Multiple Lattice Tilings in Euclidean Spaces

Yang, Qi; Zong, Chuanming

doi:10.4153/s0008439518000103

Cited by 11 publications

(12 citation statements)

References 23 publications

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“…Theorem 10 (Yang and Zong [17]). If is a twodimensional convex domain that is neither a parallelogram nor a centrally symmetric hexagon, then we have…”

Section: Multiple Lattice Tilingsmentioning

confidence: 99%

“…where equality in (17) holds if and only if (after a suitable affine linear transformation) 10 is a centrally symmetric decagon whose edge midpoints are 1 = (0, −1),…”

Section: Multiple Lattice Tilingsmentioning

confidence: 99%

“…Let Λ be the integer lattice ℤ 2 ; it can be verified that both 8 ( ) + Λ and 8 ( ) + Λ are indeed fivefold lattice tilings of Clearly, Theorem 11 follows from (14)- (18). The proofs of these inequalities are complicated, in particular (17) and (18). Their proofs rely on carefully designed area estimations by dealing with many cases.…”

Section: Multiple Lattice Tilingsmentioning

confidence: 99%

“…As noticed by Yang and Zong [17], based on the twodimensional examples, for any ≥ 3 one can construct -dimensional centrally symmetric polytopes satisfying 2 ≤ * ( ) ≤ 5 and 2 ≤ ( ) ≤ 5.…”

Section: Open Problemsmentioning

confidence: 99%

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Can You Pave the Plane with Identical Tiles?

Zong¹

2020

Notices Amer. Math. Soc.

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Everybody knows that identical regular triangles or squares can tile the whole plane. Many people know that identical regular hexagons can tile the plane properly as well. In fact, even the bees know and use this fact! Is there any other convex domain that can tile the Euclidean plane? Of course, there is a long list of them! To find the list and to show the completeness of the list is a unique drama in mathematics which has lasted for more than one century, and the completeness of the list has been mistakenly announced more than once! Up to now, the list consists of triangles, quadrilaterals, fifteen types of pentagons, and three types of hexagons. In 2017, Michaël Rao announced a computer proof for the completeness of the list. Meanwhile, Qi Yang and Chuanming Zong made a series of unexpected discoveries in multiple tilings in the Euclidean plane. For example, besides parallelograms and centrally symmetric hexagons, there is no other convex domain that can form any two-, three-, or fourfold translative tiling in the plane. However, there are two types of octagons

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“…Theorem 10 (Yang and Zong [17]). If is a twodimensional convex domain that is neither a parallelogram nor a centrally symmetric hexagon, then we have…”

Section: Multiple Lattice Tilingsmentioning

confidence: 99%

“…where equality in (17) holds if and only if (after a suitable affine linear transformation) 10 is a centrally symmetric decagon whose edge midpoints are 1 = (0, −1),…”

Section: Multiple Lattice Tilingsmentioning

confidence: 99%

Section: Multiple Lattice Tilingsmentioning

confidence: 99%

Section: Open Problemsmentioning

confidence: 99%

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Can You Pave the Plane with Identical Tiles?

Zong¹

2020

Notices Amer. Math. Soc.

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“…Clearly, one of the most important and natural problems in multiple tilings is, for given n and k, to classify or characterize all the n-dimensional k-fold lattice tiles and all the n-dimensional k-fold translative tiles (see Problems 1-4 at the end of Gravin, Robins and Shiryaev [8]). In the plane, it was proved by Yang and Zong [22,23,27,28] that, besides parallelograms and centrally symmetric hexagons, there is no other two-, three-or fourfold translative tile. However, there are three classes of other fivefold translative tiles and three classes of other sixfold lattice tiles.…”

Section: Introductionmentioning

confidence: 99%

The Three and Fourfold Translative Tiles in Three-Dimensional Space

Han¹,

Sriamorn²,

Yang³

et al. 2021

Preprint

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Equality Case in van der Corput’s Inequality and Collisions in Multiple Lattice Tilings

Averkov

2019

Discrete Comput Geom

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Van der Corput's provides the sharp bound vol(C) ≤ m2 d on the volume of a d-dimensional origin-symmetric convex body C that has 2m − 1 points of the integer lattice in its interior. For m = 1, a characterization of the equality case vol(C) = m2 d is equivalent to the well-known problem of characterizing tilings by translations of a convex body. It is rather surprising that so far, for m ≥ 2, no characterization of the equality case has been available, though a hint to the respective characterization problem can be found in the 1987 monograph of Gruber and Lekkerkerker. We give an explicit characterization of the equality case for all m ≥ 2. Our result reveals that, the equality case for m ≥ 2 is more restrictive than for m = 1. We also present consequences of our characterization in the context of multiple lattice tilings.

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Multiple Lattice Tilings in Euclidean Spaces

Cited by 11 publications

References 23 publications

Can You Pave the Plane with Identical Tiles?

Can You Pave the Plane with Identical Tiles?

The Three and Fourfold Translative Tiles in Three-Dimensional Space

Equality Case in van der Corput’s Inequality and Collisions in Multiple Lattice Tilings

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