# Visualizing hyperbolic honeycombs

@article{Nelson2015VisualizingHH, title={Visualizing hyperbolic honeycombs}, author={Roice Nelson and Henry Segerman}, journal={Journal of Mathematics and the Arts}, year={2015}, volume={11}, pages={39 - 4} }

ABSTRACT We explore visual representations of tilings corresponding to Schläfli symbols. In three dimensions, we call these tilings ‘honeycombs’. Schläfli symbols encode, in a very efficient way, regular tilings of spherical, euclidean and hyperbolic spaces in all dimensions. In three dimensions, there are only a finite number of spherical and euclidean honeycombs, but infinitely many hyperbolic honeycombs. Moreover, there are only four hyperbolic honeycombs with material vertices and material…

## 12 Citations

Fractal Images from Multiple Inversion in Circles

- Mathematics
- 2019

Images resulting from multiple inversion and reflection in intersecting circles and straight lines are presented. Three circles and lines making a triangle give the well-known tilings of spherical,…

Cohomology fractals, Cannon-Thurston maps, and the geodesic flow

- Mathematics
- 2020

Cohomology fractals are images naturally associated to cohomology classes in hyperbolic three-manifolds. We generate these images for cusped, incomplete, and closed hyperbolic three-manifolds in…

Navigating Higher Dimensional Spaces using Hyperbolic Geometry

- Computer Science, ArtArXiv
- 2021

This work introduces a novel method of interactive visualization of higher-dimensional grids, based on hyperbolic geometry, making it applicable in data visualization, user interfaces, and game design.

Iterated inversion system: an algorithm for efficiently visualizing Kleinian groups and extending the possibilities of fractal art

- MathematicsJournal of Mathematics and the Arts
- 2021

This paper proposes an efficient algorithm for visualizing some kinds of Kleinian groups: the Iterated Inversion System (IIS), which enables us to render images of Kleinan groups composed of inversions as circles or spheres in real-time.

Cohomology fractals

- Mathematics
- 2020

We introduce cohomology fractals; these are certain images associated to a cohomology class on a hyperbolic three-manifold. They include images made entirely from circles, and also images with no…

Non-euclidean Virtual Reality II: Explorations of H² ✕ E

- Computer Science
- 2017

The goal is to make three-dimensional non-euclidean spaces feel more natural by giving people experiences inside those spaces, including the ability to move through those spaces with their bodies, particularly for users who are not familiar with moving through space using “computer game” controls.

Holography on tessellations of hyperbolic space

- Physics
- 2020

We compute boundary correlation functions for scalar fields on tessellations of two- and three-dimensional hyperbolic geometries. We present evidence that the continuum relation between the scalar…

Ray-marching Thurston geometries

- MathematicsExperimental Mathematics
- 2022

Algorithms that produce accurate real-time interactive in-space views of the eight Thurston geometries using ray-marching are described and a theoretical framework for these algorithms is given, independent of the geometry involved.

Horosphere, Cyclide and 3d Hyperbolic Tilings

- Mathematics
- 2018

We explore different ways to visualize three dimensional hyperbolic tilings using two dimensional cross sections

On the use of 3 D printing for scientific visualization June 15 th , 2018

- Art
- 2018

We present the results of a case-based study in using 3D printing for applications in scientific visualization. The goal was to get insight into the practical sides of producing 3D prints from 3D…

## References

SHOWING 1-10 OF 31 REFERENCES

REGULAR HONEYCOMBS IN HYPERBOLIC SPACE

- Mathematics
- 1956

made a study of honeycombs whose cells are equal regular polytopes in spaces of positive, zero, and negative curvature. The spherical and Euclidean honeycombs had already been described by Schlaf li…

Bending Hyperbolic Kaleidoscopes

- Mathematics
- 2011

We demonstrate different ways to create and visualize tilings made with hyperbolic kaleidoscopes. We start with tiling of the hyperbolic plane and build bended kaleidoscopes in the hyperbolic space.…

Visualizing hyperbolic space: unusual uses of 4x4 matrices

- MathematicsI3D '92
- 1992

Formulas for computing reflections, translations, and rotations in hyperbolic space are presented, which emphasizes the need for graphics libraries which allow completely arbitrary 4 X 4 transformations.

Visualising the arithmetic of quadratic imaginary fields

- Mathematics
- 2014

We study the orbit of R under the Bianchi group PSL2(OK), where K is an imaginary quadratic field. The orbit, called a Schmidt arrangement SK , is a geometric realisation, as an intricate circle…

Lorentzian Coxeter systems and Boyd–Maxwell ball packings

- Mathematics
- 2015

In the recent study of infinite root systems, fractal patterns of ball packings were observed while visualizing roots in affine space. In this paper, we show that the observed fractals are exactly…

Discrete groups and visualization of three-dimensional manifolds

- MathematicsSIGGRAPH
- 1993

This paper establishes a generalization of the application of projective geometry to computer graphics, and lays the groundwork for visualization of spaces of non-constant curvature in discrete groups and topological manifolds.

Indra's Pearls: The Vision of Felix Klein

- Art
- 2002

Preface Introduction 1. The language of symmetry 2. A delightful fiction 3. Double spirals and Mobius maps 4. The Schottky dance 5. Fractal dust and infinite words 6. Indra's necklace 7. The glowing…

Three-Dimensional Geometry and Topology, Volume 1

- Mathematics
- 1997

is a recursive definition!) The first term of the 'Goodstein sequence' of a number is the number itself. The nth term of the Goodstein sequence is obtained by expressing the previous term in 'pure…

Ortho-Circles of Dupin Cyclides

- Mathematics
- 2006

We study the set of circles which intersect a Dupin cyclide in at least two difierent points orthogonally. Dupin cyclides can be obtained by inverting a cylinder, or cone of revolution, or by…