1 3 M ay 1 99 4 Average kissing numbers for non - congruent sphere packings
نویسنده
چکیده
(The appearance of the number of the beast in the lower bound is purely coincidental.) The supremal average kissing number k is defined in any dimension, as are kc, the supremal average kissing number for congruent ball packing, and ks, the maximal kissing number for a single ball surrounded by congruent balls with disjoint interiors. (Clearly, kc ≤ k and kc ≤ ks.) It is interesting that k is always finite, because a large ball can be surrounded by many small balls in a non-congruent ball packing. Nevertheless, a simple argument presented below shows that k ≤ 2ks in every dimension, and clearly ks is always finite. In two dimensions, an Euler characteristic argument shows that k ≤ 6, but it is also well-known that ks = kc = 6. One might therefore conjecture that k = kc always, or at least in dimensions such as 2, 3, 8, and 24 (and conjecturally several others) in which ks = kc [1]. Surprisingly, in three dimensions, k > 12 even though ks = kc = 12. Supported by an NSF Postdoctoral Fellowship, grant #DMS-9107908 Incumbent of the William Z. and Eda Bess Novick Career Development Chair. Supported by NSF grant #DMS-9112150
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