Exploration & Epiphany

Key Concepts:

• Incomplete Open Cubes: Hollow cubes with some edges removed.
• Rotational Equivalence: Two cubes are rotationally equivalent if one can be rotated to look identical to the other.
• Constraints: Rules that define which incomplete open cubes are considered valid (connectedness, 3D, rotational uniqueness).
• Conceptual Art: Art where the idea behind the work is more important than the physical object itself.
• Serial Art: Art created by systematically applying a set of rules or procedures.
• Lookalikes: Transformations of a cube that leave it visually unchanged.
• Family Portrait: A collection of all 24 possible orientations of a given incomplete open cube.
• Family Album: A collection of all family portraits for a given set of cubes.
• Burnside's Lemma: A theorem from group theory used to count the number of distinct objects under a group of symmetries.

1. Introduction to Variations of Incomplete Open Cubes

• The video explores Sol LeWitt's artwork "Variations of Incomplete Open Cubes," which is a digital reproduction of all possible incomplete open cubes under certain constraints.
• The goal is to investigate the sculpture and explore the mathematical ideas behind it, including visualizations, patterns, and symmetry.
• The exploration leads to a non-trivial result using a classical technique for counting cubes.

2. Defining Incomplete Open Cubes

• A cube is a regular six-sided shape. An open cube is a hollow cube framed by its 12 edges.
• An incomplete open cube is created by removing one or more edges from an open cube.
• Constraints:
  • Connectedness: The remaining edges must be connected.
  • Three-Dimensionality: The remaining edges must form an obvious cube with at least one edge along the height, width, and depth.
  • Rotational Uniqueness: Only one representative from each rotationally equivalent family is included.
• The final artwork contains 122 rotationally unique incomplete open cubes.
• Mathematically, this can be described as an enumeration of all proper subsets of a cube that are connected and span R3 modulo rotations.

3. Sol LeWitt and Conceptual Art

• Sol LeWitt was a modern artist associated with conceptual art, minimalism, and serial art.
• Conceptual Art: LeWitt's wall drawings, such as wall drawing number 118 (50 points connected by lines), exemplify conceptual art. The instructions for creating the artwork are the primary focus, emphasizing the idea over the execution.
  • Quote: "The idea becomes a machine that makes art."
• Minimalism: LeWitt's work uses minimal ornamentation, often employing flat white surfaces and simple geometric forms like the cube.
  • Quote: "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: LeWitt sought a technique where "all of the planning and decisions are made beforehand, and the execution is a perfunctory affair." His serial structures often begin with the question, "How many ways?"

4. LeWitt's Process and Notebooks

• LeWitt left behind about 50 notebook pages with sketches and working drawings that capture the development of the incomplete open cubes. These are in the archives at the Wadsworth Athenium in Connecticut.
• Rotational equivalence was the most difficult constraint to figure out.
• LeWitt made small models of the cubes out of paperclips or pipe cleaners to compare them and eliminate rotational duplicates.
• Quote: "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. I had to build a model for each one and then rotate it."
• LeWitt consulted with two mathematicians after completing the piece to confirm that he had found the right number.

5. Mathematical Exploration: Simplifying the Problem

• The mathematical exploration focuses on the problem of rotational equivalence, relaxing the other constraints (connectedness, 3D) for now.
• This means considering all possible cubes, including flat or disconnected ones, and even the full cube and the empty cube.
• The number of families found will be larger than LeWitt's final count of 122 until the other conditions are brought back at the end.

6. Two-Dimensional Case: Incomplete Open Squares

• To warm up, the video considers the two-dimensional case: incomplete open squares.
• Each edge of a square presents two choices: on or off.
• The number of possible squares is 2 raised to the number of choices (edges), which is 2^4 = 16.
• Sorting the 16 incomplete open squares into rotationally equivalent families:
  • One-part squares: Family of 4
  • Corner pieces: Family of 4
  • Parallel bars: Family of 2
  • Three-part squares: Family of 4
  • Complete square: Family of 1
  • Empty square: Family of 1
• Total: 6 families of rotationally equivalent incomplete open squares.
• Applying Sol LeWitt's other conditions:
  • Remove the non-incomplete square.
  • Remove the non-two-dimensional squares.
  • Remove the disconnected square.
• Leaves two squares.

7. Brute Force Search and the Need for a Formula

• The video illustrates the difficulty of brute force search for families of cubes, emphasizing the need for a formula.
• A cube has 12 edges that can be set to on or off, resulting in 2^12 = 4096 total incomplete open cubes.
• Comparing cubes for rotational equivalence is tricky and time-consuming.

8. LeWitt's Divide and Conquer Strategy

• LeWitt restricted his attention to cubes with a given number of edges, breaking the overall problem into several smaller problems.
• The final layout of the art piece reflects this approach, with rows dedicated to different-sized cubes.

9. Symmetry and Family Size

• The video proposes focusing on a simpler question: how to calculate a given cube's family size.
• The hunch is that family size has something to do with symmetry. More symmetrical shapes seem to have smaller families.

10. Labeling Systems and Complementary Pairs

• LeWitt devised a numbering system for the cubes and experimented with labeling the parts of the cube (corners and edges).
• He switched to numbered edges, which turned out to be a more efficient and elegant way to annotate the cubes.
• The edge numbering notation makes it easy to think about complementary pairs. Every incomplete open cube is part of a complementary pair.
• By exploiting this duality, LeWitt was able to cut his search effort in half.

11. Labeling Family Members and Transformations

• The video introduces a labeling system for family members based on transformations (rotations) of the cube.
• Rotations around face-centered axes (f1, f2, f3): 9 orientations
• Rotations around corner axes (c1 to c4): 8 orientations
• Rotations around edge axes (e1 to e6): 6 orientations
• Including the original cube (no transformation), there are 24 distinct orientations.
• These labels refer to transformations that can be applied to any given cube.

12. Family Portraits and Lookalikes

• A family portrait is a collection of all 24 possible orientations of a given incomplete open cube.
• If a shape doesn't have 24 family members, its family portrait will contain repeats.
• Lookalikes are transformations of a cube that leave it visually unchanged.
• The size of a given family times the number of lookalikes in its family portrait always equals 24.
• Family Size = 24 / Number of Lookalikes

13. Family Albums and the Number of Lookalikes

• A family album is a collection of all family portraits for a given set of cubes.
• Different portraits of the same family are just scrambled versions of each other.
• Every family album should contain 24 lookalikes.

14. Re-indexing Trick and Burnside's Lemma

• The epiphany is that instead of counting lookalikes by looking at each incomplete open cube and asking how many transformations make it look the same, we can count them by looking at each transformation and asking how many incomplete open cubes are left unchanged by this transformation.
• This is a fundamentally different kind of question that is much easier to answer in general.
• This technique is an example of Burnside's Lemma from group theory.

15. Applying the Re-indexing Trick

• For a 90-degree rotation of a cube, the top four edges, the four vertical bars, and the bottom edges must all be the same (on or off). This results in 2^3 = 8 possible lookalikes.
• For a 180-degree rotation, the cube is split into six independent choices, resulting in 2^6 = 64 lookalikes.
• For the corner axis, there are four distinct choices, giving 2^4 = 16 possible lookalikes.
• For the edge axis, there are seven independent choices, giving 2^7 = 128 distinct shapes.
• The act of doing nothing contributes 4096 lookalikes.
• Total lookalikes: 5,232
• Number of families: 5,232 / 24 = 218

16. Mirror Images and Chiral Cubes

• LeWitt discovered that some mirror image pairs are not rotationally equivalent.
• This realization was an epiphany for LeWitt, leading to a sudden change in his tables as he discovered undiscovered rotationally distinct mirror images.

17. LeWitt's Final Result and Legacy

• LeWitt's enumeration reached its final form in early 1974.
• He created 40-inch painted aluminum sculptures and a series of isometric drawings.
• His most abstract representation for the cubes compresses the isometric view down to a flat hexagonal form.
• The video creator found this minimalist table super convenient for entering the relevant data.

18. Burnside's Lemma and Generalizations

• Burnside's Lemma is a general-purpose counting technique that can be used in many situations beyond incomplete open cubes.
• Examples: incomplete open tetrahedra, coloring the edges of a cube, Rubik's Cube.

19. Conclusion and the Imperfection

• The video concludes that there are 218 families of rotationally equivalent cubes.
• A 2014 paper confirms that with all the constraints applied, Sol LeWitt's finding of 122 is the correct number.
• However, there is an error in the final sculpture: cubes labeled 104 and 105 are the same, and one cube is missing.
• The mathematical point of view contributes to the space of ideas around this piece and adds something aesthetic to how we can appreciate the work.
• Quote: "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."

Main Takeaways/Synthesis:

The video provides a deep dive into Sol LeWitt's "Variations of Incomplete Open Cubes," exploring the mathematical concepts and artistic process behind the work. It demonstrates how a seemingly simple question ("How many ways can a cube be incomplete?") can lead to complex mathematical explorations involving symmetry, group theory, and enumeration. The video highlights the importance of problem-solving strategies, labeling systems, and the role of intuition and epiphanies in both art and mathematics. It also emphasizes that the journey of discovery, with its false starts and unexpected connections, can be just as valuable as the final result. The video concludes by showing how mathematical analysis can enhance our appreciation of conceptual art by providing a framework for understanding the underlying ideas and processes.