Search Results

Such a periodic crack partitions the tiling into to "halves"; it is in some sense an infinite "digital line" cleaving the tiling in two halves. … Is there an irregular tiling by L-polyominoes under my definition of "irregular"? Q2. Are there accepted definitions of what constitutes an irregular tiling, by one tile (a monohedral tiling)? …

The tiling by $T'$ is "combinatorially equivalent." … For example, the familiar regular hexagon tiling is not rigid, because it can be "compressed" vertically:       Rigid tilers are "brittle" in that even a slight deformation changes their tiling properties …

What is the probability that a random walk on the edges of a Penrose tiling returns to its starting point? … Each step from a vertex of the tiling follows one of the incident edges chosen randomly.                     …

But I don't see how to achieve a tiling with incongruent isosceles triangles. Perhaps it is easier to answer this question: Q2. … "Tiling the plane with equilateral triangles." arXiv:1805.08840 abstract (2018). Corollary 4. …

The best upperbound is $\theta(C) \le 1.228$ due to Dan Ismailescu, based on finding special tiling "p-hexagons" in $C$. A p-hexagon has two opposite, parallel edges of the same length. Q2. …

Is there a tiling of the plane by one each of simple polygons of $n$ vertices: one triangle, one quadrilateral, one pentagon, $\ldots$ , one simple polygon of $n$ vertices, $\ldots$ ? …

More precisely, in a regular sphere packing, either the HCP or FCC lattice packing, does there exist a line $L$ disjoint from every sphere, i.e., not touching any sphere? If so, one could "look throug …

An article in the Notices of the AMS, Volume 61, Issue 10, 2014 (PDF download link), on Khot's Unique Games Conjecture, says this: Another group ... found a shape that in a certain sense lies h …

I've encountered a tiling problem with a physical constraint that might place it outside the literature on tiling. "Tiling" is a bit of a misnomer; it is a special type of cover. … Surely Yoav's new 12.5% tiling made of overlapping equilateral triangles is the optimal. …

Any (non-self-intersecting) quadrilateral tiles the plane.     (MathWorld image.) Q. What is the strongest known generalization of this statement to higher dimensions? I.e., $\mathbb{R}^d$ …

It is an unsolved problem to determine the "thinnest" $2$-fold covering of the plane by disks. The $2$-fold coverage problem by disks is to find the minimum number of congruent (unit-radius) disks su …

But a tiling of the plane by unit-area hexagons is more efficient. … Is there a more efficient net than this hexagonal tiling? …

11 unfoldings of the cube form monohedral tilings of the plane, as so well illustrated in the "Etudes" video to which Igor Pak pointed:           A polyhedron that is the prototile of a monohedral tiling …

Let $T$ be a tiling of the plane. Fix an origin and shoot a ray $r$ from the origin. Mark off points $p_i$ along $r$ separated by unit distance. … Is there a tiling $T$ such that every $b(r)$, for all origins and rays $r$, is transcendental? If the answer to Q1 is No, the following two questions are superfluous: Q2. …

My question is: Which graphs $G_{\cal T}$ can be realized by some tiling $\cal T$. … What are some graphs $G$ that cannot be realized by some tiling $\cal T$? Added. …