A Cube Tiling of Dimension Eight with No Facesharing

Mackey,

doi:10.1007/s00454-002-2801-9

Cited by 47 publications

(40 citation statements)

References 8 publications

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“…The last case was discovered by Lagarias and Shor [67], and the other cases were consequences of Mackey [73]. In general we have ξ n+1 ≤ ξ n + 1.…”

Section: Theorem 73 (Lagarias and Shormentioning

confidence: 84%

What is known about unit cubes

Zong

2005

Bull. Amer. Math. Soc.

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Abstract. Unit cubes, from any point of view, are among the simplest and the most important objects in n-dimensional Euclidean space. In fact, as one will see from this survey, they are not simple at all. On the one hand, the known results about them have been achieved by employing complicated machineries from Number Theory, Group Theory, Probability Theory, Matrix Theory, Hyperbolic Geometry, Combinatorics, etc.; on the other hand, the answers for many basic problems about them are still missing. In addition, the geometry of unit cubes does serve as a meeting point for several applied subjects such as Design Theory, Coding Theory, etc. The purpose of this article is to figure out what is known about the unit cubes and what do we want to know about them.

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“…The last case was discovered by Lagarias and Shor [67], and the other cases were consequences of Mackey [73]. In general we have ξ n+1 ≤ ξ n + 1.…”

Section: Theorem 73 (Lagarias and Shormentioning

confidence: 84%

What is known about unit cubes

Zong

2005

Bull. Amer. Math. Soc.

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“…The conjecture was proved to be true for n ≤ 6 in [Pe40]. It was proved to be false for n ≥ 10 in [LaSh92] and for n ≥ 8 in [McKa02]. The counterexample found were of class T 2 .…”

mentioning

confidence: 93%

New results on torus cube packings and tilings

Sikirić

Itoh

2015

Proc. Steklov Inst. Math.

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ABSTRACT. We consider sequential random packing of integral translate of cubes [0, N ] n into the torus Z n 2N Z n . Two special cases are of special interest:(1) The case N = 2 which corresponds to a discrete case of tilings (considered in [DIP06, DI11]) (2) The case N = ∞ corresponds to a case of continuous tilings (considered in [DI10, DI11]) Both cases correspond to some special combinatorial structure and we describe here new developments.

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“…The higher the dimension of the space, the more freedom we get. Keller's conjecture for the remaining dimensions was solved in 2002 by M a c k e y [16], who showed that the conjecture is false for n = 8, 9 as well, and finally this year M y r v o l d et al (private communication) showed, providing a computer based proof, that Keller's conjecture is true for n = 7.…”

Section: Introductionmentioning

confidence: 99%

Error-correcting codes and Minkowski’s conjecture

Horák¹

2010

Tatra Mountains Mathematical Publications

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ABSTRACT. The goal of this paper is twofold. The main one is to survey the latest results on the perfect and quasi-perfect Lee error correcting codes. The other goal is to show that the area of Lee error correcting codes, like many ideas in mathematics, can trace its roots to the Phytagorean theorem a 2 +b 2 = c 2 . Thus to show that the area of the perfect Lee error correcting codes is an integral part of mathematics. It turns out that Minkowski's conjecture, which is an interface of number theory, approximation theory, geometry, linear algebra, and group theory is one of the milestones on the route to Lee codes.

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A Cube Tiling of Dimension Eight with No Facesharing

Abstract: A cube tiling of eight-dimensional space in which no pair of cubes share a complete common seven-dimensional face is constructed. Together with a result of Perron, this shows that the first dimension in which such a tiling can exist is seven or eight.

Cited by 47 publications

References 8 publications

What is known about unit cubes

What is known about unit cubes

New results on torus cube packings and tilings

Error-correcting codes and Minkowski’s conjecture

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