Semi-regular Tilings of the Hyperbolic Plane

Datta, Basudeb; Gupta, Subhojoy

doi:10.1007/s00454-019-00156-0

Cited by 10 publications

(8 citation statements)

References 10 publications

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“…Since X has 18 squares, s 2 +s 3 +s 4 = 18. These give (s 2 , s 3 , s 4 ) = (6, 12, 0), (7, 10, 1), (8,8,2), (9,6,3), (10,4,4), (11,2,5) or (12,0,6).…”

Section: From Definition 37 and The (Geometric) Construction Of Trunc...mentioning

confidence: 99%

“…A semi-regular tiling of a surface S of constant curvature (eg., the round sphere, the Euclidean plane or the hyperbolic plane) is a semi-equivelar map on S in which each face is a regular polygon and each edge is a geodesic. It follows from the results in [4] that there exist semi-regular tilings of the hyperbolic plane of infinitely many different vertex-types. It is also shown that there exists a unique semi-regular tiling of the hyperbolic plane of vertextype [p q ] for each pair (p, q) of positive integers satisfying 1/p + 1/q < 1/2.…”

Section: Introductionmentioning

confidence: 99%

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Platonic solids, Archimedean solids and semi-equivelar maps on the sphere

Datta

Maity

2022

Discrete Mathematics

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“…Since X has 18 squares, s 2 +s 3 +s 4 = 18. These give (s 2 , s 3 , s 4 ) = (6, 12, 0), (7, 10, 1), (8,8,2), (9,6,3), (10,4,4), (11,2,5) or (12,0,6).…”

Section: From Definition 37 and The (Geometric) Construction Of Trunc...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Platonic solids, Archimedean solids and semi-equivelar maps on the sphere

Datta

Maity

2022

Discrete Mathematics

Self Cite

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“…Call this tiling T . This [n, m, n, m] pattern about each vertex of the tiling satisfies condition (1) in [8]. Therefore there exists a semi-regular tiling of H 2 , T , by regular polygons with vertex type [n, m, n, m] at every vertex, which is equivalent to T [8, Lemma 2.5].…”

Section: Totally Geodesic Checkerboard Surfacesmentioning

confidence: 99%

Right-angled links in thickened surfaces

Kaplan-Kelly¹

2022

Preprint

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Traditionally, alternating links are studied with alternating diagrams on S 2 in S 3 . In this paper, we consider links which are alternating on higher genus surfaces Sg in Sg × I. We define what it means for such a link to be right-angled generalized completely realizable (RGCR) and show that this property is equivalent to the link having two totally geodesic checkerboard surfaces, equivalent to each checkerboard surface consisting of one type of polygon, and equivalent to a set of restrictions on the link's alternating projection diagram. We then use these diagram restrictions to classify RGCR links according to the polygons in their checkerboard surfaces and provide a bound on the number of RGCR links for a given surface of genus g. Along the way, we answer a question posed by Champanerkar, Kofman, and Purcell about links with alternating projections on the torus.

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“…For unit negative curvature this manifold is the hyperbolic plane H 2 and has the Poincaré disk representation (for a review see [52]). In this case, any 4-regular graph with 3 squares and a cycle of length larger than 4 per vertex can be promoted to the 1-skeleton of a tiling of a negative curvature Cartan-Hadamard manifold by assigning a fixed length to each edge (CH) [53]. In Fig.…”

Section: Cartan-hadamard Manifolds In 2dmentioning

confidence: 99%

Emergent time, cosmological constant and boundary dimension at infinity in combinatorial quantum gravity

Trugenberger

2021

Preprint

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Combinatorial quantum gravity is governed by a discrete Einstein-Hilbert action formulated on an ensemble of random graphs. There is strong evidence for a second-order quantum phase transition separating a random phase at strong coupling from an ordered, geometric phase at weak coupling. Here we derive the picture of space-time that emerges in the geometric phase, given such a continuous phase transition. In the geometric phase, ground-state graphs are discretizations of Riemannian, negative-curvature Cartan-Hadamard manifolds. On such manifolds, diffusion is ballistic. Asymptotically, diffusion time is soldered with a manifold coordinate and, consequently, the probability distribution is governed by the wave equation on the corresponding Lorentzian manifold of positive curvature, de Sitter space-time. With this asymptotic Lorentzian picture, the original negative curvature of the Riemannian manifold turns into a positive cosmological constant. The Lorentzian picture, however, is valid only asymptotically and cannot be extrapolated back in coordinate time. Before a certain epoch, coordinate time looses its meaning and the universe is a negative-curvature Riemannian "shuttlecock" with ballistic diffusion, thereby avoiding a big bang singularity. The so-called "dimension at infinity" of negative curvature manifolds, i.e. the large-scale spectral dimension seen by diffusion processes with no spectral gap, those that can probe the geometry at infinity, is always three. As a consequence, co-diffusing observers see the conformal boundary of the negative curvature manifold as a surface of spectral dimension two independently of the interior topological dimension.

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Semi-regular Tilings of the Hyperbolic Plane

Cited by 10 publications

References 10 publications

Platonic solids, Archimedean solids and semi-equivelar maps on the sphere

Platonic solids, Archimedean solids and semi-equivelar maps on the sphere

Right-angled links in thickened surfaces

Emergent time, cosmological constant and boundary dimension at infinity in combinatorial quantum gravity

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