17 problems

Let $\gamma$ be a periodic trajectory in a cyclic quadrilateral tiling, and let $\Omega_{\gamma}$ be the domain bounded by it. Tree conjecture for cyclic quadrilateral tiling billi…

Fix the combinatorics $G$ of a periodic $2$-colored tiling of the plane, and consider the parameters of locally foldable tilings with those fixed combinatorics. Measure-zero conjec…

Fix the combinatorics of a periodic $2$-colored tiling of a plane, defined by a bipartite graph $G$ on the $2$-torus. Suppose that a tiling with these combinatorics folds into a bo…

Let $\gamma$ be a singular periodic trajectory passing through a vertex $v$ of a tiling. Let $P$ and $P_e$ be tiles intersecting along an edge $e$. Bounded flower conjecture. The t…

Let $M$ be a fixed surface, and let $H$ be a covector whose corresponding sections of the level surfaces associated with $M$ may exhibit chaotic behavior. Novikov–Maltsev conjectur…

$4n+2$ conjecture. Every periodic trajectory on a triangle tiling has period $4n+2$ for some positive integer $n$.

Rauzy-gasket escaping-trajectory conjecture. Every trajectory through the circumcenter is escaping, and trajectories passing increasingly close to the circumcenter, with…

No-dense-trajectories conjecture. Trajectories on triangle tilings never exhibit dense behavior.

Tree conjecture. Every periodic trajectory on a triangle tiling encloses a tree.

Let the trihexagonal tiling be equipped with its tiling-billiard dynamics, and consider trajectories in this system. Dense and escaping trajectories conjecture. Dense trajectories…

Trihexagonal high-period conjecture. The trihexagonal tiling has periodic trajectories of arbitrarily high period.

Triangle no-spiraling conjecture. Trajectories on triangle tilings never spiral.

$4n+2$ period conjecture. In a triangle tiling, every periodic trajectory has a period of the form

Triangle periodic-length conjecture. There exist triangle tilings with periodic trajectories of arbitrarily large length.

Right-triangle drift-periodicity conjecture. Every rational right triangle tiling has a drift-periodic trajectory.

Isosceles escaping-trajectory conjecture. An isosceles triangle tiling has an escaping trajectory if and only if its vertex angle is not of the form

Trihexagonal escaping-trajectory conjecture. The trihexagonal tiling has non-periodic escaping trajectories.