arxiv: 2604.26832 · v1 · submitted 2026-04-29 · 🧮 math.DS · cs.FL· math.MG

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Primitive Two-Dimensional Words and Iterated Pedal Triangles via Symbolic Coding

Taylor J. Smith

Pith reviewed 2026-05-07 10:48 UTC · model grok-4.3

classification 🧮 math.DS cs.FLmath.MG

keywords primitive two-dimensional wordspedal trianglessymbolic codingsorted pedal mapbranch itinerariescombinatorics on wordsdynamical systemsbijection

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The pith

A four-symbol coding of the sorted pedal map establishes a bijection between primitive two-dimensional words and triangles whose nth pedal triangle is the first similar one.

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

The paper connects combinatorics on words with the geometry of iterated pedal triangles by showing that a specific symbolic representation of the pedal map dynamics produces a one-to-one correspondence between the two sets. For each n at least 1, every primitive 2-by-n binary word maps to a unique triangle that becomes similar to its original form exactly at the nth pedal iteration, and every such triangle arises from exactly one word. This bijection is built from the branch itineraries of the four-symbol coding, which tracks how the sorted pedal map acts on the triangles. A reader would care because the numerical equality between the two families was already known; the coding now supplies an explicit matching that lets combinatorial techniques classify the triangles or lets geometric iteration reveal structure in the words.

Core claim

The central claim is that a finite four-symbol coding of the sorted pedal map is complete, injective, and surjective onto the primitive two-dimensional words while exactly preserving the condition that the nth pedal triangle is the first one similar to the original. The resulting branch itineraries therefore supply a bijection between the set of primitive 2-by-n words over a binary alphabet and the set of triangles whose first similar pedal triangle occurs at step n.

What carries the argument

The four-symbol coding of the sorted pedal map, which labels each branch of the map and generates itineraries that label both the words and the triangles while tracking the similarity condition at the nth iterate.

Load-bearing premise

The four-symbol coding must be a complete, injective, and surjective map from the primitive words onto the relevant triangles while preserving the exact location of the first similarity under pedal iteration.

What would settle it

An explicit primitive two-dimensional word of dimension 2 by n that cannot be realized as the itinerary of any triangle under the sorted pedal map, or a triangle whose nth pedal iterate is similar to the original but whose itinerary does not correspond to a primitive word.

Figures

Figures reproduced from arXiv: 2604.26832 by Taylor J. Smith.

**Figure 1.** Figure 1: A pedal triangle (in grey) constructed from a triangle (in white) view at source ↗

**Figure 2.** Figure 2: A pedal heptacycle given by Kingston and Synge [5]. Each pedal triangle view at source ↗

read the original abstract

The notion of a two-dimensional word arises naturally in the study of combinatorics on words, while the iterative construction of pedal triangles results in a rich dynamical system in the study of geometry. At first, these two classes of objects seem to be unrelated. However, it is known that for all $n \geq 1$, the number of primitive two-dimensional words of dimension $2 \times n$ over a binary alphabet agrees with the number of triangles whose first similar pedal triangle is their $n$th pedal triangle. We construct a finite four-symbol coding of the sorted pedal map and use the resulting branch itineraries to give a bijection between these two classes.

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

The paper turns the known count match into an explicit bijection by coding the sorted pedal map with four symbols and reading off branch itineraries.

read the letter

The main takeaway is that this work supplies a direct geometric-to-symbolic correspondence rather than just another enumeration. The authors partition the space of triangle shapes with a four-symbol coding on the sorted pedal map, then show that the itineraries under iteration give exactly the primitive 2D words of size 2 by n, with return times matching the first similarity of the nth pedal triangle. That step is new; the equality of counts was already observed, but the constructive bijection via itineraries was not.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs a finite four-symbol coding of the sorted pedal map on the space of triangle shapes. It then uses the branch itineraries under this coding to produce an explicit bijection between primitive two-dimensional words of dimension 2×n over a binary alphabet and the triangles whose first similar pedal triangle occurs precisely at the nth iteration, thereby accounting for the known equality of their cardinalities.

Significance. The explicit, parameter-free symbolic coding supplies a direct combinatorial-geometric correspondence that explains a previously observed numerical coincidence. Because the construction proceeds by defining symbols from the ordering of feet and vertex actions and then verifying itinerary-to-word and return-time-to-similarity mappings, the result is falsifiable and potentially extensible to other iterated geometric constructions.

minor comments (3)

[Section 3] The four-symbol alphabet is introduced in the main construction without an accompanying summary table listing the geometric meaning of each symbol; adding such a table would improve cross-referencing.
[Introduction] A few sentences in the introduction repeat the statement that the counts agree without citing the source of that numerical observation; a single reference would suffice.
[Section 2] The notation for the first-return time under the pedal map is used before its formal definition; moving the definition earlier or adding a forward reference would aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, accurate summary of the bijection via the four-symbol coding, and recommendation to accept. The significance statement correctly identifies the combinatorial-geometric correspondence as the core contribution.

Circularity Check

0 steps flagged

No significant circularity; explicit construction of bijection

full rationale

The paper states that the counts of primitive 2D words and certain pedal triangles agree (as background), then constructs a four-symbol coding of the sorted pedal map by partitioning based on ordering of feet and vertex actions. It defines symbols and itineraries explicitly, verifies injectivity/surjectivity onto primitive words of size 2×n, and shows that first-return times under the map encode the similarity condition for the nth pedal triangle. This is a direct, self-contained construction with no reduction to fitted parameters, self-definitional loops, or load-bearing self-citations; the bijection is established by explicit verification rather than by renaming or importing uniqueness from prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests primarily on standard domain assumptions from Euclidean geometry and combinatorics on words, with the main addition being the newly introduced four-symbol coding.

axioms (2)

domain assumption Standard properties of pedal triangles, iteration, and similarity in the Euclidean plane
Used to define the sorted pedal map and the condition that the nth pedal triangle is the first similar one.
domain assumption Definitions of two-dimensional words, primitivity, and binary alphabets in combinatorics on words
Used to define the set of primitive 2×n words whose count is matched by the geometric objects.

invented entities (1)

Finite four-symbol coding of the sorted pedal map no independent evidence
purpose: To encode the dynamics of iterated pedal triangles as symbolic sequences that correspond bijectively to primitive 2D words.
This coding is introduced in the paper as the key device for constructing the bijection.

pith-pipeline@v0.9.0 · 5403 in / 1404 out tokens · 66524 ms · 2026-05-07T10:48:49.761910+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references

[1]

Alexander

James C. Alexander. The symbolic dynamics of the sequence of pedal triangles.Math- ematics Magazine, 66(3):147–158, 1993

1993
[2]

Guilhem Gamard, Gwena¨ el Richomme, Jeffrey Shallit, and Taylor J. Smith. Periodicity in rectangular arrays.Information Processing Letters, 118:58–63, 2017

2017
[3]

Two-dimensional languages

Dora Giammarresi and Antonio Restivo. Two-dimensional languages. In G. Rozenberg and A. Salomaa, editors,Handbook of Formal Languages, volume 3, pages 215–267. Springer-Verlag, Berlin Heidelberg, 1997

1997
[4]

Hobson.A Treatise on Plane Trigonometry

Ernest W. Hobson.A Treatise on Plane Trigonometry. Cambridge University Press, Cambridge, 1891
[5]

Kingston and John L

John G. Kingston and John L. Synge. The sequence of pedal triangles.American Mathematical Monthly, 95(7):609–620, 1988

1988
[6]

Peter D. Lax. The ergodic character of sequences of pedal triangles.American Mathe- matical Monthly, 97(5):377–381, 1990

1990
[7]

Lothaire.Combinatorics on Words, volume 17 ofEncyclopedia of Mathematics and its Applications

M. Lothaire.Combinatorics on Words, volume 17 ofEncyclopedia of Mathematics and its Applications. Addison-Wesley, Reading, 1983

1983
[8]

M¨ obius

August F. M¨ obius. ¨Uber eine besondere Art von Umkehrung der Reihen.Journal f¨ ur die reine und angewandte Mathematik, 9:105–123, 1832
[9]

Computer Science and Applied Mathematics

Azriel Rosenfeld.Picture Languages: Formal Models for Picture Recognition. Computer Science and Applied Mathematics. Academic Press, New York, 1979

1979
[10]

Neil J. A. Sloane. The On-line Encyclopedia of Integer Sequences.http://oeis.org
[11]

Taylor J. Smith. Properties of two-dimensional words. Master’s thesis, University of Waterloo, 2017

2017
[12]

Mixing property of the pedal mapping.American Mathematical Monthly, 97(10):898–900, 1990

Peter Ungar. Mixing property of the pedal mapping.American Mathematical Monthly, 97(10):898–900, 1990

1990
[13]

¨Uber die Fußpunktdreiecke.Monatshefte f¨ ur Mathematik und Physik, 14(1):243–253, 1903

Gyula V´ alyi. ¨Uber die Fußpunktdreiecke.Monatshefte f¨ ur Mathematik und Physik, 14(1):243–253, 1903. 14

1903