13 problems

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 $\alpha$ 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…

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…