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Some numerical problems in discrete geometry. (English) Zbl 0946.52009
Summary: What is the smallest square which contains ten pairwise disjoint congruent open disks of unit diameter? It is conjectured that the minimum side is 3.373... .
This paper proves that the side is at least 3.334.
##### MSC:
 52C15 Packing and covering in $$2$$ dimensions (aspects of discrete geometry)
##### Keywords:
discrete geometry; packing of disks; lower bounds
Full Text:
##### References:
 [1] Moser, L., Problem 24 (corrected), Canad. math. bull., 3, 78, (1960) [2] Croft, H.T.; Falconer, K.J.; Guy, R.K., unsolved problems in geometry, chapter D1: packing circles or spreading points in a square, (), 108-110 [3] Goldberg, M., The packing of equal circles in a square, Math. mag., 43, 24-30, (1970) · Zbl 0188.55004 [4] Guy, R.K.; Selfridge, J.L., Optimal coverings of the square, (), 745-799 · Zbl 0311.05026 [5] Melissen, J.B.M., Dentest packings of eleven congruent circles in a circle, Geometriae dedicata, 13, 1-11, (1994) · Zbl 0807.52014 [6] Mollard, M.; Payan, C., Some progress in the packing of equal circles in a square, Discrete math., 84, 303-307, (1990) · Zbl 0715.52005 [7] de Groot, C.; Monagan, M.; Peikert, R.; Würtz, D., Packing circles in a square: A review and new results, Lect. notes contr. inf. sci., 180, 45-54, (1992) · Zbl 0789.52002 [8] C. de Groot, R. Peikert and D. Würtz, The optimal packing of ten equal circles in a square, IPS Research Report, ETH Zr̈ich. [9] Hujter, H., Combinatorial optimization problems related to geometrical packings and coverings, chapter 1:graph theoretical questions related to packings and coverings by congruent disks, () [10] R. Milano,m Configurations optimales des disques dans un polygone régulier, Mémoire de Licence, Univ. Libre de Bruxelles. [11] Peikert, R., Dichteste packungen von gleichen kreisen in einem quadrat, El. math., 49, 16-26, (1994) · Zbl 0814.52007 [12] Schaer, J., On the packing of ten equal circles in a square, Math. mag., 44, 139-140, (1971) · Zbl 0215.50603 [13] Valette, G., A better packing of ten equal circles in a square, Discrete math., 76, 57-59, (1989) · Zbl 0672.52007
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