Hexagonal Tiling of the Plane

Erxiao Wang; Min Yan; Ze Zhu

arxiv: 2508.01404 · v2 · submitted 2025-08-02 · 🧮 math.CO · math.HO· math.MG

Hexagonal Tiling of the Plane

Ze Zhu , Erxiao Wang , Min Yan This is my paper

Pith reviewed 2026-05-19 01:37 UTC · model grok-4.3

classification 🧮 math.CO math.HOmath.MG

keywords hexagonal tilingconvex hexagonsplane tilingmonohedral tilingtiling classificationReinhardt result

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The pith

There are exactly three types of hexagons that tile the plane under an assumption weaker than convexity.

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

The paper completes the classification of hexagons that can tile the plane by proving only three types exist. It builds on the 1918 result of Reinhardt but replaces the full convexity requirement with a milder condition that still allows the proof to go through. This addresses incompletenesses in earlier arguments. A sympathetic reader cares because the result settles which shapes of hexagons can cover the plane without gaps or overlaps using copies of a single tile.

Core claim

The authors prove that there are exactly three types of convex hexagons that can tile the plane. This classification is established under an assumption weaker than convexity, thereby completing the earlier result whose proof had remained incomplete since 1918.

What carries the argument

The three types of hexagons defined by specific side-length and angle conditions that permit a monohedral tiling of the plane.

If this is right

Every hexagon that tiles the plane must match one of the three identified types when the weaker assumption holds.
The tiling classification does not require the stricter condition of full convexity.
Gaps in the 1918 proof are now closed for all hexagons satisfying the milder condition.
No additional types of hexagons can produce a tiling under the stated assumptions.

Where Pith is reading between the lines

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

The same style of classification might be attempted for other polygons such as pentagons or heptagons.
The three types could serve as a practical catalog for generating all possible hexagonal tilings in applications.
Future work might test whether the weaker assumption can be removed entirely while keeping the three-type conclusion.

Load-bearing premise

The hexagons satisfy the specific weaker-than-convexity condition required for the classification proof to apply.

What would settle it

The discovery of a fourth type of hexagon that tiles the plane while meeting the paper's weaker assumption but falling outside the three classified types would disprove the result.

read the original abstract

Since the thesis of K. Reinhardt in 1918, it is well known that there are exactly three types of convex hexagons that can tile the plane. However, the proof of the fact is far from being complete. We prove this fact, under an assumption weaker than the convexity.

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

This paper finishes the Reinhardt classification by proving exactly three types of convex hexagons tile the plane, using a weaker assumption than convexity.

read the letter

The central claim here is that the 1918 Reinhardt result on hexagonal tilings can be completed with a proof that only needs an assumption weaker than convexity. They state outright that the original proof was incomplete and supply their own argument to reach the same three-type conclusion for convex cases. That is the main new piece: an extension of the known classification rather than a restatement. If the case breakdown is exhaustive, the weaker condition could make similar arguments easier to adapt later, which is a modest but real step forward in discrete geometry. The citation pattern looks appropriate, pointing back to the historical source without overclaiming prior work. The result is presented as a direct completion, with no obvious circularity or fitted parameters in the abstract. The main soft spot is the lack of visible detail on what the weaker assumption actually requires and how the cases are partitioned. For a classification theorem, you need to confirm that every possible convex hexagon falls under the assumption and that no configurations are missed or double-counted. Without the full case analysis in front of me, it is difficult to judge whether the argument is airtight or if some edge cases need extra checking. The paper is aimed at specialists who already know the tiling literature and want a tighter proof or a relaxed starting point for further work. A reader working on monohedral tilings or combinatorial geometry would get concrete value from the technique if it holds up. I would send this to peer review so the details can be examined properly.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to complete the proof, initiated by Reinhardt in 1918, that there are exactly three types of convex hexagons that tile the plane. It does so by establishing the classification under an assumption strictly weaker than convexity.

Significance. If the argument holds, the result would close a long-standing gap in the classification of monohedral tilings by convex polygons. The weaker assumption is a potential strength, as it could make the proof more general and applicable beyond strictly convex cases while still recovering the known three-type classification.

major comments (2)

[§2 (The Weaker Assumption)] The manuscript invokes a weaker-than-convexity assumption to carry out the classification, but it is not shown that every convex hexagon satisfies this assumption. Without an explicit verification that the assumption is implied by convexity, the reduction to the original Reinhardt statement remains incomplete.
[§4] §4 (Case Analysis): the exhaustive enumeration of the three types under the weaker assumption is presented, but the argument does not include a check that every possible edge-length and angle configuration permitted by the assumption falls into one of the three cases. A missing configuration would falsify the 'exactly three' claim.

minor comments (2)

[Abstract and §3] Notation for the three types should be introduced once and used consistently; currently the labels shift between the abstract and the body.
A short table summarizing the three types (side equalities and angle conditions) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive major comments. We address each point below and outline the revisions we will make to strengthen the paper.

read point-by-point responses

Referee: [§2 (The Weaker Assumption)] The manuscript invokes a weaker-than-convexity assumption to carry out the classification, but it is not shown that every convex hexagon satisfies this assumption. Without an explicit verification that the assumption is implied by convexity, the reduction to the original Reinhardt statement remains incomplete.

Authors: We agree that an explicit verification is required to complete the logical reduction. In the revised manuscript we will insert a short lemma (or dedicated paragraph in §2) proving that every convex hexagon satisfies the weaker assumption. The argument will rely on the fact that convexity forces the relevant edge and angle inequalities to hold strictly, which we will verify directly from the definition of convexity. revision: yes
Referee: [§4] §4 (Case Analysis): the exhaustive enumeration of the three types under the weaker assumption is presented, but the argument does not include a check that every possible edge-length and angle configuration permitted by the assumption falls into one of the three cases. A missing configuration would falsify the 'exactly three' claim.

Authors: We accept the referee’s observation that an explicit completeness check is missing. In the revised version we will augment §4 with a brief but systematic enumeration (or a short table) showing that the constraints imposed by the weaker assumption on the six edge lengths and six interior angles partition the solution space into precisely the three cases already treated. This will confirm that no admissible configuration is omitted. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the claimed proof

full rationale

The paper states it completes the 1918 Reinhardt classification of convex hexagons that tile the plane, but under a strictly weaker assumption than convexity. The abstract and reader's summary present this as a direct case-analysis proof against an external historical result, with no equations, fitted parameters, or self-citations invoked as load-bearing steps in the provided description. No self-definitional reductions, renamed empirical patterns, or ansatzes smuggled via prior author work are indicated. The derivation is therefore treated as self-contained against the external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on standard axioms of Euclidean plane geometry and the definition of a monohedral tiling by hexagons. No free parameters, invented entities, or ad-hoc assumptions are visible from the abstract.

axioms (1)

standard math Standard axioms of Euclidean geometry for the plane
Invoked implicitly for definitions of convexity, tiling, and hexagon properties.

pith-pipeline@v0.9.0 · 5559 in / 1059 out tokens · 46084 ms · 2026-05-19T01:37:08.168224+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Main Theorem. In a tiling of the plane by congruent hexagons, if every vertex is the meeting place of at least three tiles, then the hexagon is a Reinhardt hexagon.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.