27 problems

Let $Til(p/1)$ denote the rational tilings in the family with denominator $1$. In these tilings, the eigenvalues controlling fluctuations are all less than $1$, and in $Til(2)$ thi…

Let $Til(p/q)$ denote the rational tilings in the family of substitution systems described in the paper, with $p/q$ rational. The previously established case $Til(1/2)$ has triangl…

Consider the sea grafting (g) shown in Figure of the paper. A bi-infinite fully leafed caterpillar is a fully leafed caterpillar that extends infinitely in both directions. Extenda…

Let a Penrose P2 tiling be a tiling by kites and darts, and consider its fully leafed induced subcaterpillars, meaning fully leafed induced subtrees whose underlying tree is a cate…

Let $\alpha$ satisfy … where $\tau$ is the golden ratio, and let $\gamma$ denote the tilt angle of the growth form of the hat tiling. Hat growth-form tilt conjecture. The growth fo…

For positive $b\neq 1$, let $Tile(1,b)$ be the deformed hat tile, and let a tiling by these tiles have growth form with area $area_{growth\ form}$. The tile has area…

An L-tetromino tiling is a tiling by L-shaped tetrominoes, and its growth form is the asymptotic shape obtained from its coordination shells. L-tetromino nonpolygonality conjecture…

Convex-pentagon ASP uniqueness conjecture. In $S_t(\delta,\epsilon)$, Figure 06AD is the only case in which the ASP comprises four types of convex pentagons; when the anterior and…

Three-polygon ASP conjecture. If

A Wang tile set is a finite set of square prototiles with marked, coloured edges; tiles may meet along complete edges only when the corresponding symbols match, and translations ar…

The chair tiling is obtained as a fixed-point tiling of the plane from a single chair-tile substitution. Consider the graph formed by taking only the vertices where $90$-degree edg…

A Wang tile is a non-rotatable unit square with colored edges, and a tile set is a collection of such tiles. An infinite valid tiling is a covering of the infinite plane by copies…

Let $X$ be a subshift of finite type (SFT) on $\mathbb{Z}^3$. An aperiodic configuration is one with no nonzero period. Centers-of-aperiodicity conjecture. If $X$ contains an aperi…

Let $X$ be a $\mathbb{Z}^d$ subshift, and let $h_{d-1}(X)$ denote its entropy of dimension $d-1$, defined from the pattern-complexity function $N_X(n)$ by … Let an aperiodic config…

Let $T$ be the prototile used in the monohedral tilings discussed in this section, with its 14 translation classes. Missing translation classes conjecture. In any periodic tiling w…

For each integer $n\geq 2$, consider cyclotomic aperiodic substitution tilings (CASTs) with dense tile orientations (DTO), finite local complexity (FLC), the minimal inflation mult…

Well-balanced-class conjecture. Every well-balanced class in

Let $\Omega_4$ be the Wang shift under consideration, let $X_4\subset\Omega_4$ be the associated subshift, and let $Y_i^+$ and $Y_i^-$ be the horizontal-shift orbit closures of the…

Let $\Omega_0$ be the Jeandel–Rao Wang shift and let $X_0\subset\Omega_0$ be the minimal aperiodic subshift constructed in the paper. A shift-invariant probability measure on…

A golden tiling is a tiling by the rhombohedra associated with the icosahedral construction, and nondegenerate weak alternation means weak alternation with no infinite run of rhomb…

Let $n$ be the polygon-order parameter, let the cases be those listed in Table, and let $\eta_{\min}$ and the edge configurations be the quantities specified in Tables--. A rhombic…

Let $n$ be odd with $n\geq5$, and consider a CAST as described in the generalized Penrose CAST conjecture, with its prototiles and substitution rules. A CAST has individual dihedra…

Let $n$ be odd with $n\geq5$, and consider the CASTs described in Theorem. Let $\mu_{n,\lfloor n/2\rfloor}$ be the diagonal-based inflation multiplier used there, and let…

A cyclotomic aperiodic substitution tiling (CAST) is a substitution tiling with the cyclotomic structure considered in the paper. For odd $n$, let $D_n$ denote the dihedral symmetr…

Monteil's conjecture. For the tileset considered, the logarithm of the number of patterns of size $n\times n$ is of order $n$: