Why can a combinatorial geometry problem in three dimensions differ from a similar problem in two dimensions?

In combinatorial geometry, problems in three dimensions often differ from their two-dimensional counterparts because the geometric constraints and available space change significantly as dimensions increase. While some solutions can be readily extended to higher dimensions, others are entirely unique to their specific spatial context, requiring individual analysis for each case.1

A classic example of this divergence is the "kissing number" problem, which involves determining how many non-overlapping spheres of equal size can touch a central sphere of the same size. In two dimensions, it is easily shown that six circular disks can surround a central disk. In three dimensions, however, the problem becomes more complex: it is possible to place 12 balls around a central one, but the arrangement allows for significant free space, leading to a long-standing historical debate over whether a 13th ball could fit. This illustrates how the added degree of freedom in three-dimensional space alters the fundamental nature of packing and covering problems compared to the plane.21

Because each problem may involve distinct relations of symmetry, tangency, or extrema, researchers must often apply specialized reasoning rather than assuming a simple extension from two dimensions.1