arXiv:1612.09295v15

First Families of Regular Polygons and their Mutations

G. H. Hughes

math.DS37D4037E1052A1052C1597G4000A6651E1253A6037A10

Paper metadata

Original submission: December 29, 2016
Latest revision: August 9, 2026 · version 15
Claimed MSC: 37D40, 37E10, 52A10, 52C15, 97G40, 00A66, 51E12, 53A60, 37A10
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 81 pages with all images clickable. This paper is the capstone of H[1] to H[7] which span about 35 years. Appendix II discusses large scale invariance

Abstract

Every regular N-gon defines a canonical family of regular polygons which are conforming to the bounds of the 'star polygons' determined by N. These star polygons are formed from truncated extended edges of the N-gon and the intersection points ('star' points) determine the parameters of the family. The First Family Theorem (FFT) shows how each star[k] point defines a matching S[k] 'tile' and the GeneralizedFFT shows how every S[k] can generate their own families with an S[k+1] on the left and (if space permits) an S[k'] on the right where for N even k' = N/2-k. If such a tile does exist S[k] and S[k'] will be called a 'dual' pair and share an edge. If N is even the S[1] tile and N will form a dual pair since N = S[N/2-1] relative to the S[k]. When the outer-billiards map Tau is introduced in Section 4, this shared edge will form an barrier to the dynamics so S[k] and S[k'] will be on opposite sides of two invariant regions and the shared edge can be extended to form a separatrix between regions which is similar to an 'integral' curve when N is regarded as a harmonic oscillator H. The Invariance Conjecture implies there will be EulerPhi(N)/2) distinct Ik to match the 'primitive' star[k] with gcd(k,N) = 1. These curves (also known as Lagrangian graphs) define what we call the eigen-states of N. The Twice-Odd Lemma says that N and N/2 will share eigen-states. The assumption that each S[k] can have right and left side families in W is the Reflection Principle which says that W must evolve both right and left handed. This principle is actually a property of N itself but the Dihedral group D(N) is Non-Abelian with regard to reflections and rotations. Therefore every regular N-gon is 'chiral' and -W and +W are distinct properties of N. Mapping them to the union cannot be smooth since the Jacobian J = 0, and the resonances should correspond to the S[K] dual pairs as solitons.

AI-generated audit

Audit summary

Audited against arXiv v15

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements5 reported findingsContains unsupported statements

The exact star-point geometry, the First Family height formulas, the cyclotomic scaling-field basis, and the periodic orbit of each $S[k]$ center are correct. The manuscript appropriately labels its mutation, edge, step-sequence, and invariance programs as conjectures, and they are not treated as false claims here. Two theorem-level upgrades are not able to be verified: the text does not prove that the entire proposed $S[k]$ polygons and their generalized families survive as tiles of the limiting web, and the final assertion that every regular $N$ -gon has exactly $C_N=\varphi(N)/2$ realized dual pairs depends on the stated Invariance Conjecture and is followed only by an invalid chirality/Jacobian argument.

Theorem 2.1 and Lemma 3.1Correct

First Family construction and scaling

Pages 13–14 and 19 · First Family Theorem and First Family Scaling Lemma · arXiv:1612.09295v15

For even $N$ , reflection of the tangent function gives $s_{N/2-k}=1/s_k$ . Placing the center a half-side from $\operatorname{star}[k]$ therefore gives $\frac{h_{S[k]}}{h_N}=s_1s_k=\frac{s_1}{s_{N/2-k}},$ and the two specified star points determine the conforming polygon. For odd $N$ , passing to the conforming $2N$ -gon doubles the indices and yields $h_{S[k]}/h_N=s_2s_{2k}$ in the $2N$ notation. Dividing the $k=1$ formula by the general formula gives $h_{S[1]}/h_{S[k]}=s_1/s_k=\operatorname{scale}[k]$ . Apart from the small- $N$ wording correction reported below, these identities establish the geometric family claimed in the theorem.

Lemma 3.3 and Lemma 3.4Correct

Primitive scales form a unit basis of the maximal real cyclotomic subfield

Pages 20–22 · Scaling Field Lemma and Alternate Generators Lemma · arXiv:1612.09295v15

Writing $\zeta=e^{2\pi i/N}$ gives $\frac{\tan(\pi/N)}{\tan(k\pi/N)}=\frac{\zeta-1}{\zeta+1}\frac{\zeta^k+1}{\zeta^k-1}.$ For $\gcd(k,N)=1$ , both this element and its inverse are algebraic integers because the relevant factors are quotients of Galois-conjugate cyclotomic integers. There are $\varphi(N)/2$ such scales with $1\leq k<N/2$ . Dividing a rational dependence among them by $s_1$ would give a dependence among the primitive cotangent values, which are linearly independent; hence the scales are a basis of $\mathbb Q(\zeta+\zeta^{-1})$ . The Möbius relation between $\tan^2(\pi/N)$ and $2\cos(2\pi/N)$ also proves that the former generates the same real field.

Girstmair, Some linear relations between values of trigonometric functions at kπ/n ↗

Lemma 4.1Correct

The centers of the First Family polygons have the stated periodic orbits

Pages 25–26 · Canonical orbits of $S[k]$ centers · arXiv:1612.09295v15

The similarity called $Du$ sends the midpoint polygon of $N$ to $N$ and sends the edge orbit of $\operatorname{star}[k]$ to a vertex-supported outer-billiards orbit. The star-polygon orbit advances by $k$ vertices and has length $N/\gcd(k,N)$ . Substitution of the First Family height formula identifies $Du(\operatorname{star}[k])$ with the center $cS[k]$ , so that center has exactly the asserted itinerary and period. This statement concerns the center; it does not by itself prove that every point of the proposed polygon shares the itinerary.

Web-preservation claim for the $S[k]$ and generalized familiesNot able to verify

A center orbit does not establish survival of an entire polygonal tile

Pages 24–32 · opening of Section 4, Lemma 4.1, Evolution of the Web Parts I–II, and use of the GFFT · arXiv:1612.09295v15

The section announces that every First Family $S[k]$ exists in the complement of the limiting singularity set and that its effective star points support the GFFT family. Lemma 4.1 proves only the itinerary of $cS[k]$ . To promote this to a tile, one must show that every point in the proposed polygon remains in the same sequence of continuity domains and that no iterate of its interior meets a singular ray. The subsequent `shear and rotate' discussion argues from one-sided continuity that a portion of each base segment survives, but gives no domain inequalities over a full cycle and no induction identifying the surviving portion with all edges of the displayed $S[k]$ . Part II itself says the step evolution `should imply' the asserted surviving star points. Thus the full-tile and generalized-family conclusions remain unsupported, although the center-orbit theorem is valid.

Final unnumbered theoremNot able to verify

The count of realized dual pairs is conditional on the Invariance Conjecture

Page 80 · theorem beginning `Every regular $N$ -gon will have $C_N$ distinct dual pairs' · arXiv:1612.09295v15

The arithmetic set of primitive indices has cardinality $C_N=\varphi(N)/2$ , but the theorem concerns realized adjacent $S[k]$ – $S[k^\ast]$ pairs in the web. The manuscript's abstract explicitly says that the Invariance Conjecture implies this count, and Appendix II introduces the relevant reflection behavior as an assumption before giving extensive examples and tables. No argument proves for arbitrary $N$ that every primitive index produces an actual adjacent pair, or that no further pair occurs. The final paragraph's chirality and Jacobian assertions do not fill that obligation and are formally false as explained in the proofs findings. The statement should remain a conjecture or be made conditional on the Invariance Conjecture unless an independent existence-and-exhaustion proof is supplied.

02Proofs8 reported findingsContains incorrect or incomplete proofs

The trigonometric and cyclotomic proofs in Sections 1–3 and the center-orbit proof in Lemma 4.1 are correct. The arguments that extend center dynamics to full polygonal tiles and the symbolic calculations selected from approximate itineraries are incomplete. The proposed proof of the final dual-pair theorem is incorrect: a regular polygon has reflection symmetry, the dihedral relation is $fr=r^{-1}f$ , and every continuity branch of outer billiards has Jacobian determinant $1$ , not $0$ . Several local wording errors have uniquely determined repairs and are reported separately without changing the overall statements status.

Sections 1–3Correct and complete

Star-point, First Family, and scaling-field calculations

Pages 7–23 · Lemmas 1.1–3.5 and Theorem 2.1 · arXiv:1612.09295v15

The scaling identity is the direct tangent quotient $\frac{\tan(k\pi/N)}{\tan(jk\pi/N)}=\frac{\operatorname{scale}_N[jk]}{\operatorname{scale}_N[k]},$ the First Family formulas follow from complementary tangent indices, and the cyclotomic-unit argument plus primitive-cotangent independence supplies the required basis cardinality and independence. The small domain and labeling qualifications reported below are local and do not alter these core derivations.

GFFT and web-survival argumentIncomplete as written

The proof constructs candidate polygons but not their claimed dynamical survival

Pages 16–18 and 27–32 · Lemma 2.3 and Evolution of the Web Parts I–II · arXiv:1612.09295v15

Lemma 2.3 correctly computes the height and location of a candidate conforming polygon once a star point of $S[k]$ is declared effective. It does not prove which star points survive the singularity web. The later argument assumes the local web advances by the proposed step and then infers that the same arithmetic progression of star points survives; it does not prove that the intervening segments avoid all competing outer-billiards domains. A complete repair needs explicit half-plane/domain inequalities for every step of the cycle, followed by an induction showing that the full claimed boundary—and not merely a nonempty subsegment—survives.

Lemma 4.2Incorrect as written · verified local repair

Affine equivariance is conjugacy, not commutation

Page 26 · proof of the Twice-Odd Lemma · arXiv:1612.09295v15

The proof says that $T$ commutes with the outer-billiards map merely because both are affine. Arbitrary affine maps do not commute. The intended conclusion is nevertheless obtained by affine equivariance: if $P'=T(P)$ , support lines and midpoints are preserved, so $T\circ\tau_P=\tau_{P'}\circ T.$ Once the manuscript's displayed $T$ is checked to send the embedded $N/2$ polygon to the corresponding $N$ polygon, this conjugacy maps the local singularity webs. Replacing commutation by this conjugacy argument repairs the lemma locally.

Appendix I exact calculationsIncomplete as written

Approximate surrogate orbits do not certify the symbolic itinerary

Pages 48–53 · Example A1, parts (i)–(iii) · arXiv:1612.09295v15

The exact formulas for the candidate $M_x$ parameters are reconstructed from symbolic sequences first obtained by iterating 8- or 30-decimal approximations. For a discontinuous piecewise isometry, a single sign or atom error changes the exact reconstruction. The manuscript says errors are easy to detect and that survival is clear by inspection, but supplies neither interval enclosures nor exact signed-distance bounds proving that every one of the 150, 140, or 600 iterates stays in the selected atom. The algebra after fixing an itinerary is exact; the itinerary itself is not rigorously certified. A repair is to propagate rational or algebraic interval bounds and verify a nonzero margin from every singular boundary at every listed iterate.

Proof following the final dual-pair theoremIncorrect as written

The chirality and zero-Jacobian argument contradicts the map's defining formula

Pages 79–80 · paragraphs preceding and following the final theorem · arXiv:1612.09295v15

A regular unmarked $N$ -gon is carried to its reflected copy by an isometry already contained in its dihedral symmetry group, so clockwise and counterclockwise drawings do not provide the asserted geometric chirality. The dihedral relation is $fr=r^{-1}f$ , not $f\circ r=-r\circ f$ . More decisively, Definition 4.1 gives each continuity branch as $\tau(p)=2v-p$ , whose derivative is $-I_2$ and whose Jacobian determinant is $1$ , not $0$ . Thus the claimed failure of smoothness does not imply the existence of any dual pair, much less exactly $C_N$ pairs. Repair classification: no repair of the theorem is supplied; the manuscript would need a separate existence-and-exhaustion argument.

Lemma 1.3 and Theorem 2.1Minor formal correction

The star-point labels and the small- $N$ range must be stated

Pages 8 and 13–14 · Two-Star Lemma and First Family Theorem · arXiv:1612.09295v15

Two coordinates determine height through $h=(p_2-p_1)/(s_k\pm s_j)$ only when their indices $j,k$ and their same-side/opposite-side relation are known, as the proof itself states; two unlabeled points admit several interpretations. Also, for $N=3$ or $4$ there is no second distinct star point with which $S[1]$ can be `strongly conforming' under Definition 2.1. State the Two-Star Lemma for two labeled star points and either restrict the strong-conformity clause of Theorem 2.1 to $N\geq5$ or explicitly exempt the degenerate small cases. These local changes leave the height formulas unchanged.

Corollary 3.1Minor formal correction

The printed index range includes an undefined tangent

Page 22 · Corollary 3.1 · arXiv:1612.09295v15

The statement says every $s_j$ with $j<N$ is a rational linear combination of primitive $s_k$ . When $N$ is even, $j=N/2$ is included but $s_{N/2}=\tan(\pi/2)$ is undefined. Replace the range by $1\leq j<N/2$ ; values on the other side follow from tangent symmetry where defined. The field-basis proof then applies exactly as written.

Discussion following the Invariance ConjectureTypo

`Algebraic integer' should read `algebraic unit'

Page 64 · comparison of $s_1^2$ with $\lambda_N=2\cos(2\pi/N)$ · arXiv:1612.09295v15

The manuscript says the traditional generator $\lambda_N$ may fail to be an algebraic integer. In fact $\lambda_N=\zeta_N+\zeta_N^{-1}$ is always an algebraic integer. The surrounding comparison is about selecting a unit generator, and $\lambda_N$ need not be a unit. Replacing `integer' by `unit' gives the uniquely consistent statement and does not affect any proof.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:1612.09295v15
Authors listed: G. H. Hughes
Audit date: August 18, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. For every adverse or unverified outcome, restate the exact claim and hypotheses, identify its location in the reviewed version, and give the decisive reasoning. Do not infer that a theorem is false merely because its printed proof is defective. Assign the overall statements status by the following priority: 1. Contains wrong statements, if at least one central statement is Incorrect. 2. Contains unsupported statements, if no central statement is Incorrect but at least one is Not able to verify. 3. Correct, if every central statement is Correct, apart from any typographical or minor formal findings allowed by Part 4. PART 2 — CORRECTNESS AND COMPLETENESS OF THE PROOFS Audit the proofs of the central statements and every material internal or cited input on which they depend. For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. 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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. 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P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

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First Families of Regular Polygons and their Mutations

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

First Family construction and scaling

Primitive scales form a unit basis of the maximal real cyclotomic subfield

The centers of the First Family polygons have the stated periodic orbits

A center orbit does not establish survival of an entire polygonal tile

The count of realized dual pairs is conditional on the Invariance Conjecture

Star-point, First Family, and scaling-field calculations

The proof constructs candidate polygons but not their claimed dynamical survival

Affine equivariance is conjugacy, not commutation

Approximate surrogate orbits do not certify the symbolic itinerary

The chirality and zero-Jacobian argument contradicts the map's defining formula

The star-point labels and the small-NNN range must be stated

The printed index range includes an undefined tangent

`Algebraic integer' should read `algebraic unit'

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The star-point labels and the small- $N$ range must be stated