Raw transcript — Exploration & Epiphany | Guest video by Paul Dancstep

Source: https://www.youtube.com/watch?v=_BrFKp-U8GI Duration: 52m 11s Captured: 2026-04-20

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Imagine yourself in an art museum. As you stroll through the galleries, you come across a large low table covered in small white skeletal frames. Each looks like a random fragment of a cube, and they’re all laid out in an orderly grid. What we’re looking at is a digital reproduction of an artwork called Variations of Incomplete Open Cubes by Sol LeWitt. It represents every possible way that a cube can be incomplete, subject to a few conditions. In this video, you and I will investigate this sculpture and explore some of the mathematical ideas behind it. Although I’ve tried to make these explorations accessible and playful, in the end we’ll stumble into a highly non trivial result. A lovely classical technique, an ingenious trick for counting cubes by thinking about symmetry.

[00:01:02] By open cube, we mean a hollow cube framed by its 12 edges. We can make an open cube incomplete by selecting one of these edges and getting rid of it. The first of LeWitt’s constraints is that the existing edges must all be connected. The second constraint says that the existing edges have to form an obvious cube. To understand the final constraint, the cube must be three-dimensional and must factor out rotational equivalents.

[00:02:03] If we played around with this shape long enough, we would eventually discover that there are 24 different ways we can reposition it, each of which looks unique from the others, but all fundamentally the same shape. The final work of art shows the full collection of 122 rotationally unique incomplete open cubes. The entire definition can be put into explicit mathematical terms. In the business, you might rephrase this as an enumeration of all proper subsets of a cube that are connected and span R3 modulo rotations.

[00:03:01] So we have something interesting. An artist who effectively solved a math problem as part of his creative process. Sol LeWitt was a modern artist associated with conceptual art, minimalism, and serial art. The core idea of conceptual art can be seen in LeWitt’s series of wall drawings — he would provide a set of instructions, trained installers would follow them. Like a musician writing a score to be performed by others, LeWitt’s instructions emphasize the conceptual nature of the resulting work. It’s the idea of the piece that is the true product of the artist.

[00:04:01] Perhaps his most well known quote: “The idea becomes a machine that makes art.” Of the cube, LeWitt wrote: “The most interesting characteristic of the cube is that it is relatively uninteresting. Therefore, it is the best form to use for any more elaborate function, the grammatical device from which the work may proceed.” Serial art was a reaction to abstract expressionism. LeWitt sought instead a technique in which “all of the planning and decisions are made beforehand, and the execution is a perfunctory affair.” Many of LeWitt’s serial structures begin with the question, how many ways?

[00:05:00] In 1973, LeWitt asked: How many ways can a cube be incomplete? Sol LeWitt left behind about 50 notebook pages, which show sketches and working drawings that capture the development of the incomplete open cubes. By reviewing their contents, we effectively get to look over Sol LeWitt’s shoulder as he solves the problem. One of the things these notebooks make clear is that of the three constraints, rotational equivalence was by far the most difficult to figure out.

[00:06:02] As the story goes, he would make small models of the cubes out of paperclips or pipe cleaners. By comparing the models, he would occasionally discover that two cubes he had thought were distinct turn out to be the same shape. The red marks indicate times when he found duplicates and had to cross one off. As LeWitt explained: “I was trying to figure out a way to do it through numbers and letters logically. But in the end, it all had to be done empirically.”

[00:07:01] In this video, we’re going to solve the specific problem that eluded Sol LeWitt by showing a thought process that leads to the solution. We’ll relax constraints on connectedness and three-dimensionality for now and bring them back at the end. The total number of incomplete open cubes is 2^12 = 4,096 because each of the 12 edges has two choices.

[00:09:01] For each edge presents two choices, on or off. From this sequence of choices we get a tree of decisions which generates all possible squares. The number of outcomes at the bottom must equal 2 raised to the height of the decision tree. A cube has 12 edges that can be set to on or off. So we have 12 binary choices to make. 2 to the 12th gives us 4096 total incomplete open cubes.

[00:11:02] Working through the 2D case (incomplete open squares): six total families of rotationally equivalent incomplete open squares. We got this result using brute force search. What we’d really like is a formula, something we can also apply to the cube.

[00:13:01] Brute force on cubes is intractable. From early on, we see LeWitt restricting his attention to cubes with a given number of edges. Different pages dedicated to different sized cubes show how he broke the overall problem into several smaller problems. This divide and conquer approach is reflected in the final layout of the art piece.

[00:15:02] If every family were the same size, then it would be easy to calculate the total number of families. But if every family seems to be doing its own thing in terms of size, then we apparently have no choice but to go through and manually count them up like this, one at a time.

[00:16:01] A hunch develops: more symmetrical shapes seem to belong to smaller families. For LeWitt, the best plan was divide and conquer. For us, the plan is to delay the main question and first focus on a simpler question — how to calculate a given cube’s family size — pursuing the hunch that this has something to do with symmetry.

[00:17:01] LeWitt switched from corner labels to numbered edges. Notation enables thinking. With edge numbers, complementary pairs become natural — a 4-part cube has an 8-part complement. Finding the right labeling system can be an invaluable step in making sense of a problem.

[00:21:01] We can spin the cube around the diagonal axes (4 corner axes × 2 rotations each = 8). With six cubes from the edge axes, eight from the corner axes and nine from the face axes, we end up with 23 distinct orientations. Plus the original cube, 24 distinct orientations.

[00:23:00] Family portraits — for any cube, applying all 24 transforms generates a portrait. For a cube with only 6 family members (e.g., the table), the portrait contains exactly 4 copies of each family member. The relationship: family size × number of lookalikes = 24.

[00:25:00] A lookalike is a cube + transformation pair where the cube looks the same after applying the transformation. The size of a given family times the number of lookalikes in its family portrait always equals 24.

[00:27:01] To say a shape is symmetrical is really to say that it looks the same from various points of view. The number of lookalikes is basically a numerical measure of symmetry. Our formula captures how more symmetry means smaller families.

[00:30:03] Different portraits of the same family are just scrambled versions of each other. Pile all 24 portraits into a “family album.” Each family album contains exactly 24 lookalikes regardless of family size — because family size × lookalikes per column = 24, and total columns × lookalikes per column = 24 either way.

[00:35:00] If we counted up all the lookalikes across all families and divided by 24, this would give us the number of families. In the 2D case, working through all six families: each contributes 4 lookalikes (since 2D has 4 transformations), totaling 24 lookalikes. Six families × 4 = 24. Confirmed.

[00:36:00] But this technique is impractical for cubes — counting lookalikes is no easier than counting family sizes. All we’ve shown is that the number of families is related to lookalikes which are also hard to count. Disappointment is part of doing business.

[00:37:00] Epiphany: I don’t have to count lookalikes by going through each shape. I can count them by going through each transformation. A lookalike is a (cube, transformation) pair such that the cube is unchanged. We can ask: for each transformation, how many cubes are unchanged? This is a fundamentally different question that can be answered with a formula.

[00:39:04] For 90° rotation: top edge being on forces the right edge to be on, which forces the bottom edge, which forces the left. Whatever choice we make is contagious across all sides. So lookalikes for a 90° rotation are determined by a single overall choice. 2^1 = 2 lookalikes.

[00:40:00] For 180° rotation: splits the square into two independent subsets — horizontal pair and vertical pair. Two choices = 2^2 = 4 lookalikes.

[00:41:01] For 3D, 90° rotation around vertical axis: top 4 edges exchanged (must all be the same), 4 vertical bars (must all be the same), 4 bottom edges (must all be the same). 3 independent choices = 2^3 = 8 lookalikes.

[00:42:00] 180° rotation around vertical axis: top square splits into two pairs, vertical bars split into two pairs, bottom square splits into two pairs. 6 choices = 2^6 = 64 lookalikes.

[00:43:02] Corner-axis 120° rotation: 4 independent choices = 2^4 = 16 lookalikes per corner-axis rotation. Edge-axis 180° rotation: 7 independent choices = 2^7 = 128 lookalikes per edge-axis rotation. Identity transformation contributes 4096 lookalikes.

[00:43:02 final tally] Total lookalikes: 5,232. Divided by 24: 218. There are 218 rotationally unique families of incomplete open cubes (without LeWitt’s other constraints).

[00:44:01] LeWitt’s epiphany came from holding a cube up to a mirror. Some incomplete open cubes are NOT rotationally equivalent to their mirror images. He had to add the rotationally distinct mirror images he’d missed. His tables undergo a dramatic growth spurt.

[00:45:01] Sometime in early 1974, LeWitt’s enumeration had reached its final form. He produced 40 inch painted aluminum sculptures of each variation, isometric drawings, and an art book. His most abstract representation compresses the isometric view down to a flat hexagonal form — the “raw data” of the incomplete open cubes.

[00:48:00] This is an example of a much more general counting technique from group theory known as Burnside’s Lemma. Same logic applies to incomplete open tetrahedra (12 families), to colored cube edges with 3 colors (22,815 families), to Rubik’s Cube positional possibilities. Burnside’s Lemma is a general purpose counting technique you can use anytime you find yourself decorating a symmetrical object and wanting to know exactly how many options you have.

[00:50:00] A 2014 paper from Kansas State University confirms LeWitt’s count of 122 is correct under all constraints. But there’s an error in the final sculpture: the cubes labeled 104 and 105 are actually the same cube, a duplicate. Meanwhile, this cube [shown] is nowhere to be found. There’s something elegant about this little grace note of imperfection right at the end.

[00:51:00] LeWitt’s biographer wrote that LeWitt’s primary impact on contemporary art was the idea that the product of the mind is more significant than the product of the hand. The mathematical point of view contributes to the space of ideas around this piece. In LeWitt’s own words: “All intervening steps, scribbles, sketches, drawings, failed works, models, studies, thoughts, conversations are of interest. Those that show the thought process of the artist are sometimes more interesting than the final product.”