamental domain. A Wallpaper Group is a classification of symmetry groups in the 2D plane. Extract its Wallpaper Group and fundamental domain.More specifically, given a picture of a floor, we: Render the visualization on the screen.Find a proper mapping from a fundamental domain and a symmetry group in Euclidean space to Hyperbolic space.Classify the symmetry group of the floor pattern.The general approach for this project consists of three steps: The technique is inspired by Hyperbolization of Euclidean Ornaments by Martin von Gagern and combines images sourced from post describes the project in more detail, going through several examples and implementation details. The result is a novel way to look at floor tilings and their infinite nature. Hyperbolic Floors is an interactive visualization that maps hundreds of floors across the world to tessellations in Hyperbolic space.
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