Conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles

Experimental Mathematics

Volumn 9, Issue 2, 2000, Pages 241-250

  Nurmela, Kari J ^a  

^a AALTO UNIVERSITY   (Finland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034348622     PISSN: 10586458     EISSN: 1944950X     Source Type: Journal    
DOI: 10.1080/10586458.2000.10504649     Document Type: Article

References (16)

1. A. Ben-Israel, “A Newton-Raphson method for the solution of systems of equations”, J. Math. Anal. Appl. 15 (1966), 243-252.
2. H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved problems in geometry, Springer, New York, 1991.
3. D. M. Gay, “Algorithm 611: Subroutines for unconstrained minimization using a model/trust- region approach”, ACM Trans. Math. Software 9: 4 (1983), 503-524.
4. R. L. Graham, B. D. Lubachevsky, K. J. Nurmela, and P. R. J. Ostergard, “Dense packings of congruent circles in a circle”, Discrete Math. 181: 1-3 (1998), 139-154.
5. R. H. Hardin, N. J. A. Sloane, and W. D. Smith, “Spherical codes”. In preparation.
6. A. Heppes and H. Melissen, “Covering a rectangle with equal circles”, Period. Math. Hungar. 34: 1-2 (1997), 65-81.
7. A. Lengyel and I. A. Veres, “Egysegnegyzet lefedese egybevago korokkel (Covering the unit square with congruent circles)”, competition essay, Technical University Budapest, 1996.
8. H. Melissen, “Densest packings of congruent circles in an equilateral triangle”, Amer. Math. Monthly 100: 10 (1993), 916-925.
9. H. Melissen, “Loosest circle coverings of an equilateral triangle”, Math. Mag. 70 (1997), 119-125.
10. H. Melissen, Packing and covering with circles, Ph.D. thesis, Universiteit Utrecht, the Netherlands, 1997.
11. J. B. M. Melissen and P. C. Schuur, “Improved coverings of a square with six and eight equal circles”, Electron. J. Combin. 3: 1 (1996), R32.
12. J. B. M. Melissen and P. C. Schuur, “Covering a rectangle with six and seven circles”, Discrete Appl. Math. 99: 1-3 (2000), 149-156.
13. K. J. Nurmela and P. R. J. Ostergard, “Packing up to 50 equal circles in a square”, Discrete Comput. Geom. 18: 1 (1997), 111-120.
14. K. J. Nurmela and P. R. J. Ostergard, “Covering a square with up to 30 equal circles”, Research Report A62, Laboratory for Theoretical Computer Science, Helsinki University of Technology, 2000. See http://www.tcs.hut.fi/Publications/reports/A62abstract.html.
15. T. Tarnai and Z. Gaspar, "Covering a square by equal circles", Elem. Math. 50: 4 (1995),167-170.
16. C. T. Zahn Jr., "Black box maximization of circular coverage", J. Res. Nat. Bur. Standards Sect. B 66B (1962), 181-216.