A convex polyhedron without Rupert's property (2508.18475v1)

Abstract: A three-dimensional convex body is said to have Rupert's property if its copy can be passed through a straight hole inside that body. In this work we construct a polyhedron which is provably not Rupert, thus we disprove a conjecture from 2017. We also find a polyhedron that is Rupert but not locally Rupert.

• The paper disproves the conjecture that all convex polyhedra have Rupert's property by explicitly constructing the Noperthedron.
• It utilizes a partitioning of a five-dimensional parameter space and sharp analytical bounds to rigorously eliminate possible solutions.
• The study introduces a polyhedron that is Rupert but not locally Rupert, offering new insights into passage properties in convex geometry.

A Convex Polyhedron Without Rupert's Property

Introduction and Historical Context

The paper "A convex polyhedron without Rupert's property" (2508.18475) addresses a longstanding conjecture in discrete geometry regarding the so-called Rupert's property. This property, first observed for the cube by Prince Rupert of the Rhine and proved by John Wallis in 1685, asserts that a copy of a three-dimensional convex body can be passed through a straight hole cut inside the body itself. The property is equivalently formulated in terms of orthogonal projections: for a convex polyhedron $\PPP$, there exist two projections such that one fits strictly inside the other.

Previous work established that all Platonic solids possess Rupert's property, and the conjecture extended to all convex polyhedra in $\mathbb{R}^3$. The present paper disproves this conjecture by constructing an explicit counterexample, the Noperthedron ($\NOP$), and rigorously proving that it does not have Rupert's property. Additionally, the authors construct a polyhedron that is Rupert but not locally Rupert, further refining the landscape of the property.

Mathematical Framework and Definitions

The authors formalize Rupert's property for point-symmetric convex polyhedra using a projection-based criterion. For a polyhedron $\PPP$ and parameters $(\theta_1, \varphi_1, \theta_2, \varphi_2, \alpha)$, the property holds if

$R(\alpha) M(\theta_1, \varphi_1)\PPP \subset (M(\theta_2, \varphi_2)\PPP)^\circ,$

where $R(\alpha)$ is a planar rotation, $M(\theta, \varphi)$ is the orthogonal projection in direction $X(\theta, \varphi)$, and the inclusion is strict in the interior of the convex hull.

The Noperthedron is constructed as the union of orbits of three rational points under the action of a cyclic group $\mathcal{C}_{30}$, yielding a highly symmetric, point-symmetric polyhedron with 90 vertices. The symmetries of $\NOP$ allow a drastic reduction in the parameter search space for possible Rupert solutions.

Figure 1: Two projections of the unit cube, illustrating the classical Rupert's property.

Figure 2: Two projections of the Octahedron, demonstrating the projection-based formulation of Rupert's property.

Analytical and Algorithmic Proof Strategy

The proof that $\NOP$ is not Rupert is based on a partitioning of the five-dimensional parameter space into small regions and systematically excluding solutions in each region. The authors develop two main exclusion theorems:

• Global Theorem: If a vertex of the "smaller" projection is strictly outside the "larger" projection, then no solution exists in a neighborhood, quantified using operator norm bounds and derivatives of the projection matrices.
• Local Theorem: For regions where projections are nearly congruent, the authors use geometric and algebraic criteria (spanning and locally maximally distant points) to exclude solutions, leveraging the structure of the convex hull and the behavior of distances under small perturbations.

Sharp bounds are established for the operator norms of differences between rotation and projection matrices, e.g.,

$$\| M(\theta, \varphi) - M(\overline{\theta}, \overline{\varphi}) \| \leq \sqrt{2}\varepsilon,$$

and

$$\| R(\alpha) M(\theta, \varphi) - R(\overline{\alpha}) M(\overline{\theta}, \overline{\varphi}) \| \leq \sqrt{5}\varepsilon,$$

for small parameter perturbations $\varepsilon$.

Rigorous Computer-Assisted Verification

To ensure the correctness of the exclusion arguments, the authors develop rational versions of the global and local theorems, replacing all real arithmetic with rational approximations and bounding rounding errors. The entire parameter space is partitioned into approximately 18 million regions, each verified using these rational theorems.

The verification is implemented in R and SageMath, with the solution tree and all code made publicly available. The process involves:

• Generating a tree structure representing the partitioned parameter space.
• For each leaf node, applying either the rational global or local theorem to exclude solutions.
• Verifying that the union of all excluded regions covers the entire search space.

This approach provides a fully rigorous, computer-assisted proof that $\NOP$ does not possess Rupert's property.

Figure 3: The Ruperthedron, a polyhedron that is Rupert but not locally Rupert.

Construction of a Rupert but Not Locally Rupert Polyhedron

The authors further construct the Ruperthedron ($\RUP$), a polyhedron with 90 vertices, which is Rupert but not locally Rupert. This demonstrates that local Rupertness is strictly stronger than Rupertness, answering another open question in the field. The verification for $\RUP$ is similar but restricted to local neighborhoods in parameter space.

Implications and Future Directions

The explicit construction of a convex polyhedron without Rupert's property refutes a widely held conjecture and clarifies the limitations of projection-based passage properties in convex geometry. The methodology—combining sharp analytical bounds, geometric criteria, and rigorous computer-assisted verification—sets a new standard for such existence proofs in discrete geometry.

The distinction between Rupert and locally Rupert polyhedra opens new avenues for classification and understanding of passage properties. The authors highlight several open problems, including the development of more efficient exclusion methods, optimality proofs for Rupert solutions, and the determination of Nieuwland numbers for classical solids.

Conclusion

This work provides a definitive counterexample to the conjecture that all convex polyhedra are Rupert, using a combination of group-theoretic construction, analytical bounds, and exhaustive computer verification. The introduction of the Noperthedron and Ruperthedron refines the taxonomy of passage properties in convex polyhedra and suggests new directions for research in geometric combinatorics and computational geometry. The techniques developed herein are broadly applicable to other problems involving high-dimensional parameter exclusion and rigorous computer-assisted proofs.