By P. K. Jain, Ahmed Khalid

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**Example text**

Ci ; cj / D 2. ci C rO B/ and let cO ij be the intersection of the line segment ci cj with SOi for all j 2 Ti . , ˛ with 0 < ˛ < 2 and cos ˛ D 1rO ). , ˛) and where Sarea. / refers to the spherical area measure on S2 . uij ; ˛/ < 0:89332 : In order to estimate svol2 bd n [ i D1 !! 2) 22 2 Proofs on Unit Sphere Packings from above let us assume that m members of P have 12 touching neighbours in P and k members of P have at most 9 touching neighbours in P. Thus, n m k members of P have either 10 or 11 touching neighbours in P.

O1 ; q/ D rO , where dist. ; / denotes the Euclidean distance between the corresponding two points. Clearly, cos ˛ D 1rO with ˛ < 3 (Fig. 1). 2. Let T be the convex hull of the points t2 ; t3 ; : : : ; t13 . Then the radius of the circumscribed circle of each face of the convex polyhedron T is less than sin ˛. f2; 3; : : : ; 13g Proof. Let F be an arbitrary face of T with vertices tj ; j 2 IF and let cF denote the center of the circumscribed circle of F . Clearly, the triangle 4o1 cF tj is a right triangle with a right angle at cF and with an acute angle of measure ˇF at o1 for all j 2 IF .

Thus, n m k members of P have either 10 or 11 touching neighbours in P. ) Without loss of generality we may assume that 4 Ä k Ä n m. 1 2 / and Á p 3 2 /Or . cj C rO B/. 2) imply the following estimate. 8. ci C rO B/ < 24:53902 3 Proof. n m !! ci C rO B/ i D1 ! n 3 ! 6. 8)). i i / is based p on the new parameter value rN WD 2 (replacing rO D 1:81383). The details are as follows. First, recall that if f cc denotes the face-centered cubic lattice with shortest nonzero lattice vector of length 2 in E3 and we place unit balls centered at each lattice point of f cc , then we get the fcc lattice packing of unit balls, labelled by Pf cc , in which each unit ball is touched by 12 others such that their centers form the vertices of a cuboctahedron.

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