Decomposition of a Cube into Nearly Equal Smaller Cubes. Issue 10 (1st December 2017)

• Record Type: Journal Article
• Authors: Frankl, Peter; Meir, Amram; Pach, János
• Is Part Of: American Mathematical Monthly. Volume 124:Issue 10(2017)
• Page Start: 895
• Page End: 904
• Publication Date: 2017-12-01
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.4169/amer.math.monthly.124.10.895
• Languages: English
• ISSNs: 0002-9890

Abstract

Abstract: Let d be a fixed positive integer and let ε >0 . It is shown that for every sufficiently large n ≥ n 0( 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 ≥ n 0, 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 three different sizes.