What is known about unit cubes

Zong, Chuanming

doi:10.1090/S0273-0979-05-01050-5

What is known about unit cubes

Author: Chuanming Zong
Journal: Bull. Amer. Math. Soc. 42 (2005), 181-211
MSC (2000): Primary 05A15, 05B20, 05B25, 05B45, 11H31, 11J13, 15A33, 20K01, 28A25, 52B05, 52C20, 52C22
DOI: https://doi.org/10.1090/S0273-0979-05-01050-5
Published electronically: January 26, 2005
MathSciNet review: 2133310
Full-text PDF Free Access

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Abstract: Unit cubes, from any point of view, are among the simplest and the most important objects in -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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