research/AMR-005-0010.md

---
id: AMR-005-0010
classification: PARTIAL-PROGRESS
wording_corrected: yes
---

# AMR-005-0010 — Unbounded unicycle tracks (Finn's construction)

## Problem (corrected statement if needed)

Source: S. Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1 (2015), 59–67, DOI 10.1007/s40598-014-0001-3 (Section 9 "The Unicycle Problem", Conjecture 3). Original wording, verified against the published article:

> The following construction is due to D. Finn. Let $\gamma(t), t\in[0,L]$ be an arc length parameterized smooth curve in the plane which coincides with all derivatives, for $t=0$ and $t=L$, with the $x$-axis at points $(0,0)$ and $(1,0)$, respectively. One uses $\gamma$ as a "seed" trajectory of the rear wheel of a bicycle. Then the new curve $\Gamma = T(\gamma) = \gamma + \gamma'$ is also tangent to the horizontal axis with all derivatives at its end points $(1,0)$ and $(2,0)$. One can iterate this procedure yielding a smooth infinite forward bicycle trajectory $\mathcal{T}$ such that the tracks of the rear and the front wheels coincide.
> **Conjecture 3.** Unless $\gamma$ is a straight segment, the amplitude of the curve $\mathcal{T}$ is unbounded, i.e., $\mathcal{T}$ is not contained in any horizontal strip; $\mathcal{T}$ is not a graph [sic: "grap;" in the published text]; and $\mathcal{T}$ is not embedded, that is, it starts to intersect itself.

Corrections to the dataset transcription: essentially faithful; the dataset silently fixed the published typo "not a grap;" to "not a graph". Note that the conjecture has **three distinct clauses**: (i) *vertical* amplitude unbounded (not contained in any horizontal strip); (ii) $\mathcal{T}$ is eventually not a graph of a function $y=f(x)$; (iii) $\mathcal{T}$ is not embedded (self-intersections appear). Their literature status differs (see below), which the single-sentence transcription obscures.

## Status / Literature

All references verified via Crossref, the arXiv API, and publisher pages (abstracts/full text seen verbatim).

- **Origin of the construction.** D. L. Finn, "Can a Bicycle Create a Unicycle Track?", *College Math. J.* 33 (2002), 283–292, DOI 10.1080/07468342.2002.11921954 (verified via Crossref). Finn's seed-and-iterate construction of "unicycle tracks".
- **Oscillation growth (proved 2009).** M. Levi, S. Tabachnikov, "On bicycle tire tracks geometry, hatchet planimeter, Menzin's conjecture and oscillation of unicycle tracks", *Exp. Math.* 18 (2009), 173–186, DOI 10.1080/10586458.2009.10128894, arXiv:0801.4396 (DOI seen in the Crossref reference list of the source article; abstract seen via arXiv API). Establishes: each next arc $\gamma_n$ of $\mathcal{T}$ has strictly more intersections with the $x$-axis, more local extrema of the height function, and more inflection points than the previous one; also that a unicycle track cannot be extended backward indefinitely.
- **"Not a graph" clause — PROVED (2025 preprint).** I. Molodyk, "On the Complexity of Horizontal Unitracks", arXiv:2510.10388 (v1, 12 Oct 2025; abstract and full HTML text read verbatim). Theorem 4.2: unless $\gamma_0$ is a straight segment, the iterates $\gamma_n = \varphi^n(\gamma_0)$ cannot all remain graphs of smooth functions. Theorem 4.3: the *horizontal* amplitude $H(\gamma_n)$ is non-decreasing and grows linearly, $n - c_1 \le H(\gamma_n) \le 2n - c_2$ for constants $c_1,c_2$ depending on $\gamma_0$; consequently the length of $\gamma_n$ tends to infinity. Proof idea for Theorem 4.2: assuming all $\gamma_n$ are graphs, the horizontal coordinates $x_n(t)$ form a decreasing sequence with a monotone limit $L$; slope estimates via $s_n(x) = 1 - \cos\arctan f_n'(x)$ and the length-monotonicity $\operatorname{Len}(\varphi(\gamma)) \ge \operatorname{Len}(\gamma)$ (strict unless $\gamma$ is straight) force $\operatorname{Len}(\gamma_0) \le 1$, which for a curve joining $(0,0)$ to $(1,0)$ forces $\gamma_0$ to be the straight segment. The author acknowledges Tabachnikov as advisor. **Caveat: preprint, not yet peer-reviewed as of this writing.**
- **Vertical amplitude and self-intersection clauses — still OPEN.** In the same preprint these are stated explicitly as open: Conjecture 4.4 ($V(\gamma_n)$ unbounded — exactly clause (i) of Tabachnikov's Conjecture 3), Conjecture 4.5 (some $\gamma_n$ has self-intersections) and the weaker Conjecture 4.6 (the full track $\mathcal{T}$ self-intersects — clause (iii)). Molodyk does prove $V(\gamma_n)$ is (strictly) increasing for non-trivial seeds.
- **Related recent work.** S. Wagon, "A Spiral Bicycle Track that Can Be Traced by a Unicycle", arXiv:2503.11847 (2025; abstract seen via arXiv API): numerical evidence (unibike error $<10^{-7}$) that iterating Finn's construction on the polar square-root curve converges to a spiral-shaped unibike curve — consistent with, but not resolving, the growth conjectures.
- **Discrete analogues.** A SUMMER@ICERM 2012 undergraduate report, "On Bicycle Uni-track Path Efficiency: Bicycle 'Unicycle' Paths" (icerm.brown.edu/summerug/2012/cmj_bicycle_unicycle_paths.pdf), claims proofs of amplitude growth and failure of embedding for *discrete* unicycle paths built from line segments and circle arcs. I could not extract the PDF (fetch failed twice) and could not verify authorship or details; treat as unverified supporting evidence.

## Work done

- Retrieved the original statement from the published AMJ article (publisher HTML at amj.math.stonybrook.edu) and confirmed the dataset transcription is faithful (modulo the "grap;" typo fix). Verified the source article's metadata via Crossref (DOI 10.1007/s40598-014-0001-3).
- Verified Finn (2002) and Levi–Tabachnikov (2009) via Crossref; verified arXiv records 0801.4396, 2503.11847, 1602.06455, 1211.2345 via the arXiv API.
- Read the full text of arXiv:2510.10388 (Molodyk, Oct 2025), which is the decisive recent progress, and mapped its Theorems 4.2/4.3 and Conjectures 4.4–4.6 onto the three clauses of Tabachnikov's Conjecture 3.
- Reasoned about the remaining open clause (i): the signed area between $\gamma_n$ and the $x$-axis is preserved under iteration (Theorem C of Molodyk's introduction), while the number of zeros of $\gamma_n$ strictly increases (Levi–Tabachnikov). These two facts are *consistent* with bounded vertical amplitude (ever denser oscillations of bounded height), so the area invariant alone cannot force $V(\gamma_n)\to\infty$; any proof must exploit finer structure (e.g. curvature blow-up near the vertical tangencies whose existence Molodyk proves).

## Result

The conjecture is **partially resolved in the literature** as of October 2025:

1. **Clause (ii) "not a graph" — settled affirmatively** (Molodyk, arXiv:2510.10388, Theorem 4.2): every non-trivial seed eventually produces an iterate that is not a graph of a function.
2. **A strong quantitative substitute for unboundedness** (ibid., Theorem 4.3): the horizontal amplitude grows linearly, $n-c_1 \le H(\gamma_n) \le 2n-c_2$, so in the horizontal direction the track escapes every vertical strip; lengths of the arcs tend to infinity.
3. **Clause (i) "not contained in any horizontal strip" — open**: $V(\gamma_n)$ is known to be strictly increasing, but unboundedness is Conjecture 4.4 of the 2025 preprint.
4. **Clause (iii) "not embedded" — open**: even the weaker statement that the full track $\mathcal{T}$ self-intersects is listed as open (Conjectures 4.5–4.6 ibid.). Levi–Tabachnikov's growth of zeros, extrema and inflection points, plus numerical evidence (Wagon 2025), strongly support it.

## What remains

- Prove $V(\gamma_n) \to \infty$ (vertical amplitude), the literal "amplitude" clause of Conjecture 3. Obstacle identified above: the preserved signed area and the growing oscillation count do not by themselves preclude bounded height.
- Prove self-intersection of $\mathcal{T}$ (Molodyk's Conjectures 4.6, and the stronger 4.5). Molodyk's vertical-tangency mechanism (the leftmost point of $\gamma_n$ has vertical tangent for all large $n$ and marches left by 1–2 units per step) looks like the natural entry point: the track folds back over earlier arcs, but a rigorous intersection argument is missing.
- Peer review / publication status of arXiv:2510.10388 should be checked before citing clause (ii) as a theorem in the strongest sense.
- The analogous conjecture for the *circular* version (Conjecture 4 of the source: iterates of $\gamma \mapsto$ endpoints of unit tangent segments all convex $\Rightarrow$ circle) is a separate open item, not treated here.