Results in Mathematics

, 74:158

# On Parallel Packing and Covering of Squares and Cubes

Article

## 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

52C15 52C17

## References

1. 1.
Bezdek, K., Khan, M.A.: The geometry of homothetic covering and illumination. In: Discrete Geometry and Symmetry. Springer Proceedings Mathematics & Statistics, vol. 234, pp. 1–30. Springer, Cham (2018)Google Scholar
2. 2.
Boltyanski, V., Martini, H., Soltan, P.S.: Excursions Into Combinatorial Geometry. Springer, Berlin (1997)
3. 3.
Dostert, M., Guzmn, C., de Oliveira, F., Fernando, M., Vallentin, F.: New upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry. Discrete Comput. Geom. 58(2), 449–481 (2017)
4. 4.
Füredi, Z.: Covering a triangle with homothetic copies. In: Discrete Geometry. Monographs and Textbooks in Pure and Applied Mathematics, vol. 253, pp. 435–445. Dekker, New York (2003)
5. 5.
Januszewski, J.: Covering by sequences of squares. Studia Sci. Math. Hung. 39, 179–188 (2002)
6. 6.
Januszewski, J.: A note on translative packing a triangle by sequences of its homothetic copies. Period. Math. Hung. 52, 27–30 (2006)
7. 7.
Januszewski, J.: Translative packing of a convex body by sequences of its positive homothetic copies. Acta Math. Hung. 117(4), 349–360 (2007)
8. 8.
Januszewski, J.: Parallel packing and covering of an equilateral triangle with sequences of squares. Acta Math. Hung. 125, 249–260 (2009)
9. 9.
Januszewski, J.: Optimal translative packing of homothetic triangles. Studia Sci. Math. Hung. 46(2), 185–203 (2009)
10. 10.
Lu, M., Su, Z.: Parallel covering of isosceles triangles with squares. Acta Math. Hung. 155(2), 266–297 (2018)
11. 11.
Moon, J.W., Moser, L.: Some packing and covering theorems. Colloq. Math. 17, 103–110 (1967)
12. 12.
Moser, W., Pach, J.: Research Problems in Discrete Geometry. Privately published collection of problems (1994)Google Scholar
13. 13.
Martini, H., Richter, C., Spirova, M.: Illuminating and covering convex bodies. Discrete Math. 337, 106–118 (2014)
14. 14.
Vásárhelyi, É.: Über eine Überdeckung mit homothetischen Dreiecken. Beitr. Algebra Geom. 17, 61–70 (1984)
15. 15.
Zong, C.: Packing, covering and tiling in two-dimensional spaces. Expo. Math. 32(4), 297–364 (2014)

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Miao Fu
• 1
• Yanlu Lian
• 1
• Yuqin Zhang
• 1
1. 1.School of MathematicsTianjin UniversityTianjinPeople’s Republic of China