Incomplete open cubes

2025-09-07 · Source: 3Blue1Brown · Field: Science & Research — Mathematics & Computational Sciences · Depth: Novice, quick

Summary

The concept of "incomplete open cubes" explores the number of distinct configurations possible when some edges are removed from a cube, with configurations considered identical if one can be rotated to match another. This mathematical puzzle served as the premise for Sol LeWitt's 1974 modern art piece. A recent guest video, commissioned via Patreon funds and created by Paul Dancstep, delves into the artwork's story while simultaneously illustrating a problem-solving process that leads to answering this counting question. This process effectively involves rediscovering elements of group theory, specifically Burnside's Lemma, making the content accessible to a wide audience from middle schoolers to PhD students.

Key takeaway

For mathematicians or art historians interested in the intersection of abstract concepts and artistic expression, exploring Sol LeWitt's 1974 work through the lens of incomplete cubes offers a unique perspective. You should consider how group theory, specifically Burnside's Lemma, provides a rigorous framework for analyzing such combinatorial problems, enriching both your mathematical understanding and appreciation of conceptual art.

Key insights

Counting distinct incomplete cube configurations involves group theory and Burnside's Lemma.

Principles

• Rotational symmetry defines equivalence.
• Group theory aids combinatorial counting.

Method

The problem-solving process involves rediscovering group theory and Burnside's Lemma.

In practice

• Apply Burnside's Lemma to symmetry problems.
• Explore combinatorial puzzles with group theory.

Topics

• Incomplete Cubes
• Combinatorics
• Group Theory
• Burnside's Lemma
• Sol LeWitt Art

Best for: Research Scientist, General Interest

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Editorial summary, takeaway, and curation by AIssential. Original article published by 3Blue1Brown.