A DENSITY RESULT FOR RANDOM PACKINGS OF UNIT DISKS

作者：Richter Christian^*

来源：SIAM Journal on Discrete Mathematics, 2014, 28(1): 122-126.

DOI：10.1137/13091110X

• random packing of disks
• intensity
• dual tessellation
• density
• periodic arrangement

This paper answers an open question posed in 1993 concerning the least dense arrangement of a %26apos;full%26apos; packing of equal-sized disks. A disk D in the packing is full if it touches at least four other disks and these disks leave no gap large enough for any other disk to touch D. A packing is called full if every disk is full. In the context of disks with unit diameter, Cowan [J. Appl. Probab., 30 (1993), pp. 263-268] showed that the intensity of disk centers (the average number per unit area) for any full packing must be greater than 12/(4 + root 3 + 5tan(3 pi/10)) = 0.95132 ... and questioned if intensities less than 1 were possible. In this paper, we answer his question affirmatively by showing that, for every intensity from [0.95257 ..., 1], there exists a full packing of unit-diameter disks. We also establish that a full packing exists for all higher intensities up to the obvious maximum of 2/root 3. Specific packings are constructed to demonstrate these existence issues.

• 出版日期2014