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Perfect packing of cubes

  • A. Joós
Article

Abstract

It is known that \({\sum_{i =1}^\infty {1/ i^2}={\pi^2/6}}\). We can ask what is the smallest \({\epsilon}\) such that all the squares of sides of length \({1, 1/2, 1/3, \ldots}\) can be packed into a rectangle of area \({{\pi^2/6}+\epsilon}\). A packing into a rectangle of the right area is called perfect packing. Chalcraft [4] packed the squares of sides of length \({1, 2^{-t}, 3^{-t}, \ldots}\) and he found perfect packings for \({1/2 < t \le 3/5}\). We generalize this problem and pack the 3-dimensional cubes of sides of length \({1, 2^{-t}, 3^{-t}, \ldots}\) into a right rectangular prism of the right volume. Moreover we show that there is a perfect packing for all t in the range \({0.36273 \le t \le 4/11}\).

Key words and phrases

packing cube rectangular prism 

Mathematics Subject Classification

52C17 52C22 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.University of DunaújvárosDunaújvárosHungary

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