arxiv: 2604.26531 · v1 · submitted 2026-04-29 · 🧮 math.MG · cs.CG· math.CO

Recognition: unknown

A stellated tetrahedron that is probably not Rupert

Tony Zeng

Pith reviewed 2026-05-07 10:39 UTC · model grok-4.3

classification 🧮 math.MG cs.CGmath.CO

keywords Rupert polyhedronstellated tetrahedronconvex polyhedrapolygon fittinglinear programmingnumerical evidencerotation sampling

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The pith

Numerical checks indicate a stellated tetrahedron is not Rupert.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates whether certain convex polyhedra allow a hole to be cut so that an identical copy can pass through, a property called being Rupert. It targets a stellated tetrahedron as a candidate for the simplest such non-Rupert example, simpler than the known counterexamples. The authors reduce the 3D passage question to a series of 2D polygon fitting problems and solve them with linear programs across many random orientation pairs. They report that over 88 percent of the sampled pairs from the space of rotations yield no fitting at full size.

Core claim

By sampling many pairs of orientations from SO(3) times SO(3) and solving linear programs to determine the largest scaling factor allowing one 2D projection to translate into the other, the authors find that more than 88 percent of such pairs do not allow a scaling of at least 1, suggesting the stellated tetrahedron admits no Rupert passage.

What carries the argument

Linear program solvers that compute the largest scaling factor for which one polygon can be translated to fit inside another polygon, applied to the 2D projections of the 3D polyhedron in chosen orientations.

If this is right

The stellated tetrahedron would serve as a simpler counterexample than those previously constructed.
The same linear-program technique can test the Rupert property for other polyhedra of low face count.
The Rupert property appears sensitive to the precise geometry even for highly symmetric shapes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Denser sampling or an exhaustive search over orientations might convert the numerical evidence into a rigorous proof of non-Rupert status.
The same method could be used to check whether any tetrahedron, or any simplicial polyhedron, fails to be Rupert.
Similar optimization reductions may apply to other questions about whether one rigid body can pass through a hole in a copy of itself.

Load-bearing premise

The sampled orientations are representative of all possible passages and the reduction from 3D passage to 2D polygon fitting is complete.

What would settle it

An explicit pair of orientations together with a linear-program result showing a maximum fitting scale of 1 or greater would demonstrate that a passage exists for that case.

Figures

Figures reproduced from arXiv: 2604.26531 by Tony Zeng.

**Figure 1.** Figure 1: An illustration that the cube is Rupert. Rupert’s question was answered in the affirmative by Wallis himself, who showed that the flat projection of the cube into a square can fit inside the regular hexagonal projection of the cube along its space diagonal (see view at source ↗

**Figure 2.** Figure 2: The Noperthedron of Steininger and Yurkevich. 1.2. State of the Art. Currently, it is known that all 5 Platonic solids, 11 out of 13 Catalan solids, and 87 of the 92 Johnson solids have the Rupert property. Note that proving that a polyhedron is Rupert is a far easier task than showing that it is not. For the former, one need only exhibit two projections and verify containment, while for the latter, it is … view at source ↗

**Figure 3.** Figure 3: The stellated tetrahedron P11/20. Faces with similar colors share the same base tetrahedral face prior to stellation. We conjecture this convex polyhedron to not be Rupert view at source ↗

**Figure 4.** Figure 4: Numerical Nieuwland numbers for Pa plotted against a computed using the techniques of [2]. 1.4. Main Result. The space of all pairs of orientations, that is SO(3) × SO(3), is a 6-dimensional manifold. Since we can encode an element of SO(3) using three angles α, θ, ϕ, we can define a parameter space Λ0 as a 6-dimensional space, i.e. [0, 2π] 6 over which to search for Rupert passages. We discuss in §3 how Λ… view at source ↗

**Figure 5.** Figure 5: P does not fit inside Q, but P ′ does fit inside Q. For a convex polygons P, we say that a δ-perturbation of P is any polygon P ′ such that ∥pi − p ′ i ∥ < δ for all i. Then we have the following lemma view at source ↗

**Figure 6.** Figure 6: The construction of polygons P ′ (left) and Q′ (right) as in the Corollary view at source ↗

**Figure 7.** Figure 7: A 2-dimensional illustration of the algorithm. not Rupert as we lack a way to show that P11/20 is not locally Rupert. That is to say, we do not have a result of the form for all R1 ∈ SO(3), there is no Rupert passage R1, R2 for any orientation R2 that is ε-close to R1. Such a statement exists in [6] as their polyhedron has a particular nice structure, and perhaps most importantly, is centrally symmetric. T… view at source ↗

read the original abstract

A convex polyhedron is Rupert if a hole can be cut into it (making its genus $1$) such that an identical copy of the polyhedron can pass through the hole. Resolving a conjecture of Jerrard-Wetzel-Yuan, Steininger and Yurkevich recently constructed a convex polyhedron which is not Rupert. We propose a search for the simplest possible non-Rupert polyhedron and provide numerical evidence suggesting that a particular stellated tetrahedron is not Rupert. The computational techniques utilize linear program solvers to compute the largest possible scalings of polygons that can be translated to fit in other polygons. The relative simplicity of the stellated tetrahedron as compared to other polyhedra allows this more rudimentary check to be computationally tractable. In particular, we show that over 88% of a particular encoding of $\text{SO}(3) \times \text{SO}(3)$ equipped with the standard measure does not yield a Rupert passage.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Numerical sampling on a stellated tetrahedron gives suggestive evidence it fails the Rupert test in most orientations, but the finite sample and reduction completeness are the main limits.

read the letter

The main point is that this paper tests a relatively simple stellated tetrahedron and finds that over 88% of sampled orientation pairs from SO(3) x SO(3) produce no Rupert passage when checked with linear programming on the 2D projections. It positions this shape as a candidate for a minimal non-Rupert polyhedron, building on the recent constructions by Steininger and Yurkevich but aiming for something easier to verify computationally.

Referee Report

2 major / 2 minor

Summary. The paper claims to provide numerical evidence that a particular stellated tetrahedron is probably not Rupert. It does so by sampling pairs of orientations from a parametrization of SO(3) × SO(3) and using linear-program solvers to check whether the orthogonal projections admit a translation-fitting scaled copy at factor 1; the authors report that more than 88% of the sampled pairs yield no such passage.

Significance. If the Monte-Carlo sampling is representative and the 2D-projection reduction is complete, the result would identify a relatively simple convex polyhedron as a non-Rupert example, advancing the search for the minimal counterexample to the Jerrard–Wetzel–Yuan conjecture. The computational approach—reducing the passage check to linear programs on 2D polygons—is appropriate and exploits the low complexity of the stellated tetrahedron to remain tractable.

major comments (2)

[Abstract] Abstract: the central claim that “over 88% … does not yield a Rupert passage” is stated without sample size, variance estimate, or error bounds on the Monte-Carlo measure; this omission prevents assessment of whether the observed 12% failure rate could be explained by undersampling of a positive-measure set of passages.
[The computational techniques] The computational method: the reduction of the 3D Rupert-passage problem to checking fixed-orientation 2D polygon-fitting conditions via linear programs is used without an argument that every possible passage (including those involving non-prismatic holes or intermediate rotations during transit) must correspond to some pair of orthogonal projections that satisfy the translation condition at scale 1.

minor comments (2)

[Abstract] The specific parametrization chosen for the “particular encoding of SO(3) × SO(3)” is not described in sufficient detail to allow independent reproduction of the sampling measure.
[Abstract] The abstract refers to “the standard measure” on SO(3) × SO(3) without citing the normalization or Haar-measure convention employed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below and will revise the manuscript to improve clarity and acknowledge limitations where appropriate.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that “over 88% … does not yield a Rupert passage” is stated without sample size, variance estimate, or error bounds on the Monte-Carlo measure; this omission prevents assessment of whether the observed 12% failure rate could be explained by undersampling of a positive-measure set of passages.

Authors: We agree that additional statistical details would strengthen the abstract. In the revised manuscript we will specify the sample size used for the Monte Carlo sampling over the parametrization of SO(3) × SO(3) and, where computable from the existing data, include variance estimates or confidence intervals for the reported proportion. revision: yes
Referee: [The computational techniques] The computational method: the reduction of the 3D Rupert-passage problem to checking fixed-orientation 2D polygon-fitting conditions via linear programs is used without an argument that every possible passage (including those involving non-prismatic holes or intermediate rotations during transit) must correspond to some pair of orthogonal projections that satisfy the translation condition at scale 1.

Authors: The referee correctly notes that our reduction checks fixed-orientation projection pairs for translation fitting at scale 1, which corresponds to a necessary condition for passages through prismatic holes with constant orientation. We do not possess a complete argument showing that every conceivable Rupert passage, including those with intermediate rotations or non-prismatic holes, must manifest as such a pair. The manuscript presents the 88 % figure as numerical evidence that the stellated tetrahedron is probably not Rupert rather than a rigorous proof. In revision we will add an explicit discussion of this scope and limitation. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical sampling and LP checks

full rationale

The paper's central claim is a Monte Carlo statistic obtained by sampling orientations in a parametrization of SO(3) × SO(3) and using linear-program solvers to test whether one 2D projection can be translated to fit inside the other at scale 1. This computation applies standard geometric optimization directly to the input polyhedron without fitting any free parameters to the reported percentage, without renaming known results, and without load-bearing self-citations or imported uniqueness theorems. The reduction to 2D polygon fitting is presented as an explicit computational procedure rather than an algebraic identity that collapses to the inputs by construction. Consequently the derivation chain remains independent of the final statistic.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard facts from convex geometry and optimization; no new free parameters, axioms, or invented entities are introduced beyond the choice of the specific polyhedron.

axioms (1)

domain assumption The Rupert property for a convex polyhedron reduces to checking whether one projected polygon can be scaled and translated to fit inside another for some pair of orientations.
This reduction is invoked to justify the use of 2D linear programs.

pith-pipeline@v0.9.0 · 5456 in / 1143 out tokens · 41129 ms · 2026-05-07T10:39:08.174356+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 2 canonical work pages

[1]

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Bernard Chazelle. The polygon containment problem.Advances in Computing Research, 1:1– 33, 1983

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[2]

Optimizing for the Rupert property.The American Mathematical Monthly, 131(3):255–261, 2023

Albin Fredriksson. Optimizing for the Rupert property.The American Mathematical Monthly, 131(3):255–261, 2023

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[3]

Platonic passages.Mathematics Maga- zine, 90(2):87–98, 2017

Richard P Jerrard, John E Wetzel, and Liping Yuan. Platonic passages.Mathematics Maga- zine, 90(2):87–98, 2017

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[4]

Two sufficient conditions for a polyhedron to be (locally) Rupert.arXiv preprint arXiv:2208.12912, 2022

Evan Scott. Two sufficient conditions for a polyhedron to be (locally) Rupert.arXiv preprint arXiv:2208.12912, 2022

work page arXiv 2022
[5]

An algorithmic approach to rupert’s problem.Math- ematics of Computation, 92(342):1905–1929, 2023

Jakob Steininger and Sergey Yurkevich. An algorithmic approach to rupert’s problem.Math- ematics of Computation, 92(342):1905–1929, 2023

1905
[6]

A convex polyhedron without Rupert’s property.arXiv preprint arXiv:2508.18475, 2025

Jakob Steininger and Sergey Yurkevich. A convex polyhedron without Rupert’s property.arXiv preprint arXiv:2508.18475, 2025

work page arXiv 2025
[7]

Jean Henri van Swinden.Grondbeginsels der meetkunde. P. den Hengst en zoon, 1816
[8]

E Theatro, Sheldoniano, Oxoniae, 1685

John Wallis.De Algebra Tractatus; Historicus & Practicus, volume 2. E Theatro, Sheldoniano, Oxoniae, 1685. Department of Mathematics, University of W ashington, Seattle, W A 98195, USA Email address:txz@uw.edu