Quad Cluster Verifications

This document correlates the numbering of the paper Sphere Packings III with the numbering used int the verifications. The comments of this file contain various observations about the verifications.

Each calculation has four cases: Split, Asymmetric, Octahedron, Truncated. The flat and asymmetric cases are both made of two flat quarters. In the split case, the diagonal of the flat quarters has an endpoint at the corner at which the dihedral angle is computed. The inequalities that the two quarters are to satisfy are identical in this case. In the asymmetric case the diagonal runs in the other direction, so that the corner with the dihedral angle lies on one quarter but not on the other. In this case, the inequalities are different for the two cases.

In the mixed cases it is enough to verify that the inequality holds when scored by vor(S,1.255). Since this is an upper bound to vor(S,sqrt(2)), when this bound passes it completes the verifications. When it fails, there is a fifth case: If 1.255-truncation doesn't go through we need sqrt(2)-truncation as well. In the asymmetric verifications, we have divided the verification into several cases depending on the scoring type of the front and back quarters. (Call the front quarter the one with the dihedral angle term.)

A few html comments have been spliced into the computer output from the various verifications to make them easier to read.

In the truncated verifications, we drop case R0 when by symmetry it covers the same ground as R7.






The octahedral cases of Proposition 4.2 were verified by the partitioning method described in Sphere Packings III, Appendix 2. (A fourth inequality is partially verified in these files. It is not relevant and should be ignored.) Here is a summary of these calculations.