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Results in Mathematics

, 74:158 | Cite as

On Parallel Packing and Covering of Squares and Cubes

  • Miao Fu
  • Yanlu LianEmail author
  • Yuqin Zhang
Article
  • 72 Downloads

Abstract

Suppose that T is a right triangle with leg lengths 1 and \(\sqrt{2}\), \(T_{r}\) is a trirectangular tetrahedron with three right-angle edges lengths 1, 1 and \(\sqrt{2}\). And let {\(S_{n}\)} be a sequence of the homothetic copies of a square S with a side parallel to some leg of T, {\(C_{n}\)} be a sequence of the homothetic copies of a cube C with a face parallel to the base face of \(T_{r}\). We first determine the bound of sums of areas of squares from the sequence {\(S_{n}\)} that permits a parallel packing of T. Then we show the bound of sums of volumes of cubes from the sequence {\(C_{n}\)} that permits a parallel covering of \(T_{r}\).

Keywords

Packing covering homothetic copies tetrahedron 

Mathematics Subject Classification

52C15 52C17 

Notes

Acknowledgements

This research was supported by National Natural Science Foundation No. 11801410.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of MathematicsTianjin UniversityTianjinPeople’s Republic of China

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