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An more extensive and up-to-date tabulation is maintained by Sloane and Nebe. In nonlattice packings, the kissing number may vary from sphere to sphere, so the largest value is given below (Conway and Sloane 1993, p. The following table gives the largest known kissing numbers in DIMENSION D for lattice (L) and nonlattice (NL) packings (if a nonlattice packing with higher number exists). Odlyzko and Sloane (1979) found the exact value for 24-D. Exact values for lattice pacbings are known for n = 1 to 9 and n = 24 (Conway and Sloane 1992, Sloane and Nebe). More concise proofs were published by Schiitte and van der Waerden (1953) and Leech (1956).
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Newton correctly believed that the kissing number in 3-D was 12, but the first proofs were not produced until the 19th century (Conway and Sloane 1993, p. The number of equivalent HYPERSPHERES in n-D which can touch an equivalent HYPERSPHERE without any intersections, also sometimes called the NEWTON NUMBER, CONTACT NUMBER, COORDINATION NUMBI~R,~~ LIGANCY. “Solution of Kirkman’s Schoolgirl Problem.” Combinatorics, Pruc. 1Mafhemahca2 Recreations and Essays, 13th ed. “Kirkman Triple Systems.” $1.6.3 in The CRC Handbook of Combinatorial Designs (Ed. For n = 1, there is a single unique (up to an isomorphism) solution, while there are 7 different systems for n = 2 (Mulder 1917, Cole 1922, Ball and Coxeter 1987). Earlier editions of Ball and Coxeter (1987) gave constructions of Kirkman triple systems with 9 5 w 5 99. Ray-Chaudhuri and Wilson (1971) showed that there exists at least one Kirkman triple system for every NONNEGATIVE order n. Solution to KIRKMAN'S SCHOOLGIRL PROBLEM requires construction of a Kirkman triple system of order n = 2. Kiss Surface STEINER TRIPLE SYSTEMS of order 3 and 9 are Kirkman triple systems with 12 = 0 and 1.
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