Rupert’s Tunnel Problem and the Noperhedron: Computer-Assisted Breakthroughs in 3D Geometry

In this episode, Quanta explores a centuries-old geometry puzzle known as Rupert’s tunnel problem, where two identical solids are manipulated to pass one through the other. Erika Clarich explains how mathematicians moved from intuition-based guesses to computer-assisted experimentation, identifying holdouts among common shapes and ultimately constructing the Noperhedron, a 152-faced polyhedron that challenges the conjecture for all convex polyhedra. The conversation walks through the geometry of shadows, the idea of orientation space, and how a rigorous proof requires ruling out every possible orientation. It also highlights how modern computation can expand the reach of mathematical insight, while leaving room for deeper human understanding. The episode closes with reflections on the tactile beauty of 3D geometry and ongoing questions in the field.

Introduction

The Quanta podcast opens a window into a long-standing geometric puzzle, Rupert’s tunnel problem, and a modern counterexample that has reshaped how mathematicians think about proving such questions. Erika Clarich, a longtime Quanta contributor, guides listeners through the history, the challenge of turning visual intuition into a proof, and the role of computers in discovering new shapes and insights.

"This is a problem in geometry that you could explain to a child and they would understand." - Erika Clarich

From Intuition to Computation

The episode recounts how Rupert originally posed a seemingly simple question about passing one cube through another with a rim, a problem that scales to any convex polyhedron. Clarich explains that shapes that look irregular often yield Rupert tunnels, while more sphere-like shapes resist such a passage, prompting mathematicians to conjecture a universal truth. The host stresses that a counterexample would overturn the conjecture, but proving such a counterexample must be rigorous and exhaustive, not just a clever example.

"Imagine that you have 2 cubes and this time we're going to think of them as solid." - Erika Clarich

Cataloging the Shapes

The discussion moves to the landscape of convex polyhedra, Platonic solids, and Archimedean solids, describing how earlier mathematicians found Rupert tunnels for many shapes and how the modern focus shifted to more regular, highly symmetric objects. The idea is that if a Rupert tunnel exists for a broad class of shapes, the conjecture might hold; conversely, identifying a true counterexample requires ruling out all potential orientations. The conversation also explains how computers extended the search far beyond what pencil-and-paper methods could handle, testing hundreds of millions of shapes and orientations.

"Two copies of the same shape and passing one through the other, Rupert tunnels." - Erika Clarich

The Noperhedron: A Breakthrough Pursued by A Pair of Mathematicians

Steininger and Yurkovich, two relatively young researchers, developed a new approach that leveraged a very specific geometric criterion: the shape must satisfy conditions about shadows seen from all angles and a three-point boundary property. When looking for a shape that would defeat the Rupert tunnel method, they used computers to search for a structure that fit those criteria but would still block all potential passages. The result of this search is the Noperhedron, a shape with 152 faces (two large 15-gon bases and a triangular framework in between). The shape is described visually as a rotund crystal vase with a wide middle and flat top and bottom, offering a vivid counterpoint to the cube example used in the original puzzle.

"The noperhedron is a shape with 152 faces." - Erika Clarich

How the Method Works: Shadows, Angles, and the Parameter Space

The core of the analysis rests on shadow comparisons: orient one copy of the shape, project its shadow in a fixed direction, and compare it with the other copy’s shadow. If, for every orientation, one shadow cannot fit inside the other, a Rupert tunnel cannot exist. The hosts describe how ruling out an orientation is not just about a single position but about an entire neighborhood in a high-dimensional orientation space. By parameterizing orientation with angles and treating the problem as ruling out regions in a multi-dimensional space, the team could exclude large swaths of orientations, turning a potentially infinite task into a finite one assisted by rigorous theorems.

"Imagine holding the cube in a particular direction and looking at the shadow that it casts straight down." - Erika Clarich

Gaps and Open Questions: Holdouts and the Need for New Methods

Despite the breakthrough, the story emphasizes that some holdout shapes remain unresolved with this method, notably certain Archimedean solids like the snub cube and the rhombic dodecahedron-based shapes. The discussion makes clear that while the Noperhedron demonstrates that Rupert tunnels do not exist for all shapes under the used criteria, it does not close the book on Rupert tunnels for every possible shape. The researchers acknowledge that different techniques would be required to settle the status of those remaining shapes, suggesting a fertile ground for future work in geometry and computation.

"There may be many more noperts out there." - Erika Clarich

Reflections and the Path Forward

The episode closes with reflections on the tactile, puzzle-like nature of 3D geometry and the collaborative, playful energy mathematicians bring to challenging problems. Clarich notes that the discovery process—balancing computer-aided experiments with human insight—offers a model for how future mathematical breakthroughs might occur. The conversation also hints at broader implications for the use of computation in proving geometric statements and the ongoing search for new methods to address stubborn shapes that resist current techniques.

"There may be many more noperts out there." - Erika Clarich

Closing Note: A Touch of Cultural Context

As a final note, the episode segues to a lighter reflection on Jane Austen’s 250th birthday, inviting listeners to read Emma as a way to unwind after a deep dive into a challenging mathematical topic. The blend of rigorous math with human storytelling reinforces the podcast’s core mission: exploring the frontiers of science and math while remaining accessible and engaging.

"There may be many more noperts out there." - Erika Clarich