Reserve in packing cubes into the unit cube

Packing of points into the unit 6-dimensional cube

The best known estimate for the packing of points into the 6-dimensional unit cube in mutual distances at least 1 is improved. The following problem was stated in [10] and later repeated e.g. in [1], [6], [11], [12], [13]. “Let f(n) denote the maximum number of points that can be arranged in the n-dimensional unit cube so that all mutual distances are at least 1. Obviously, f(n) = 2n for n ≤ 3....

On Decompositions of a Cube into Cubes and Simplexes

Some combinatorial results concerning finite decompositions (dissections) of a k-dimensional cube into cubes (respectively, simplexes) of the same dimension are presented in the paper. In connection with such decompositions, the notion of a decomposability number is introduced and the problem of description of all these numbers is discussed. 2000 Mathematics Subject Classification: 52A15, 52B05...

Decomposition of a cube into nearly equal smaller cubes

Let d be a fixed positive integer and let ε > 0. It is shown that for every sufficiently large n ≥ n0(d, ε), the d-dimensional unit cube can be decomposed into exactly n smaller cubes such that the ratio of the side length of the largest cube to the side length of the smallest one is at most 1 + ε. Moreover, for every n ≥ n0, there is a decomposition with the required properties, using cubes of...

Translative Packing of Unit Squares into Squares

Cube polynomial of Fibonacci and Lucas cubes

The cube polynomial of a graph is the counting polynomial for the number of induced k-dimensional hypercubes (k ≥ 0). We determine the cube polynomial of Fibonacci cubes and Lucas cubes, as well as the generating functions for the sequences of these cubes. Several explicit formulas for the coefficients of these polynomials are obtained, in particular they can be expressed with convolved Fibonac...