# Questions tagged [tiling]

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### Decidability of completing Penrose tilings

Is the following problem known to be un/decidable? Problem: Given a finite configuration of Penrose tiles in the plane, determine if there is an extension of the configuration tiling the whole plane.
186 views

### Does $\mathbb{Z}\times\mathbb{Z}$ have an aperiodic monotile?

For any set $S\subseteq \mathbb{Z}\times\mathbb{Z}= \mathbb{Z}^2$ and $a\in \mathbb{Z}^2$, we set $a+S = \{a+s: s\in S\}$, where $+$ is the componentwise addition in $\mathbb{Z}^2$. Moreover, for any ...
37 views

### Draw an arbitrary line on a Penrose tiling. Determine a sequence of tiles can it intersect

Let us consider a Penrose tiling of $\mathbb R^2$. Starting with an arbitrary point on the tiling, draw an arbitrary straight line. Assume that this straight line never overlaps perfectly with a ...
81 views

### Tiling the plane with quadrilaterals that are mutually non-congruent and affine equivalent

Question: Can the plane be tiled with convex quadrilaterals that are (1) mutually non-congruent in a Euclidean sense and (2) mutually affine-equivalent? Remark: Every trapezoid is affine equivalent to ...
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### References and upper bounds for the SONNAT tiling game?

In a video released about a month ago, Pembesita describes a tiling game called SONNAT: Same Orientation Neighbour Not Allowed, Tiling. In the single-player game, the player may employ two rhombi. The ...
1 vote
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### Periodic tilings in finite type tiling spaces and substitution tiling spaces

I was reviewing the following statement from a survey by E. Arthur Robinson about tilings in $\mathbb{R}^d$ to better understand geometric tiling rather than tilings over symbols. I consider the ...
27 views

### Local complexity of tilings under substitutions

I am trying to read this survey of E. Arthur Robinson, about tilings of $\mathbb{R}^d$. I have some familiarity with 'symbolic' tilings, but I don't think I have a good intuition on 'geometric' ...
1 vote
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56 views

### Tiling with non-congruent triangles all of which have an equal angle and equal area

Reference 1: an earlier question on tiling with pair-wise non-congruent tiles: Tiling with triangles of same circumradius and inradius Reference 2: Triangulation of polygons with all triangles having ...
377 views

### Is "Escherian metamorphosis" always possible?

$\DeclareMathOperator\int{int}\DeclareMathOperator\diam{diam}\DeclareMathOperator\area{area}\DeclareMathOperator\cl{cl}\DeclareMathOperator\ran{ran}\DeclareMathOperator\dom{dom}$This is a tweaked ...
188 views

### Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles

Definition: Let us refer to obtuse triangles with the largest angle strictly above a given cutoff value as 'strongly obtuse' - the definition is parametrized by the cutoff value. Likewise, strongly ...