15 problems

Let $\Gamma$ be a prism of a hexagonal tessellation of the Klein bottle, and suppose its cutout contains an $m\times n$ array of tiles, with $m$ and $n$ even. Assume that both type…

Let $G$ be a Lie group, let $H$ be a closed subgroup, and let $d(H)$ denote the invariant used in the paper to measure the dimension relevant to tessellations of $G/H$. One-dimensi…

Nonexistence conjecture. None of these homogeneous spaces has a tessellation. These are presented as three special cases of Kobayashi's general conjecture; the supplied source give…

Optimality conjecture. The configuration space of proper decorations is optimal if and only if all cone angles at vertices in $V_{-1}$ satisfy

Strict tessellation conjecture. $\Omega$ has a complete set of trigonometric eigenfunctions for the Laplace eigenvalue equation with the Dirichlet boundary condition if and only if…

Honeycomb conjecture. The hexagonal lattice is the globally minimal solution among tessellations of the plane with equal cell size.

Let a stationary isotropic STIT tessellation be given in a convex window in Euclidean space of dimension greater than two, and consider the inradii of its cells and their largest o…

Let $\sigma_1$ and $\sigma_2$ be valence sequences with $\sigma_1<\sigma_2$, and let $\underline{\lambda}_\sigma$ denote the lower growth-rate bound among face-homogeneous tessella…

Let $\sigma_1$ and $\sigma_2$ be valence sequences with $\sigma_1<\sigma_2$, and let $\underline{\lambda}_\sigma$ and $\overline{\lambda}_\sigma$ denote the lower and upper bounds…

Extremal growth-rate conjecture. The sets $\mathscr{L}_\sigma$ and $\mathscr{H}_\sigma$ are nonempty. Thus the lower and upper growth-rate bounds are realized by tessellations with…

Let $\sigma$ be the valence sequence of a face-homogeneous tessellation. Suppose that $\sigma$ contains $[p,q,r]$ as a subsequence, where $q$ is odd and $p\neq r$. Odd-valence subs…

Evenness conjecture. Every tetrahedral link complement has an even number of tetrahedra; equivalently, a corresponding combinatorial tetrahedral tessellation has an even number of…

Infinitude conjecture. There are infinitely many minimal combinatorial tetrahedral tessellations.

Let $\Gamma$ be a group acting on hyperbolic three-space $\mathbb H^3$, and let $D_x(\Gamma)$ be the Dirichlet tessellation with respect to a generic point $x$. Assume the action h…

Let $F$ be the field used in the construction, let $F_q$ be its relevant extension field, let $\Omega^+$ be the specified subgroup of the normaliser of the Coxeter group, and let…