Incomplete Open Cubes

Examples of Sol LeWitt's Incomplete Open Cubes on display at City Hall Park in New York City.

A cube has six faces, eight vertices, and twelve edges. In his series titled Incomplete Open Cubes, conceptual and minimalist artist Sol LeWitt (1928-2007) chose to work with cubes represented as frameworks.

LeWitt started by removing one edge from an open cube, then two edges, and so on, as an exploration of how many variations of an incomplete open cube exist and what they look like. One key constraint was that the remaining edges had to be joined.

Removing one edge results in just one possible configuration for an incomplete open cube.

By experiment, LeWitt identified 122 unique variations of open cubes with three edges (the minimum number needed to suggest three dimensions) to eleven edges. Nine of these frameworks, rendered in painted aluminum, were recently on display in the exhibition "Sol LeWitt: Structures, 1965-2006" at City Hall Park in New York City.

Three connected edges is the minimum number needed to suggest three dimensions.

In 2000, the San Francisco Museum of Modern Art exhibited all 122 incomplete open cubes. A representation of these variations is shown here.

One interesting mathematical question is whether LeWitt found all the possible variations of incomplete open cubes that met his criteria? Did he miss any? How would you find all the possibilities and prove that no others exist?

Scott Kim wondered the same thing in a "Bogglers" column in the April 2003 issue of Discover. As an "easy" question, he asked, "Can you make six different shapes that each contain four edges? The edges in each shape must all connect to form a single figure. Mirror images of the same shape are considered different, but rotations are not."

One example of a configuration made from four connected edges.

More difficult: How many distinct shapes can you make using five connected edges of a cube? The answer is 14. Can you find them? Is there a formula or some systematic way to determine the number of possibilities, from three to eleven edges.

It turns out that LeWitt did not include examples in which three or four edges are all lie in the same plane. Are other configurations missing?

For another mathematical puzzle concerning a LeWitt artwork, see "Puzzling Lines." For articles about other LeWitt artworks, see "Thirteen Geometric Shapes" and "LeWitt’s Pyramid."

Photos by I. Peterson