Kissing numbers

How many equal, non-overlapping spheres can touch a central sphere? The answer depends on dimension. A numerical configuration supplies a lower bound only when its geometric constraints are justified.

The kissing number K(d) is the largest number of equal spheres that can touch one equal central sphere in d-dimensional Euclidean space without overlaps. Equivalently, seek many directions on the unit sphere whose pairwise angles are at least 60 degrees.

A feasible set of directions gives a lower bound. A proof that no larger set is possible gives an upper bound. Numerical coordinates near the feasibility threshold need careful treatment: rounding can turn a nearly valid configuration into an invalid one. Exact coordinates or certified error bounds help make a construction checkable.