Visualizing Hyperbolic Honeycombs

Nelson, Roice; Segerman, Henry

doi:10.48550/arxiv.1511.02851

2015

DOI: 10.48550/arxiv.1511.02851

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Visualizing Hyperbolic Honeycombs

Roice Nelson¹,

Henry Segerman²

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2017

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“…A surprising feature of this honeycomb is that the cubes are no longer of finite size -it turns out that the vertices must be infinitely far away. See [4] for more on this phenomenon.…”

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confidence: 99%

Non-euclidean virtual reality I: explorations of $\mathbb{H}^3$

Hart,

Hawksley,

Matsumoto

et al. 2017

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We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulations of three-dimensional hyperbolic space are available at h3.hypernom.com. 1 Figure 1: A view from H 3 .We are used to living in three-dimensional euclidean space, and our day-to-day experiences of curvature centre around surfaces embedded in E 3 . In the study of topology, the closed two-dimensional surfaces are the sphere, the torus, the two-holed torus, the three-holed torus, and so on. Thinking of these surfaces topologically, they don't come with a particular choice of geometry -that is, we can think about a surface as if they were made from plasticine -without knowing what lengths and angles mean on the surface. There are however particularly nice geometries for these surfaces: isotropic geometries, meaning that the geometry is the same everywhere in the space, facing in every direction. We have spherical geometry for the sphere, euclidean geometry for the torus, and hyperbolic geometry for all of the others. In three dimensions, the story is more complicated. Thurston's geometrization conjecture, proved by Perelman [5], gives eight geometries 1 The code is available at github.com/hawksley/hypVR.

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“…A surprising feature of this honeycomb is that the cubes are no longer of finite size -it turns out that the vertices must be infinitely far away. See [4] for more on this phenomenon.…”

mentioning

confidence: 99%

Non-euclidean virtual reality I: explorations of $\mathbb{H}^3$

Hart,

Hawksley,

Matsumoto

et al. 2017

Preprint

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Visualizing Hyperbolic Honeycombs

Cited by 1 publication

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Non-euclidean virtual reality I: explorations of $\mathbb{H}^3$

Non-euclidean virtual reality I: explorations of $\mathbb{H}^3$

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