Four lines in three-dimensional space. Their Voronoi diagram — the partition of space into regions closest to each line — has a complete topological classification: exactly fifteen distinct topological types, with a bijection between nearest-site and farthest-site diagrams.
Papadopoulou and Wang enumerate every possible topology by analyzing how the bisector surfaces between pairs of lines intersect. Each bisector is a quadric surface; the intersections between three bisectors produce Voronoi edges; four-fold intersections produce vertices. The vertex count is always even, ranging from zero to eight, and this constraint severely limits the possible configurations.
The classification reveals a structural feature they call “twists” — local topological modifications that can be inserted into a diagram without disrupting its global structure. A twist changes the connectivity of edges near a vertex without changing the number of vertices or the overall topology of the diagram. Full twists preserve the topological type but change the geometric realization; removing the possibility of twists reduces the classification to a clean finite enumeration.
Four lines is the smallest case where Voronoi diagrams in three dimensions exhibit genuinely three-dimensional behavior — three lines produce diagrams that are essentially two-dimensional in structure. And four lines is complex enough that the classification is non-trivial: fifteen types is a large enough number to require systematic enumeration but small enough to be complete.
The complete classification of a simple geometric object — four lines, Euclidean distance — takes a full paper. The geometry of proximity, even in its simplest three-dimensional cases, is richer than intuition suggests.