Pentagonal Subdivision

Yan, Min

doi:10.37236/8624

Cited by 6 publications

(8 citation statements)

References 3 publications

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“…Thus, by iteratively applying operations 1-3, we have created a subdivision scheme that creates pentagon meshes from an arbitrary input. So far, the operations described here match the approach of Bowers and Stephenson [3], Yan [25], and Akleman et al [1] combinatorially. However, embeddings of the resulting meshes, when following the steps as outlined above, generally lack an important geometric property, which we will address in the following.…”

Section: Insertion Of Vertices and Edgesmentioning

confidence: 87%

“…Hence, two vertices are added as well. The first and last vertex of the Z-triplet correspond to the two vertices of e. A variant of this subdivision scheme including the first three operations was proposed by Bowers and Stephenson in 1997 [3] and discussed by Akleman et al [1] while combinatorial aspects are investigated by Yan [25,Sec. 3.2].…”

Section: The Pentagon Snub Subdivision Schemementioning

confidence: 99%

“…Also, some of the new faces are not convex any more (see Figure 7b, non-convex faces colored in blue). Because of the focus on combinatorial aspects, both the work of Bowers and Stephenson [3] and that of Yan [25] do not address this. Akleman et al [1] do propose a second, different procedure that ensures convexity, which we will discuss after presenting our approach to preserving convexity of the faces.…”

Section: The Smoothing Operationmentioning

confidence: 99%

“…13(a)]). This combinatorial structure is investigated by Yan [25]. For both works [3,25], it is unclear how to embed the obtained meshes, while our approach creates a corresponding embedding on the fly.…”

Section: The Smoothing Operationmentioning

confidence: 99%

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Weaving patterns inspired by the pentagon snub subdivision scheme

Lipschütz,

Reitebuch,

Skrodzki

et al. 2021

Preprint

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Various computer simulations regarding, e.g., the weather or structural mechanics, solve complex problems on a two-dimensional domain. They mostly do so by splitting the input domain into a finite set of smaller and simpler elements on which the simulation can be run fast and efficiently. This process of splitting can be automatized by using subdivision schemes.Given the wide range of simulation problems to be tackled, an equally wide range of subdivision schemes is available. They create subdivisions that are (mainly) comprised of triangles, quadrilaterals, or hexagons. Furthermore, they ensure that (almost) all vertices have the same number of neighboring vertices.This paper illustrates a subdivision scheme that splits the input domain into pentagons. Repeated application of the scheme gives rise to fractal-like structures. Furthermore, the resulting subdivided domain admits to certain weaving patterns. These patterns are subsequently generalized to several other subdivision schemes.As a final contribution, we provide paper models illustrating the weaving patterns induced by the pentagonal subdivision scheme. Furthermore, we present a jigsaw puzzle illustrating both the subdivision process and the induced weaving pattern. These transform the visual and abstract mathematical algorithms into tactile objects that offer exploration possibilities aside from the visual.

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Section: Insertion Of Vertices and Edgesmentioning

confidence: 87%

Section: The Pentagon Snub Subdivision Schemementioning

confidence: 99%

Section: The Smoothing Operationmentioning

confidence: 99%

Section: The Smoothing Operationmentioning

confidence: 99%

See 2 more Smart Citations

Weaving patterns inspired by the pentagon snub subdivision scheme

Lipschütz,

Reitebuch,

Skrodzki

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Further efforts have been put into study of tilings by quadrilaterals by Ueno and Agaoka in [15], by Akama et al in [1], [2], [3], [4], [6], which include quadrilaterals of the types that are equilateral or can be divided into two triangles. On the other hand, Yan et al are on course to give a complete classification of tilings by pentagons in [11], [5], [16], [17], [7], [12], [18], [19], [20]. The problems remain open are the tilings by quadrilaterals with exactly two equal edges and those with exactly three equal edges.…”

Section: Introductionmentioning

confidence: 99%

Tilings of the Sphere by Congruent Quadrilaterals with Exactly Two Equal Edges

Cheung¹,

Ping²

2021

Preprint

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In this paper we give a classification of tilings of the sphere by congruent quadrilaterals with exactly two equal edges. The tilings are the earth map tilings, (p, q)-earth map tilings and their flip modifications, and quadrilateral subdivisions of the cube and the triangular prism. We described the ranges of values of the edges and angles for the tile to be geometrically realisable through extrinsic parameters. The symmetry groups of the tilings are also determined.

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Rational Angles and Tilings of the Sphere by Congruent Quadrilaterals

Luk,

Cheung

2024

Ann. Comb.

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We apply Diophantine analysis to classify edge-to-edge tilings of the sphere by congruent almost equilateral quadrilaterals (i.e., edge combination $$a^3b$$ a 3 b ). Parallel to a complete classification by Cheung, Luk, and Yan, the method implemented here is more systematic and applicable to other related tiling problems. We also provide detailed geometric data for the tilings.

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Pentagonal Subdivision

Abstract: We develop a theory of simple pentagonal subdivision of quadrilateral tilings, on orientable as well as non-orientable surfaces. Then we apply the theory to answer questions related to pentagonal tilings of surfaces, especially those related to pentagonal or double pentagonal subdivisions.

Cited by 6 publications

References 3 publications

Weaving patterns inspired by the pentagon snub subdivision scheme

Weaving patterns inspired by the pentagon snub subdivision scheme

Tilings of the Sphere by Congruent Quadrilaterals with Exactly Two Equal Edges

Rational Angles and Tilings of the Sphere by Congruent Quadrilaterals

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