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What is known about unit cubes. (English) Zbl 1067.52010

Summary: 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.

MSC:

52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
05A15 Exact enumeration problems, generating functions
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05B25 Combinatorial aspects of finite geometries
05B45 Combinatorial aspects of tessellation and tiling problems
11H31 Lattice packing and covering (number-theoretic aspects)
11J13 Simultaneous homogeneous approximation, linear forms
15B33 Matrices over special rings (quaternions, finite fields, etc.)
20K01 Finite abelian groups
28A25 Integration with respect to measures and other set functions
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)

Software:

01poly
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Full Text: DOI

References:

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