Regular triangle unions with maximal number of sides
Pith reviewed 2026-05-18 04:39 UTC · model grok-4.3
The pith
The maximum number of sides in a union of n triangles with cyclic boundary vertices grows linearly and is combinatorially realizable.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Any regular union of n triangles can be modeled by starting with a triangulation of a regular (n+1)-gon and applying a sequence of triangulation shifts. These shifts are dynamical mutations that maintain the cyclic ordering of the boundary vertices and ensure the resulting figure remains a simple polygon. Using this model, the maximum number of sides is shown to satisfy linear bounds, with the upper bound achieved for the sequence beginning 3, 12, 22, 33, 45, 56, 67, 80, 91.
What carries the argument
triangulation shifts: dynamical mutations applied to triangulations of a regular (n+1)-gon that preserve cyclic boundary conditions and simplicity
If this is right
- The maximal side count for n-triangle unions satisfies explicit linear lower and upper bounds.
- The upper bound is realized for all n through combinatorial constructions based on triangulation shifts.
- This provides a new method for analyzing the complexity of unions of triangles with cyclic boundaries.
- The initial terms of the maximal sequence are 3, 12, 22, 33, 45, 56, 67, 80, 91.
Where Pith is reading between the lines
- The combinatorial constructions may extend to realizations in pseudoline geometry.
- If realizable with pseudolines, the arrangements might be stretchable to the Euclidean plane.
- Similar shift techniques could help address related problems such as the Kobon triangle problem or lower envelope complexities.
Load-bearing premise
Any such union of n triangles admits a representation as a triangulation of a regular (n+1)-gon modified by a sequence of triangulation shifts that preserve the cyclic order on the boundary and the simplicity of the polygon.
What would settle it
A regular n-triangle union with more sides than the established linear upper bound, or one that cannot be derived from any sequence of triangulation shifts from a regular (n+1)-gon triangulation.
read the original abstract
Fix an integer n>=1. Suppose that a simple polygon is the union of n triangles whose vertices along the common boundary are arranged cyclically. How many sides can such a union -- to be called regular -- have at most? This gives OEIS sequence A375986, a recent entry. It will be shown here that the sequence begins 3, 12, 22, 33, 45, 56, 67, 80, 91, and satisfies linear lower and upper bounds. The latter is not merely an estimate: it is realizable combinatorially. This leads to two further questions: can the same combinatorics be realized in pseudoline geometry, and if so, can such a realization be stretched? The paper is largely expository, with excursions into neighboring topics (union complexity, the Zone Theorem, stretchability, the Kobon triangle problem, Davenport-Schinzel sequences, lower envelopes of line segments). However, it adds a new tool tailored for studying regular unions; namely, triangulation shifts. In essence, this is a method to represent any such n-union by a triangulation of a regular (n+1)-gon and its dynamical mutation.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the maximum number of sides of a simple polygon formed as the union of n triangles with cyclically arranged boundary vertices (termed a regular n-triangle union). It reports the initial segment of the maximal side-count sequence as 3, 12, 22, 33, 45, 56, 67, 80, 91, establishes linear lower and upper bounds on this quantity, and shows that the upper bound is achieved by an explicit combinatorial construction. The construction represents each such union via a triangulation of a regular (n+1)-gon together with a sequence of dynamical mutations called triangulation shifts; the paper is largely expository but introduces these shifts as a tailored tool and discusses connections to union complexity, the Zone Theorem, stretchability, the Kobon triangle problem, and Davenport-Schinzel sequences.
Significance. If the central claims hold, the work supplies concrete, tight bounds together with a realizable combinatorial model for the maximal complexity of regular triangle unions. The explicit sequence, the parameter-free linear bounds, and the constructive realization of the upper bound constitute clear strengths. The introduction of triangulation shifts provides a new representational tool that may extend to neighboring questions in combinatorial geometry, including pseudoline realizations and stretchability.
major comments (1)
- [Section introducing triangulation shifts and the construction of the upper bound] The validity of the enumerated maxima (including the term 91) and the combinatorial realization of the upper bound both rest on the claim that every sequence of triangulation shifts preserves simplicity of the boundary polygon and the cyclic order of its vertices. A dedicated subsection or lemma should supply an inductive argument or exhaustive verification that no admissible shift sequence produces a self-intersection or violates cyclicity; without this, the reported counts cannot be confirmed to correspond to actual regular unions.
minor comments (2)
- [Abstract] The abstract lists the initial terms but does not state the corresponding values of n; adding this mapping (e.g., for n = 1 to 9) would improve immediate readability.
- [Main body] Notation for the side-count function and for the shift operation should be introduced once and used consistently; occasional redefinition of symbols interrupts the flow.
Simulated Author's Rebuttal
We thank the referee for the careful reading and the recommendation of major revision. The central concern is the absence of an explicit argument that sequences of triangulation shifts preserve simplicity of the boundary and cyclic vertex order. We agree that this property requires a dedicated, self-contained treatment and will revise the manuscript to supply it.
read point-by-point responses
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Referee: The validity of the enumerated maxima (including the term 91) and the combinatorial realization of the upper bound both rest on the claim that every sequence of triangulation shifts preserves simplicity of the boundary polygon and the cyclic order of its vertices. A dedicated subsection or lemma should supply an inductive argument or exhaustive verification that no admissible shift sequence produces a self-intersection or violates cyclicity; without this, the reported counts cannot be confirmed to correspond to actual regular unions.
Authors: We acknowledge that the manuscript introduces triangulation shifts as combinatorial mutations on a triangulation of the regular (n+1)-gon but does not isolate a formal invariance proof in its own subsection. Each shift is a local edge rotation or triangle replacement performed inside the current non-crossing triangulation; by construction the operation only affects edges that remain internal to the union or lie on the boundary without crossing existing ones. Nevertheless, we accept the referee’s point that an explicit inductive argument is needed to confirm that every admissible sequence yields a simple polygon whose vertices stay in cyclic order. We will insert a new lemma (or short subsection) immediately after the definition of triangulation shifts. The lemma states: for any n, the initial triangulation of the (n+1)-gon corresponds to a simple regular 1-triangle union; if a configuration obtained after k admissible shifts is simple and cyclically ordered, then any single admissible shift preserves both properties. The inductive step follows from the planarity of the underlying triangulation and the regularity condition that prevents three diagonals from meeting at an interior point. This addition will rigorously justify the reported initial segment, including the value 91, as arising from geometrically realizable regular unions. revision: yes
Circularity Check
No significant circularity detected
full rationale
The paper defines regular n-triangle unions directly as simple polygons that are unions of n triangles with cyclically arranged boundary vertices. It then introduces triangulation shifts as an explicit representational tool to model any such union via a triangulation of a regular (n+1)-gon together with mutations. The reported sequence values and the linear lower/upper bounds are obtained from explicit combinatorial constructions that realize the upper bound; these constructions are not obtained by fitting parameters to a subset of the same data or by redefining the target quantity in terms of itself. No load-bearing step reduces by the paper's own equations or citations to an input that is definitionally equivalent to the claimed output. The derivation is therefore self-contained.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The union of the n triangles is a simple polygon whose vertices along the common boundary are arranged cyclically.
invented entities (1)
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triangulation shifts
no independent evidence
discussion (0)
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