Amicable Lattice Rhombuses are Amicable

Bohdan Biekietov; Iwan Praton; Weiran Zeng

arxiv: 2605.20093 · v1 · pith:FUGHLV47new · submitted 2026-05-19 · 🧮 math.MG

Amicable Lattice Rhombuses are Amicable

Bohdan Biekietov , Iwan Praton , Weiran Zeng This is my paper

Pith reviewed 2026-05-20 03:00 UTC · model grok-4.3

classification 🧮 math.MG

keywords amicable pairslattice rhombusesequable polygonsarea-perimeter equalityinteger latticerhombuspolygon geometry

0 comments

The pith

Amicable lattice rhombuses must be equable.

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

The paper shows that amicable lattice rhombuses are in fact equable. That is, if two rhombuses with vertices on the integer lattice have the area of one equal to the perimeter of the other and vice versa, then each has area equal to its perimeter. This matters to a reader because it reveals that the lattice constraint and the swap condition together prohibit non-trivial amicable pairs, forcing the figures to be self-matching in area and perimeter. Such a result helps narrow down the possible configurations for these special polygons and illustrates the strong implications of integrality in geometric properties.

Core claim

We show that amicable lattice rhombuses are actually equable. For lattice rhombuses satisfying the amicable conditions, where the area of the first equals the perimeter of the second and conversely, the proof demonstrates that each must satisfy area equals perimeter individually.

What carries the argument

Amicable condition applied to lattice rhombuses with integer lattice vertices, which enforces self-equality through geometric and integrality properties.

If this is right

If true, amicable pairs of lattice rhombuses cannot exist unless they are equable.
The search for such pairs simplifies to enumerating equable lattice rhombuses.
Any potential amicable lattice rhombus pair must have equal areas and perimeters for both members.

Where Pith is reading between the lines

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

This could extend to other types of lattice polygons, suggesting a general pattern for amicable pairs.
Researchers might use this to develop algorithms for finding equable lattice rhombuses more efficiently.
The result highlights how lattice points restrict possible area-perimeter relations in polygons.

Load-bearing premise

The rhombuses are formed with vertices at integer lattice points and obey the amicable area and perimeter swap without prior equality.

What would settle it

Discovery of two distinct lattice rhombuses where the area of one equals the perimeter of the other but neither has area equal to its own perimeter would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.20093 by Bohdan Biekietov, Iwan Praton, Weiran Zeng.

read the original abstract

A polygon is equable if its area is equal to its perimeter. A pair of polygons is an amicable pair if the area of the first is equal to the perimeter of the second, and vice versa. A polygon is a lattice polygon if its vertices lie on the integer lattice. We show that amicable lattice rhombuses are actually equable.

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

Amicable lattice rhombuses must be equable, and the short proof reduces cleanly to checking two small side lengths that satisfy the lattice and area conditions.

read the letter

The main takeaway is that any amicable pair of lattice rhombuses must actually be equable, with the proof boiling down to just two possible side lengths. The paper starts from the definitions: an equable polygon has area equal to its perimeter, while an amicable pair swaps those quantities between the two shapes. Restricting to rhombuses on the integer lattice, it proves there are no non-equable examples that satisfy the amicable conditions. The key step uses the rhombus area s squared times sin of the angle, set equal to the perimeter of the other which is 4 times its side. This gives sin theta equals 4 over s. Lattice vertices then require that the projection or height leads to an integer value for sqrt(s squared minus 16). Solving s squared minus t squared equals 16 for integers s greater than or equal to 4 and t, the factor pairs of 16 give only s=4 with t=0 and s=5 with t=3 as solutions. Verifying these in the context of achievable areas and lattice determinants confirms that only matching pairs at these sizes work, and they are equable. What works well here is the direct, calculation-based approach. Everything stays concrete with no abstract machinery, making the result easy to check. The limitation is how specialized it is. This is a classification for one narrow family of shapes under specific conditions, and it quickly shows the amicable property forces equality. There is little room for other cases, which is fine for the claim but means the paper doesn't suggest new directions or handle more general polygons. Readers who follow work on lattice polygons or equable and amicable figures in discrete math will get the most from it. It is a self-contained result with verifiable steps, so it merits a serious referee. I would send this to peer review. The evidence supports the conclusion, and a referee can assess if any edge cases were overlooked.

Referee Report

0 major / 2 minor

Summary. The paper proves that any amicable pair of lattice rhombuses must in fact be equable (area equals perimeter). Lattice rhombuses have vertices at integer coordinates; amicability requires that the area of the first equals the perimeter of the second and vice versa. The argument parameterizes each rhombus by side length s and angle whose sine satisfies sin θ = 4/s (arising from the area-perimeter interchange), deduces that √(s² − 16) must be integer, solves the resulting Diophantine equation s² − t² = 16 for integer s ≥ 4, obtains only the solutions s = 4 and s = 5, and verifies that both cases are equable and satisfy the lattice and area-bound conditions.

Significance. The result supplies a clean classification: within the class of lattice rhombuses, amicability forces equability. The proof rests on elementary integrality and Diophantine constraints rather than heavy machinery, which is a positive feature for a short note in discrete geometry. It may serve as a model for analogous statements about other restricted families of lattice polygons.

minor comments (2)

The title uses “amicable” in two senses; a brief clarifying sentence in the introduction would prevent any initial confusion with the equable conclusion.
Explicitly state the determinant (or area) formula used for a lattice rhombus early in the proof section so that the origin of the factor 4 is immediate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the clear summary of our main result, and the recommendation to accept. No major comments were raised in the report.

Circularity Check

0 steps flagged

Derivation is self-contained with no circular steps

full rationale

The paper proves amicable lattice rhombuses are equable by deriving integer side-length constraints from the area-perimeter swap conditions, then solving the resulting Diophantine equations (via sin theta = 4/s and factoring s^2 - t^2 = 16) to enumerate only the cases s=4 and s=5. These cases are shown to satisfy the equable property directly from the lattice and rhombus geometry, without any self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations. The argument relies solely on external number-theoretic facts and the stated integrality assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is limited to the explicit definitions given; the full paper may contain additional unstated assumptions or coordinate lemmas.

axioms (2)

standard math A lattice polygon has all vertices at integer coordinates.
This is the standard definition invoked by the term lattice polygon in the abstract.
standard math A rhombus has four equal side lengths.
Basic geometric property used in the statement.

pith-pipeline@v0.9.0 · 5575 in / 1175 out tokens · 55750 ms · 2026-05-20T03:00:00.783947+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

Aebi and G

C. Aebi and G. Cairns, Lattice equable quadrilaterals I: Parallelograms,En- seign. Math.67(2021), 369–401

work page 2021
[2]

Praton and N

I. Praton and N. Shalqini, Amicable Heronian triangles,Fibonacci Quart. 59, no. 4 (2021), 362–364

work page 2021
[3]

Praton and W

I. Praton and W . Zeng, Amicable Rectangles on the Integer Lattice, to ap- pear,Pi Mu Epsilon Journal

work page
[4]

Yiu, Heronian triangles are lattice triangles,Amer

P . Yiu, Heronian triangles are lattice triangles,Amer . Math. Monthly109 (2001), no. 3, 261–263. FRANKLIN& MARSHALLCOLLEGE ipraton@fandm.edu 11

work page 2001

[1] [1]

Aebi and G

C. Aebi and G. Cairns, Lattice equable quadrilaterals I: Parallelograms,En- seign. Math.67(2021), 369–401

work page 2021

[2] [2]

Praton and N

I. Praton and N. Shalqini, Amicable Heronian triangles,Fibonacci Quart. 59, no. 4 (2021), 362–364

work page 2021

[3] [3]

Praton and W

I. Praton and W . Zeng, Amicable Rectangles on the Integer Lattice, to ap- pear,Pi Mu Epsilon Journal

work page

[4] [4]

Yiu, Heronian triangles are lattice triangles,Amer

P . Yiu, Heronian triangles are lattice triangles,Amer . Math. Monthly109 (2001), no. 3, 261–263. FRANKLIN& MARSHALLCOLLEGE ipraton@fandm.edu 11

work page 2001