322 problems

Voronoi's conjecture. Every parallelohedron is an affine transformation of a Dirichlet–Voronoi cell of some lattice.

Consider a tiling of $n$-dimensional Euclidean space by identical hypercubes. Keller's conjecture. Two of the hypercubes in the tiling have an $(n-1)$-dimensional face in common. T…

Nivat's conjecture. Then $c$ is $1$-periodic.

Let $q$ be a Wang tile set, and let $W_q$ be its tiling space. A tiling $x\in W_q$ is doubly periodic if … for some $n_0,n_1>0$. Wang's conjecture. Every Wang tile set $q$ such tha…

Let Euclidean space of any dimension be tiled by hypercubes whose positions lie in a lattice. Minkowski's conjecture. Some pair of hypercubes must meet face-to-face. This is the ge…

Periodic orbit-closure conjecture. The orbit closure

Let $A$ be a finite set of integers, and write its mask polynomial as $A(X)=\sum_{a\in A}X^a$. Let $S_A$ be the set of prime powers $p^\alpha$ such that $\Phi_{p^\alpha}(X)$ divide…

Let $C$ denote the $n$-dimensional unit cube, let $k$ be a positive integer, and let $\Lambda$ be an $n$-dimensional lattice. A $k$-fold lattice tiling $C+\Lambda$ is a $k$-fold tr…

Łaba–Wang conjecture. If the self-similar measure $\mu_{b,D}$ is a spectral measure, then $\alpha D$ is an integer tile for some $\alpha\in\mathbb{R}$.

Let $d>1$ and let $cOmegacsubset cmathbb{Z}^d$ be a finite set. Lagarias–Wang's conjecture. If $cOmega$ tiles $cmathbb{Z}^d$ by translation, then it admits a periodic tiling. Here…

A tiling of ${\mathbb R}^3$ into cells of unit volume is being considered, and the surface area of a cell is the quantity to minimize. The proposed minimizing cell is a slightly mo…

Propp's conjecture. The $(a,b)$-benzel has

Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$, and define $m_1(\mathbb{R}^n,\|\cdot\|)$ as the supremum of the upper asymptotic densities of measurable unbounded subsets containing n…

Continuous periodic tiling conjecture. The tiling equation $\Omega\oplus X=_{\mathrm{a.e.}}\mathbb{R}^d$ is not aperiodic. Equivalently, if…

Let the truncated octahedron denote its associated graph, let $B_3$ be the relevant lower-bound quantity for three tile types, and let $T_3$ be the minimum number of tile types. Fo…

For $p\ge6$, consider the regular tiling ${p,3}$ and the circles inscribed in its faces. A circle packing is strongly solid if it remains solid after any one of its circles is remo…

A Hexagonal Dungeon is the hexagonal counterpart of an Aztec Dungeon introduced by Matt Blum. Blum's conjecture. The number of tilings of a Hexagonal Dungeon is always given by a p…

Let $\Omega$ be the tiling space, let $\Xi$ be its canonical transversal, and let $\mu$ be an invariant probability measure on $\Omega$ inducing a transverse measure $\mu^t$ on…

Let $k$ be a positive integer and write $S(k)=\{1,2,\dots,k\}$. Let $G$ be a finite abelian group. Say that $S(k)$ splits $G$ if there is a subset $T\subseteq G$ such that the tran…

Let $\mu$ be a rigid tile and let $d$ be its associated positive integer. For each $n\geq 1$, let $P_{\mu,d,n}(t)$ denote the tiling polynomial, and write $f(t)\ll g(t)$ when $f$ i…

Higher-dimensional parallelogram decomposition conjecture. Every surface in

Let $ABCD$ be an ideal trapezoid, and let $(a,b,c)$ be a triple of pairwise coprime integers satisfying … Write $x$ and $y$ for the side parameters of the ideal trapezoid as in the…

Consider the Kelvin problem in dimension $d=2$: partition the Euclidean plane into cells of equal area while minimizing the total interface length. The Honeycomb tiling is the regu…

Flip-and-trit connectivity conjecture. Any two tilings of a box can be connected by a finite sequence of flips and trits.

Let $\mathcal{M}$ be a normal, convex mosaic in $\mathbb R^3$ whose cells have unit volume, and let $\underline{\rho}_1(\mathcal{M})$ denote its lower edge density. Lower edge-dens…