Baker's Dozen — Unbounded unicycle tracks

Definitions and mathematical statement

Choose a smooth regular plane arc $\gamma_0:[0,L_0]\to\mathbb R^2$ parametrized by arclength, with endpoints $(0,0)$ and $(1,0)$. At both endpoints its first derivative is $(1,0)$ and all higher derivatives vanish. For any oriented regular arc $\alpha$, define the tangent-offset arc \[ T(\alpha)(s)=\alpha(s)+\frac{\alpha'(s)}{\|\alpha'(s)\|}. \] Set $\gamma_{j+1}=T(\gamma_j)$, reparametrizing by arclength before the next iteration. The endpoints of $\gamma_j$ are $(j,0)$ and $(j+1,0)$, and their endpoint jets agree. Concatenating the arcs in this order gives a smooth forward curve $\mathcal T$. The unit tangent, rather than an arbitrary-speed derivative, is essential to this geometric construction.

Unless $\gamma_0$ is the horizontal straight segment, the source conjectures all three conclusions: the heights on $\mathcal T$ are not bounded in any horizontal strip; its image is not the graph of a single-valued function of the horizontal coordinate; and its parametrization is not injective, so two different traversal times give a self-intersection. The last condition is stronger than simply ceasing to be a graph.

Short English statement

Repeatedly move an arc one unit along its tangent and join the resulting arcs. Must every nonstraight seed eventually produce unbounded vertical excursions, multiple heights at one horizontal position, and self-intersection?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 600010 (AMR-005-0010), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Tabachnikov - A Baker's Dozen of Problems (2015), Conjecture 3. Original source indexed by AMR-005-0010. https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/.