Angle Combinations in Spherical Tilings by Congruent Pentagons

Luk, Hoiping; Yan, Min

doi:10.48550/arxiv.1308.4207

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Append(valid Angle Values Angle Values)1

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2022

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Cited by 2 publications

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“…Then the other vertices must have at least as many γ as δ, which are β 2 γ 3 δ, β 6 γ 2 δ 2 . So the only remaining vertex is β 15 . We obtain the third AVC.…”

Section: Append(valid Angle Values Angle Values)mentioning

confidence: 99%

See 1 more Smart Citation

Rational Angles and Tilings of the Sphere by Congruent Quadrilaterals

Cheung¹,

Ping²

2022

Preprint

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We use rational solutions of trigonometric diophantine equations to classify edge-to-edge tilings of the sphere by congruent almost equilateral quadrilaterals (i.e., edge combination a 3 b). Although the classification is completed in a paper by Cheung, Luk and Yan, the method used here is more systematic and is applicable to other related tiling problems that cannot be solved by the method in the other paper. We also provide detailed geometric data for the tilings.

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“…Then the other vertices must have at least as many γ as δ, which are β 2 γ 3 δ, β 6 γ 2 δ 2 . So the only remaining vertex is β 15 . We obtain the third AVC.…”

Section: Append(valid Angle Values Angle Values)mentioning

confidence: 99%

“…The first one extends Coolsaet's method to non-convex almost equilateral quadrilaterals. The second is a method for classifying tilings when some angles are non-rational by using ideas from [15]. As a result of our classification, we also know the tilings that every angle is rational as well as those otherwise.…”

Section: Introductionmentioning

confidence: 99%

Rational Angles and Tilings of the Sphere by Congruent Quadrilaterals

Cheung¹,

Ping²

2022

Preprint

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Combinatorial Tilings of the Sphere by Pentagons

Yan

2013

Electron. J. Combin.

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A combinatorial tiling of the sphere is naturally given by an embedded graph. We study the case that each tile has exactly five edges, with the ultimate goal of classifying combinatorial tilings of the sphere by geometrically congruent pentagons. We show that the tiling cannot have only one vertex of degree > 3. Moreover, we construct earth map tilings, which give classifications under the condition that vertices of degree > 3 are at least of distance 4 apart, or under the condition that there are exactly two vertices of degree > 3. * Research was supported by Hong Kong RGC General Research Fund 605610 and 606311.

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Angle Combinations in Spherical Tilings by Congruent Pentagons

Cited by 2 publications

References 7 publications

Rational Angles and Tilings of the Sphere by Congruent Quadrilaterals

Rational Angles and Tilings of the Sphere by Congruent Quadrilaterals

Combinatorial Tilings of the Sphere by Pentagons

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