Decomposition of a cube into nearly equal smaller cubes

2015/11/17 by Peter Frankl, Frankl, Peter, Amram Meir +3
Mathematics · #05B45 #52C22 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05B45 #msc:52C22

paper · pdf · doi:10.48550/arxiv.1511.05301

13 pages, 4 figures

arxiv created 2015/11/17 · arxiv updated 2015/11/18

Abstract

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 at most d+2 different side lengths. If we drop the condition that the side lengths of the cubes must be roughly equal, it is sufficient to use cubes of two different sizes.

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