On the Packing of Cubes and Other Objects

 Access to full text

 Full (PDF)

1. Adamko, P., On the volume of points at distance at least 1 in the unit four-dimensional cube, . J. Geom. Graph. 23 (2019), 1–3. (2019) MR3982404
2. Adamko, P., Bálint, V., Universal asymptotical results on packing of cubes, . Stud. Univ. Žilina Math. Ser. 28 (2016), 5–16. (2016) 
3. Ament, P., Blind, G., Packing equal circles in a square, . Studia Sci. Math. Hungar. 36 (2000), 313–316. (2000) MR1798737
4. Andreescu, T., Mushkarov, O., A note on the Malfatti problem, . Math. Reflections 4 (2006), 1–7. (2006) 
5. Anstreicher, K. M., 10.1007/s00454-003-0819-2, . Discrete Comput. Geom. 31 (2004), 613–625. (2004) MR2053501DOI10.1007/s00454-003-0819-2
6. Bálint, V., Poznámka k jednému ukladaciemu problému, . Práce a Štúdie Vysokej školy dopravy a spojov v Žiline, séria Mat.–Fyz. 8 (1990), 7–12. (1990) 
7. Bálint, V., A packing problem and the geometrical series, . In: Nešetřil, J., Fiedler, M. (eds.): Fourth Czechoslovakian symposium on combinatorics, graphs and complexity, held in Prachatice, Czechoslovakia, 1990. Proceedings. Annals of Discrete Mathematics, vol. 51. North-Holland, Amsterdam, 1992, 17–21. (1992) MR1206238
8. Bálint, V., 10.1016/S0012-365X(97)81831-6, . Discrete Math. 178 (1998), 233–236. (1998) MR1483753DOI10.1016/S0012-365X(97)81831-6
9. Bálint, V., Maximization of the sum of areas, . Stud. Univ. Žilina Math. Ser. 24 (2010), 1–8. (2010) MR2829522
10. Bálint, V., Dva typy najlepších uložení systému štvorcov v obdĺžniku, . Proceedings of Symposium on Computer Geometry, STU, Bratislava, 2011, 13–16. (2011) 
11. Bálint, V., Adamko, P., Minimalizácia objemu kvádra pre uloženie troch kociek v dimenzii 4, . G, Slov. Čas. Geom. Graf. 12 (2015), 5–16. (2015) 
12. Bálint, V., Adamko, P., Minimization of the container for packing of three cubes in dimension 4, . Proceedings of Slovak–Czech Conference on Geometry and Graphics, STU, Bratislava, 2015, 13–24. (2015) 
13. Bálint, V., Adamko, P., Minimization of the parallelepiped for packing of three cubes in dimension 6, . Proceedings of APLIMAT 2016 – 15th Conference on Applied Mathematics, Bratislava, 2016, 44–55. (2016) 
14. Bálint, V., Bálint, V., jr., Unicity of one optimal arrangement of points in the cube, . Proceedings of Symposium on Computer Geometry, Bratislava, 2001, 8–10. (2001) 
15. Bálint, V., Bálint, V., jr., On the volume of points at distance at least one in the unit cube, . Geombinatorics 12 (2003), 157–166. (2003) MR1972054
16. Bálint, V., Bálint, V., jr., Horný odhad pre rozmiestňovanie bodov v kocke, . Sborník 5. konference o matematice a fyzice na VŠT, Brno, 2007, 32–35. (2007) 
17. Bálint, V., Bálint, V., jr., 10.1007/s10998-008-7083-2, . Periodica Math. Hung. 57 (2008), 83–91. (2008) MR2448399DOI10.1007/s10998-008-7083-2
18. Bálint, V., Bálint, V., jr., Umiestňovnie bodov do jednotkovej kocky, . G, Slov. Čas. Geom. Graf. 5 (2008), 5–12. (2008) 
19. Bálint, V., Bálint, V., jr., 10.1007/s10998-012-2275-3, . Period. Math. Hungar. 65 (2012), 1–16. (2012) MR2970062DOI10.1007/s10998-012-2275-3
20. Bálint, V., Bálint, V., jr., Packing of points into the unit 6-dimensional cube, . Contrib. Discrete Math. 7 (2012), 51–57. (2012) MR2956337
21. Bálint, V., Kaukič, M., Peško, Š., Solving one maximization problem using a computer, . Abstracts of the 3rd Croatian Conference on Geometry and Graphics, http://www.grad.hr/sgorjanc/supetar/abstracts.pdf 
22. Bezdek, A., Fodor, F., Extremal triangulations of convex polygons, . Symmetry: Culture and Science 21 (2010), 333–340. (2010) 
23. Böröczky, K., The Newton-Gregory problem revisited, . In: Bezdek, A. (ed.): Discrete Geometry, Marcel Dekker, New York, 2003, 103–110. (2003) MR2034712
24. Böröczky, K., jr., Finite packing and covering, . Cambridge Univ. Press, 2004. (2004) MR2078625
25. Brass, P., Moser, W. O. J., Pach, J., Research problems in discrete geometry, . Springer, New York, 2005. (2005) MR2163782
26. Cohn, H., Elkies, N. D., New upper bounds on sphere packings I, . Ann. of Math. (2) 157 (2003), 689–714. (2003) MR1973059
27. Croft, H. T., Falconer, K. J., Guy, R. K., Unsolved problems in geometry, . 2nd ed., Springer-Verlag, New York–Berlin–Heidelberg, 1994. (1994) MR1316393
28. Edel, Y., Rains, E. M., Sloane, N. J. A., On kissing volumes in dimensions 32 to 128, . Electron. J. Combin. 5 (1988), #R22. (1988) MR1614304
29. Erdős, P., 10.1007/BF02414146, . Ann. Mat. Pura Appl., Ser. IV 103 (1975), 99–108. (1975) MR0411984DOI10.1007/BF02414146
30. Erdős, P., Some more problems on elementary geometry, . Austral. Math. Soc. Gaz. 5 (1978), 52–54. (1978) MR0509363
31. Fejes Tóth, L., Remarks on a theorem of R. M. Robinson, . Studia Sci. Math. Hung. 4 (1969), 441–445. (1969) MR0254744
32. Fejes Tóth, L., Lagerungen in der Ebene, auf der Kugel und im Raum, . 2. Auflage, Springer-Verlag, 2003. (2003) MR0353117
33. Fejes Tóth, G., Kuperberg, W., Packing and covering with convex sets, . In: Gruber, P. M. et al. (ed.): Handbook of convex geometry, Volume B, North-Holland, Amsterdam, 1993, 799–860. (1993) MR1242997
34. Ferguson, S. P., Hales, T. C., The Kepler conjecture: The Hales–Ferguson proof, . Springer, New York, 2011. (2011) MR3075372
35. Fodor, F., 10.1023/A:1005091317243, . Geom. Dedicata 74 (1999), 139–145. (1999) MR1674049DOI10.1023/A:1005091317243
36. Fodor, F., The densest packing of 12 congruent circles in a circle, . Beitr. Algebra Geom. 21 (2000), 401–409. (2000) MR1801430
37. Fodor, F., Packing 14 congruent circles in a circle, . Stud. Univ. Žilina Math. Ser. 16 (2003), 25–34. (2003) MR2065745
38. Fodor, F., The densest packing of 13 congruent circles in a circle, . Beitr. Algebra Geom. 21 (2003), 431–440. (2003) MR2017043
39. Gauss, C. F., Recension der Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seber, . J. Reine Angew. Math. 20 (1840), 312–320. (1840) MR1578241
40. Graham, R. L., Lubachevsky, B. D., 10.37236/1223, . Electron. J. Combin. 2 (1995), #A1. (1995) MR1309122DOI10.37236/1223
41. Graham, R. L., Lubachevsky, B. D., Nurmela, K. J., Östergård, P. R. J., 10.1016/S0012-365X(97)00050-2, . Discrete Math. 181 (1998), 139–154. (1998) MR1600759DOI10.1016/S0012-365X(97)00050-2
42. Groemer, H., Covering and packing properties of bounded sequences of convex sets, . Mathematica 29 (1982), 18–31. (1982) MR0673502
43. Guy, R. K., Problems, . In: Kelly, L. M. (ed.): The geometry of metric and linear spaces. Proceedings of a conference held at Michigan State University, East Lansing, June 17–19, 1974, Springer-Verlag, 1975, 233–244. (1975) MR0388240
44. Hadwiger, H., 10.1007/BF01899995, . Arch. Math. 8 (1957), 212–213. (1957) MR0091490DOI10.1007/BF01899995
45. Hales, T. C., 10.1016/0377-0427(92)90052-Y, . J. Comput. Appl. Math. 44 (1992), 41–76. (1992) MR1199253DOI10.1016/0377-0427(92)90052-Y
46. Hales, T. C., 10.1007/BF03024356, . Math. Intelligencer 16 (1994), 47–58. (1994) MR1281754DOI10.1007/BF03024356
47. Hales, T. C., 10.1007/BF02770863, . Discrete Comput. Geom. 17 (1997), 1–51. (1997) MR1418278DOI10.1007/BF02770863
48. Hales, T. C., 10.1007/PL00009312, . Discrete Comput. Geom. 18 (1997), 135–149. (1997) MR1455511DOI10.1007/PL00009312
49. Hales, T. C., Cannonballs and honeycombs, . Notices Amer. Math. Soc. 47 (2000), 440–449. (2000) MR1745624
50. Hales, T. C., Ferguson, S. P., The Kepler conjecture, . Discrete Comput. Geom. 36 (2006), 1–269. (2006) MR3075372
51. Hortobágyi, I., Über die Scheibenklassen bezügliche Newtonsche Zahl der konvexen Scheiben, . Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 18 (1975), 123–127. (1975) MR0425775
52. Horvát, G. Á., Packing points into a unit cube in higher space, . Stud. Univ. Žilina Math. Ser. 24 (2010), 23–28. (2010) MR2829525
53. Hougardy, S., On packing squares into a rectangle, . Tech. Report 101007. Forschungsinstitut für Diskrete Mathematik, March 2010. (2010) MR2805963
54. Hsiang, W.-Y., 10.1142/S0129167X93000364, . Internat. J. Math. 4 (1993), 739–831. (1993) MR1245351DOI10.1142/S0129167X93000364
55. Hsiang, W.-Y., A rejoinder to T. C. Hales’ article: The status of the Kepler conjecture, . Math. Intelligencer 17 (1994), 35–42. (1994) MR1319992
56. Januszewski, J., 10.1007/s10998-015-0083-2, . Period. Math. Hungar. 72 (2016), 90–101. (2016) MR3470807DOI10.1007/s10998-015-0083-2
57. Jennings, D., 10.1016/0097-3165(94)90116-3, . J. Combin. Theory, Ser. A 68 (1994), 465–469. (1994) MR1297183DOI10.1016/0097-3165(94)90116-3
58. Jennings, D., 10.1016/0012-365X(94)00211-Z, . Discrete Math. 138 (1995), 293–300. (1995) MR1322104DOI10.1016/0012-365X(94)00211-Z
59. Joós, A., Pontok elhelyezése egységkockában, . PhD tézisek, 2008. (2008) 
60. Joós, A., On the volume of points at distance at least 1 in the 5-dimensional unit cube, . Acta Sci. Math. 76 (2010), 217–231. (2010) MR2668418
61. Joós, A., Bálint, V., 10.1007/s00022-018-0464-9, . J. Geom. 110 (2019), article no. 10. (2019) MR3895357DOI10.1007/s00022-018-0464-9
62. Kabatjanskij, G. A., Levenshtein, V. I., Bounds for packings on a sphere and space, . Problemy Peredachi Informatsii 14 (1978), 3–24. (1978) MR0514023
63. Kepler, J., Strena seu de nive sexangula, . Tampach, Frankfurt, 1611. English translation: The six-cornered snowflake. Oxford, 1966. (1611) 
64. Kleitman, D. J., Krieger, M. M., 10.1111/j.1749-6632.1970.tb56476.x, . Ann. New York Acad. Sci. 175 (1970), 253–262. (1970) MR0264519DOI10.1111/j.1749-6632.1970.tb56476.x
65. Kleitman, D. J., Krieger, M. M., An optimal bound for two dimensional bin packing, . Proceedings of the 16th Annual Symposium on Foundations of Computer Science, IEEE Computer Society, 1975, 163–168. (1975) MR0423195
66. Kosiński, A., 10.4064/cm-4-2-216-218, . Colloq. Math. 4 (1957), 216–218. (1957) MR0086324DOI10.4064/cm-4-2-216-218
67. Leech, J., 10.2307/3610264, . Math. Gaz. 40 (1956), 22–23. (1956) MR0076369DOI10.2307/3610264
68. Leech, J., 10.4153/CJM-1964-065-1, . Canad. J. Math. 16 (1964), 657–682. (1964) MR0167901DOI10.4153/CJM-1964-065-1
69. Levenshtein, V. I., On bounds for packings in $n$-dimensional Euclidean space, . Soviet Math. Dokl. 20 (1979), 417–421. (1979) MR0529659
70. Lubachevsky, B. D., Graham, R. L., Stikkinger, F. H., 10.1023/A:1004284826421, . Period. Math. Hungar. 34 (1997), 123–142. (1997) MR1608310DOI10.1023/A:1004284826421
71. Malfatti, G., Memoria sopra un problema sterotomico, . Memorie di Matematica e di Fisica della Societa Italiana delle Scienze 10 (1803), 235–244. (1803) 
72. Markót, M. Cs., 10.1023/B:NUMA.0000049472.75023.0a, . Numer. Algorithms 37 (2004), 253–261. (2004) MR2109911DOI10.1023/B:NUMA.0000049472.75023.0a
73. Mauldin, R. D., The Scottish Book, . Birkhäuser, 1981. (1981) MR0666400
74. Meir, A., Moser, L., 10.1016/S0021-9800(68)80047-X, . J. Combin. Theory 5 (1968), 126–134. (1968) MR0229142DOI10.1016/S0021-9800(68)80047-X
75. Melissen, J. B. M., 10.1080/00029890.1993.11990512, . Amer. Math. Monthly 100 (1993), 916–925. (1993) MR1252928DOI10.1080/00029890.1993.11990512
76. Melissen, J. B. M., Densest packing of six equal circles in a square, . Elem. Math. 49 (1994), 27–31. (1994) MR1261756
77. Melissen, J. B. M., 10.1007/BF01263647, . Geom. Dedicata 50 (1994), 15–25. (1994) MR1280791DOI10.1007/BF01263647
78. Melissen, J. B. M., Densest packing of eleven congruent circles in an equilateral triangle, . Acta Math. 65 (1994), 389–393. (1994) MR1281448
79. Melissen, J. B. M., Schuur, P. C., 10.1016/0012-365X(95)90139-C, . Discrete Math. 145 (1995), 333–342. (1995) MR1356610DOI10.1016/0012-365X(95)90139-C
80. Moon, J., Moser, L., 10.4064/cm-17-1-103-110, . Colloq. Math. 17 (1967), 103–110. (1967) Zbl0152.39502MR0215197DOI10.4064/cm-17-1-103-110
81. Moser, L., Poorly formulated unsolved problems of combinatorial geometry, . 1963. (1963) 
82. Moser, W. O. J., 10.1016/0166-218X(91)90071-4, . Discrete Appl. Math. 31 (1991), 201–225. (1991) MR1106701DOI10.1016/0166-218X(91)90071-4
83. Moser, W. O. J., Pach, J., Research problems in discrete geometry, . McGill University, Montreal, 1986, 1993. (1986) MR1106701
84. Musin, O. R., 10.1070/RM2003v058n04ABEH000651, . Russian Math. Surveys 58 (2003), 794–795. (2003) MR2042912DOI10.1070/RM2003v058n04ABEH000651
85. Musin, O. R., The kissing volume in four dimensions, . Ann. of Math. (2) 168 (2008), 1–32. (2008) MR2415397
86. Novotný, P., A note on packing of squares, . Studies Univ. Žilina Mat.-Phys. Ser. A 10 (1995), 35–39. (1995) MR1437834
87. Novotný, P., On packing of squares into a rectangle, . Arch. Math. (Brno) 32 (1996), 75–83. (1996) MR1407340
88. Novotný, P., On packing of four and five squares into a rectangle, . Note Mat. 19 (1999), 199–206. (1999) MR1816873
89. Novotný, P., Využitie počítača pri riešení ukladacieho problému, . Proceedings of Symposium on Computational Geometry, STU, Bratislava, 2002, 60–62. (2002) 
90. Novotný, P., Pakovanie troch kociek, . Proceedings of Symposium on Computer Geometry, STU, Bratislava, 2006, 117–119. (2006) 
91. Novotný, P., Najhoršie pakovateľné štyri kocky, . Proceedings of Symposium on Computer Geometry, STU, Bratislava, 2007, 78–81. (2007) 
92. Novotný, P., Ukladanie kociek do kvádra, . Proceedings of Symposium on Computer Geometry, STU, Bratislava, 2011, 100–103. (2011) 
93. Nurmela, K. J., Östergård, P. R. J., 10.1007/PL00009472, . Discrete Comput. Geom. 22 (1999), 439–457. (1999) MR1706578DOI10.1007/PL00009472
94. Odlyzko, A. M., Sloane, N. J. A., 10.1016/0097-3165(79)90074-8, . J. Combin. Theory Ser. A 26 (1979), 210–214. (1979) MR0530296DOI10.1016/0097-3165(79)90074-8
95. Oler, N., 10.4153/CMB-1961-018-7, . Canad. Math. Bull. 4 (1961), 153–155. (1961) MR0133065DOI10.4153/CMB-1961-018-7
96. Paulhus, M., 10.1006/jcta.1997.2836, . J. Combin. Theory Ser. A 82 (1998), 147–157. (1998) MR1620857DOI10.1006/jcta.1997.2836
97. Payan, Ch., 10.1016/S0012-365X(96)00201-4, . Discrete Math. 165–166 (1997), 555–565. (1997) MR1439300DOI10.1016/S0012-365X(96)00201-4
98. Peikert, R., Würtz, D., Monagan, M., de Groot, C., Packing circles in a square: A review and new results, . In: Kall, P. (ed.): System modelling and optimization. Proceedings of the 15th IFIP conference, Zurich, Switzerland, September 2–6, 1991, Springer-Verlag, Berlin, 1992, 45–54. (1992) MR1182322
99. Pirl, U., 10.1002/mana.19690400110, . Math. Nachr. 40 (1969), 111–124. (1969) MR0253164DOI10.1002/mana.19690400110
100. Rogers, C. A., 10.1112/plms/s3-8.4.609, . Proc. Lond. Math. Soc. (3) 8 (1958), 609–620. (1958) MR0102052DOI10.1112/plms/s3-8.4.609
101. Sedliačková, Z., Packing three cubes in 8-dimensional space, . J. Geom. Graph. 22 (2018), No. 2, 217–223. (2018) MR3919006
102. Schaer, J., 10.4153/CMB-1965-018-9, . Canad. Math. Bull. 8 (1965), 273–277. (1965) MR0181938DOI10.4153/CMB-1965-018-9
103. Schaer, J., 10.4153/CMB-1966-033-0, . Canad. Math. Bull. 9 (1966), 265–270. (1966) MR0200797DOI10.4153/CMB-1966-033-0
104. Schaer, J., Meir, A., 10.4153/CMB-1965-004-x, . Canad. Math. Bull. 8 (1965), 21–27. (1965) MR0175029DOI10.4153/CMB-1965-004-x
105. Schütte, K., van der Waerden, B. L., 10.1007/BF01343127, . Math. Ann. 125 (1953), 325–334. (1953) MR0053537DOI10.1007/BF01343127
106. Thue, A., On the densest packing of congruent circles in the plane, . Skr. Vidensk.-Selsk. Christiana 1 (1910), 3–9. (1910) MR2994977
107. Vardy, A., 10.1007/BF01884292, . Invent. Math. 121 (1995), 119–133. (1995) MR1345286DOI10.1007/BF01884292
108. Zalgaller, V. A., Los, G. A., 10.1007/BF01249514, . J. Math. Sci. (N.Y.) 72 (1994), 3163–3177. (1994) MR1267528DOI10.1007/BF01249514
109. Zong, C., 10.1112/S0024609397003408, . Bull. Lond. Math. Soc. 30 (1998), 1–10. (1998) MR1479030DOI10.1112/S0024609397003408