Source-linked AI summary
An explicit mono-monostatic polyhedron
Tancredi Schettini Gherardini
TL;DR
Explicit homogeneous mono-monostatic polyhedra had not been published, despite a proof that they exist. The paper constructs adaptive tangent covers and certifies a 56,946-face example with signature (1, 0, 1), using rational data and exact verification.
Problem
Although existence of homogeneous mono-monostatic polyhedra was proved, no explicit example appeared in the literature to the author's knowledge.
Source: §1 · p. 1Method
The paper builds adaptive tangent covers of a smooth mono-monostatic body and certifies the resulting rational half-space polytope with exact comparisons or rigorous error bounds.
Source: §1 · p. 2Results
56,946 faces yield a polytope with exact-centroid equilibrium signature (1, 0, 1), one stable facet and one unstable vertex, with explicit nondegeneracy margins.
Source: §1 · p. 2Takeaways & Limitations
The finite certificate turns a numerical construction into a mathematical result grounded in convex geometry and a verifiable algorithm.
Source: §1 · p. 2Within tangent covers of smooth mono-monostatic bodies, the ridge constraint means this route cannot go far below ~5 · 10^4 faces unless a seed with radically larger ridge tilt exists.
Source: §6 · p. 10
Abstract
from arXiv · showhide
A convex body is mono-monostatic if, resting under gravity on a horizontal plane, it has exactly one stable and one unstable equilibrium position. Smooth mono-monostatic homogeneous bodies exist (the G\"omb\"oc of Domokos and V\'arkonyi), and L\'angi proved that (homogeneous) mono-monostatic polyhedra exist; although no explicit example appears to have been published, to the author's knowledge. We construct explicitly two such mono-monostatic polytopes, the smaller one having $56946$ faces; importantly, we certify them: the polytope is presented as an intersection of half-spaces with rational data, and a certifying verification establishes, using exact rational arithmetic for every decisive comparison, that the body has equilibrium signature $(S,H,U)=(1,0,1)$ with respect to its own exact centroid, with explicit nondegeneracy margins. We describe the geometric obstructions that make naive discretisations of smooth mono-monostatic bodies fail, the adaptive construction that overcomes them, and the certification strategy. While the result itself is not strikingly novel, we emphasise the non-standard (but increasingly more common) methodology: the entire programme, i.e. experiments, constructions and the verifier itself, was implemented by AI agents under human mathematical direction. We argue that exact certification of numerically discovered objects is the natural contract between such AI-assisted workflows and mathematical standards of rigour.
1. Introduction
Although Lángi proved that homogeneous mono-monostatic polyhedra exist, no explicit example was known to the author. This paper constructs and certifies explicit examples, including a 56,946-face polytope.
1. Introduction
- Lángi proved existence, but no explicit homogeneous mono-monostatic polyhedron appeared in the literature to the author's knowledge.The minimum face count for such bodies remains unknown.Source: §1 · pp. 1–2
- Adaptive tangent covers of a smooth mono-monostatic body overcome a near-critical ridge requiring fine facets and a sag dipole that displaces the centroid.These obstructions defeat naive discretisation.Source: §1 · p. 2
- 56,946 faces, 170,832 edges, and 113,888 vertices define the smaller polytope, whose exact-centroid equilibrium signature is (1, 0, 1).Its stable facet and unstable vertex have certified margins, and every equilibrium is nondegenerate.Source: §1 · p. 2
- The construction treats 56,946 rational half-space pairs as the polytope's definition and verifies its signature with exact arithmetic or rigorous error bounds.The larger 272,971-face construction has an analogous certification, reproducible in minutes on a laptop.Source: §1 · p. 2
- The theorem is a finite statement about a rational object, relying on elementary convex geometry and an approximately 700-line verifier [15].IEEE-754 floating point is used only within proven error budgets.Source: §1 · p. 2
- The author directed the work while AI agents designed and ran experiments, discovered obstructions, and repaired them.The paper presents exact certification as the way to turn plausible numerical artefacts into mathematics.Source: §1 · p. 3
2. Equilibria of convex polytopes
For a convex polytope, equilibria are characterized by support geometry and correspond to critical points of the distance from the chosen interior point. Facets, edges, and vertices carry stable, saddle, and unstable equilibria, respectively.
2. Equilibria of convex polytopes
- An equilibrium occurs where the support plane through a boundary point is orthogonal to its displacement from the interior point; these are critical points of boundary distance.The paper uses the uniform-density centroid as the interior point.Source: §2 · p. 3
- Facets carry stable equilibria when the centroid's projection lies inside; edges carry saddles under interior cone tests; vertices carry unstable equilibria under normal-cone tests.For simple vertices, the unstable-equilibrium test reduces to three strict inequalities against neighbouring vertices.Source: §2 · p. 3
- The verifier tests strict conditions and aborts on exact ties, detecting degenerate configurations rather than misclassifying them.Source: §2 · p. 3
- For a nondegenerate (1, 0, 1) signature, distance has one minimum and one maximum on the sphere, and gradient flow carries every non-maximal point to the stable facet.Source: §2 · pp. 3–4
- Quasi-static rolling strictly lowers the centroid at each step and terminates on the unique stable facet, but the rolling graph remains uncomputed.Source: §2 · p. 4
3. The construction
The construction starts from a numerically optimized smooth (1, 0, 1) seed and converts it into tangent-plane polytopes using adaptive covers designed around two geometric obstructions.
3.1. The smooth seed.
- Starting from Sloan's parameterization with A=0.12 and k=5, the authors optimize a 32-coefficient smooth seed with signature (1, 0, 1) and βmin≈6.2·10−5 rad.The seed K* is nonconvex, but the cover excludes planes over its dimple, and no certified claim refers to K*.Source: §3.1 · p. 5
3.2. Why naive faceting fails, quantitatively.
- Near the ridge, spacing must reach about 10−4 rad; adaptive spacing alone still shifts the centroid ≈3.5·10−4, beyond the ridge tolerance ≪6.6·10−5.The decisive cone-test sign flip occurs at the geometric threshold, not at a rounding scale.Source: §3.2 · p. 6
3.3. The adaptive equal-sag cover.
- The cover uses graded Poisson-disc directions and probe-ring gap checks, with equal-sag spacing off-ridge and ridge-adaptive spacing; the stable equilibrium gets a facet and the unstable one an apex.Direction- and normal-space exclusions let one cap plane represent the stable-equilibrium neighborhood.Source: §3.3 · p. 6
- The corrected adaptive cover yields the 56,946-face body with centroid displacement ≈1.5·10−6; the ridge band alone requires about 5·10^4 faces.The conservative parameters without compensation yield 272,971 faces, while pruning experiments place the floor near 5.5·10^4 faces.Source: §3.3 · p. 6
- Floating-point equilibrium counts and diagnostics guide the search, but are conjecture generators rather than certified claims.The certifier is applied only after the classifier robustly reports (1, 0, 1).Source: §3.3 · p. 7
4. Exact certification
The paper certifies a rational polytope by establishing its boundary structure, centroid, and equilibrium counts with quantified margins. The verifier uses an untrusted geometric oracle, exact decisions, and bounded floating-point filters.
4.1. The object and the claim.
- The 56,946 rational half-spaces define the object being certified; the smooth seed is only scaffolding, and no certified claim depends on its properties.The verification establishes the polytope’s boundary structure, exact centroid, and strict, quantified equilibrium tests.Source: §4.1 · p. 7
4.2. Architecture: an untrusted oracle and exact decisions.
- The verifier uses a convex-hull oracle only to propose boundary triples; each subsequent decision is exact or uses a proven error bound to retain candidates for exact adjudication.A floating-point filter may certify that a test is far from tight, but may not reject a potential equilibrium.Source: §4.1 · p. 7, §4.2 · p. 7, §4.3 · p. 7
4.3. The boundary complex.
- The boundary-complex check certifies that the proposed triples, edges, and facets exhaust the polytope’s boundary.The verifier checks roughly 6.5 billion feasibility inequalities, deciding borderline cases exactly, and also checks strict local convexity at every edge.Source: §4.3 · pp. 7–8
4.4. The centroid.
- The centroid is enclosed by exact tetrahedral volume and moment accumulation, then equilibrium tests at the rational enclosure point are guarded against centroid error.The enclosure radius is about 2 · 10^-93, and the verifier records the smallest guarded decisive quantity for reproducibility.Source: §4.4 · p. 8
4.5. The three counts.
- The certified counts are one stable facet, one unstable vertex, and no saddle, with all decisive tests nondegenerate.The exact adjudication leaves one stable candidate and one unstable candidate; none of the 25,326 surviving saddle candidates qualifies.Source: §4.5 · p. 8
- The certificate is bound to the input data and rests on the verifier, exact rational arithmetic, bounded IEEE-754 operations, and elementary geometry.Consumers can rerun the verifier rather than trust shipped certificates or construction software; formal verification of the verifier is identified as a possible next assurance step.Source: §4.5 · p. 9, §4.6 · p. 9
5. The human–AI workflow
The human–AI workflow paired human direction and standards of rigour with an AI agent that carried out the experiments, construction, and certification. Numerical experiments redirected the search toward geometric rather than precision fixes, while adversarial review exposed verifier flaws before acceptance.
5. The human–AI workflow
- The human author set the problem, supplied literature, standards, resources, and strategic direction; an AI agent conducted and debugged the remaining experimental and technical work.The agent also wrote the first draft, while the author reviewed summaries and directed work at session granularity.Source: §5.1 · p. 9
5.1. Division of labour.
- A ridge experiment found identical double- and 50-digit margins with a sign change at finite cell size, redirecting the search from more precision to more faces in the right places.This led to the cell law used in the construction.Source: §5.1.1 · p. 9
5.1.1. Hypothesis testing.
- Adversarial review found three soundness gaps in a verifier that had passed all phases, prompting repairs before its certificate was accepted.The gaps concerned an ill-conditioned filter, unverified oracle adjacency, and the transfer bound from approximate to exact centroid.Source: §5.1.2 · p. 9
5.1.2. Verifying the verifier.
- A reported 19-face solution was an unbounded, combinatorially broken region, so its equilibrium counts were meaningless and the few-face search remained unsuccessful.The authors identify hard invariant gates during search and exact certification before belief as the reliable antidote to such traps.Source: §5.1.3 · p. 10
5.2. Cost.
- The programme took days, at most a couple of weeks, using a laptop and some cluster allocations, while certification itself ran in minutes.Source: §5.2 · p. 10
6. Towards a better G¨omb¨ocedron
The paper argues that tangent-cover constructions face a substantial face-count floor, while attempts to find much smaller mono-monostatic polyhedra remain unresolved. It identifies both rigidity in a 14-face monostable example and persistent obstacles in searches near 17 faces.
6. Towards a better G¨omb¨ocedron
- Within tangent covers of smooth mono-monostatic bodies, the ridge band forces Θ(σ^2/βmin) faces, and βmin ≲ 10^-3 appears unavoidable, placing this route near a floor of 5 · 10^4 faces.The authors say few-face mono-monostatic polyhedra, if found, will not resemble discretised G¨omb¨ocs.Source: §6 · p. 10
- Reshetov’s 14-face unistable polyhedron has signature (1, 1, 2), with a surplus saddle–maximum pair whose persistence is roughly 24 in unit-volume terms.The authors report that cancelling the pair would require undoing elongation despite 13 foot-ejection constraints, each with relative margin about 10^-3.Source: §6 · p. 10
- Structured searches around 17 faces repeatedly found signatures (2, 3, 3) and (3, 3, 2), but final cancellations resisted several hundred optimisation runs.Whether the obstacle is structural or a search failure remains open.Source: §6 · p. 10
- Whether a homogeneous mono-monostatic polyhedron exists with at most 20 faces, or with fewer than 103 faces, remains an open question.The authors state that any candidate at these sizes can be certified by their verifier in seconds, which they hope lowers the barrier to record attempts.Source: §6 · p. 10
7. Data and reproducibility
The paper’s accompanying artefacts include the data and certificates for both certified polytopes, the verifier, and the construction pipeline. The certification chain is deterministic and regenerates its certificate from the half-space data.
7. Data and reproducibility
- The accompanying repository includes both polytopes’ half-space data, certificates, verifier source, construction pipeline, and smooth-seed coefficients.The certification chain deterministically regenerates the certificate from the half-space data alone.Source: §7 · p. 11
Appendix A. Algorithmic details
The adaptive sampling process refines directions around tilt minima and requires a coverage audit to catch gaps missed during construction.
Appendix A. Algorithmic details
- The candidate pool is refined until local density exceeds the target, then thinned greedily by finest cells first into a maximal graded sample.The sample uses pairwise separation at least ρ.Source: §A.1 · p. 11
- Refinement must keep each local minimum of β in its frontier; an earlier omission left a hole twenty times the critical size.Source: §A.1 · p. 11
- The coverage audit probes rings around every tilt minimum and is non-optional because it detects gaps missed by refinement.Source: §A.1 · p. 11
A.1. Cover generation.
The cover-generation algorithm places tangent planes at adaptively selected directions, audits coverage, compensates for centroid displacement, and classifies the resulting polytope.
A.1. Cover generation.
- The equal-sag adaptive tangent cover refines candidate directions to local spacing ρ, selects separated directions, and places tangent planes plus a stable cap plane.Source: §A.1 · p. 11
- The audit probes rings around tilt minima, checks for gaps exceeding local ρ, and adds tangent planes at uncovered probes.Source: §A.1 · p. 11
- Centroid displacement is compensated by adjusting plane offsets, with ζ resolved by secant iteration over two to three builds.Source: §A.1 · p. 11
- The search-time classifier tests the polar-dual boundary complex using facet-foot, vertex-cone, and edge-wedge checks.It was validated on small polyhedra, including Reshetov’s signature (1, 1, 2), later certified exactly.Source: §A.2 · p. 11
A.2. The floating-point classifier.
The verifier uses a seven-phase pipeline with exact or rigorously bounded checks to certify the polytope’s geometry and equilibrium counts.
A.2. The floating-point classifier.
- Boundary-complex certification combines exact orientation and independence determinants with a bulk feasibility filter and exact adjudication of borderline pairs.The bulk filter threshold is 10^-10, against a proven error bound below 10^-14.Source: §A.3 · p. 11
- The centroid is enclosed using exact tetrahedral moments and fixed-point accumulation at 2^-320, with enclosure radius about 2 · 10^-93.Source: §A.3 · p. 11
- Stable count is filtered before its single candidate is checked exactly against all planes.Source: §A.3 · p. 11
- Unstable and saddle counts use local inequality and foot tests, respectively, with all filter survivors adjudicated exactly.Source: §A.3 · p. 11
A.3. The certifier.
The certifier records guarded equilibrium counts and can check different expected signatures; the paper reports runtimes and certified margins for both constructed polytopes.
A.3. The certifier.
- The verifier defaults to signature (1, 0, 1), checks the Euler relation S − H + U = 2, and records transfer guards for exact decisions.The expected signature is configurable, and every exact decision has |q| > 10^-60.Source: §A.3 · p. 12
- The verifier runs in under ten minutes for the larger body and under four minutes for the smaller body on a laptop.The same chain produces Reshetov’s (1, 1, 2) certificate in under a second.Source: §A.3 · p. 12
- Table 1 reports certified minimal slacks rounded down, defining stable margins by competing-plane clearance and unstable margins by normal-cone slack.Source: Table 1 · p. 12