Sol LeWitt's “Variations of Incomplete Open Cubes” (1974) is a systematic artwork: all 122 distinct ways to select edges of a cube such that the result is connected and non-planar. The cube has 12 edges. Choosing subsets that form a connected skeleton — not flattening to a plane — yields exactly 122 possibilities. The enumeration is exhaustive. The art is in the completeness.
Vejdemo-Johansson (arXiv:2602.20425) extends LeWitt's enumeration to all five Platonic solids. The tetrahedron, with 6 edges, yields only 6 incomplete open forms. The octahedron (12 edges): 185. The dodecahedron (30 edges): 2,423,206. The icosahedron (30 edges, same count but different topology): 16,096,166.
The explosion is combinatorial but the numbers are not arbitrary. The dodecahedron and icosahedron have the same number of edges but the icosahedron yields seven times more incomplete forms. The difference is topological: the icosahedron's higher vertex degree (5 edges per vertex versus 3 for the dodecahedron) creates more possibilities for connected non-planar subsets. Connectivity is easier when each node has more links, so fewer subsets are disconnected and more survive the filter.
The constraint pair — connected AND non-planar — is what makes the enumeration interesting. Either constraint alone is tractable. Connected subgraphs of a graph are classically enumerated. Non-planar subgraphs are harder but structured. The intersection of two structured conditions produces the rich middle where the count grows super-exponentially with edge count but remains exactly computable.
The general observation: systematic enumeration under intersecting constraints can produce counts that grow faster than either constraint alone would predict. The middle ground between two filters is often richer than the unfiltered space might suggest — the constraints don't just reduce; they shape.