Exploration & Epiphany | Guest video by Paul Dancstep
Audio Brief
Show transcript
This episode explores Sol LeWitt's artwork "Variations of Incomplete Open Cubes," analyzing it as both a conceptual art masterpiece and a complex combinatorial mathematics problem.
There are three key takeaways from this discussion.
First, art can be a rigorous, systematic process, mirroring mathematical or computational approaches.
Second, solving complex counting problems often requires a change in perspective, such as counting symmetries rather than individual objects.
Third, the process of discovery, whether artistic or mathematical, is often as valuable as the final result itself.
Sol LeWitt's "Variations of Incomplete Open Cubes" exemplifies conceptual, minimalist, and serial art. The artwork comprises 122 sculptures systematically cataloging incomplete cube frames under specific rules. These rules require shapes to be connected, three-dimensional, and rotationally unique. LeWitt's philosophy posited that the idea itself acts as "a machine that makes art."
The challenge of identifying rotationally unique shapes highlights a powerful mathematical tool: Burnside's Lemma. Instead of manually enumerating and de-duplicating all 4,096 possible incomplete cubes, Burnside's Lemma counts how many shapes remain unchanged by each of the 24 possible cube rotations. Summing these counts and dividing by 24 efficiently determines the total number of unique families.
LeWitt himself manually built and rotated models to solve the problem, arriving at 122 variations. Applying Burnside's Lemma rigorously calculates 218 unique families. This difference underscores the intersection of artistic intuition and mathematical proof, valuing the journey of discovery itself.
This exploration bridges the worlds of art and mathematics, demonstrating the elegant logic underlying creative expression.
Episode Overview
- The episode explores Sol LeWitt's 1974 artwork, "Variations of Incomplete Open Cubes," framing it as both a masterpiece of conceptual art and a complex combinatorial math problem.
- It delves into LeWitt's artistic philosophy, where a systematic idea or set of rules acts as a "machine that makes art," using the simple cube as a neutral grammatical unit for exploration.
- The podcast follows a mathematical journey to solve the problem posed by the artwork: determining the exact number of unique, incomplete open cubes according to LeWitt's constraints.
- A powerful mathematical tool, Burnside's Lemma, is introduced as the key to efficiently solving the problem by changing the counting perspective from individual shapes to the symmetries across all possible shapes.
- The final mathematical answer of 218 unique variations is calculated and compared to LeWitt's own count of 122, highlighting the intersection of artistic intuition and rigorous mathematical proof.
Key Concepts
- "Variations of Incomplete Open Cubes": A Sol LeWitt artwork consisting of 122 sculptures that systematically catalog all possible incomplete frame structures of a cube, based on a specific set of rules.
- The Three Constraints: To be included, each shape must be (1) Connected, with all edges forming a single piece; (2) 3-Dimensional, not flat; and (3) Rotationally Unique, meaning it cannot be the same as another shape simply by rotating it.
- Sol LeWitt's Artistic Philosophy: The piece exemplifies key art movements, including Conceptual Art (the idea is the art), Minimalism (using simple geometric forms), and Serial Art (creating work through a systematic, exhaustive process).
- Rotational Symmetries of a Cube: A cube has 24 distinct rotational orientations, derived from axes passing through its faces, corners, and edges. This number is fundamental to identifying unique shapes.
- Family Portrait and Lookalikes: A "family portrait" is the set of all 24 orientations of a single shape. A "lookalike" is an orientation that appears identical to the original after a rotation, and the number of lookalikes for a shape is a measure of its symmetry.
- Burnside's Lemma (The Counting Epiphany): Instead of counting each of the 4,096 possible shapes and their duplicates, this method involves counting how many shapes are left unchanged (fixed) by each of the 24 rotations. The sum of these counts, divided by 24, gives the total number of unique families.
- Final Calculation: The mathematical process concludes that there are 218 unique families of incomplete open cubes, a result derived by applying Burnside's Lemma.
Quotes
- At 4:04 - "The idea becomes a machine that makes art." - LeWitt's famous quote explaining the core principle of conceptual art, where the initial concept dictates the final form without subjective interference.
- At 6:24 - "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 describing his own manual, trial-and-error process for solving the problem of rotational duplicates.
- At 33:00 - "total lookalikes / 24 = # of families" - The presentation of the final, powerful formula derived from Burnside's Lemma, which is the key to solving the problem efficiently.
- At 43:47 - "Dividing by 24 gives us our final answer. It seems that there are 218 rotationally unique families of incomplete open cubes." - The conclusive announcement of the solution to the central mathematical question.
- At 51:38 - "Those that show the thought process of the artist are sometimes more interesting than the final product." - A quote from Sol LeWitt that encapsulates the video's theme, emphasizing the value of the creative and intellectual journey itself.
Takeaways
- Art can be a rigorous, systematic process, where a conceptual framework and a set of rules can generate an entire body of work in a way that mirrors a mathematical or computational approach.
- The key to solving complex counting problems often lies in a change of perspective; it can be far more efficient to count symmetries across a system than to enumerate and deduplicate individual objects.
- The process of discovery, whether through an artist's empirical exploration or a mathematician's formal proof, is often as valuable and insightful as the final result itself.
