# Questions tagged [tiling]

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### Can a row of five equilateral triangles tile a big equilateral triangle?

Can rotations and translations of this shape perfectly tile some equilateral triangle? I originally asked this on math.stackexchange where it was well received and we made some good progress. Here's ...
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### Does every polyomino tile R^n for some n?

This is a question posed by Adam Chalcraft. I am posting it here because I think it deserves wider circulation, and because maybe someone already knows the answer. A polyomino is usually defined to ...
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### Lecture notes by Thurston on tiling

I am looking for a copy of the following W. Thurston, Groups, tilings, and finite state automata, AMS Colloquium Lecture Notes. I see that a lot of papers in the tiling literature refer to it but I ...
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### Can the sphere be partitioned into small congruent cells?

On the unit $2$-sphere ${\mathbb S}^2$ furnished with the geodesic distance, a subset homeomorphic to a planar disk is called a cell. A finite family of cells is a tiling if their interiors are ...
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### Tiling the plane with incongruent isosceles triangles

It is not difficult to tile the plane with incongruent triangles. One could tile with equilateral triangles, and then partition each equilateral into three triangles, displacing their common ...
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### What can be tiled by T-tetrominoes?

The T-tetromino is a T-shaped figure made of four unit squares. An $m\times n$ rectangle can be tiled by T-tetrominoes if and only if both $m$ and $n$ are multiples of 4. This was proved in a 1965 ...
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### Covering of a surface of a cube $n\times n \times n$ by pieces of paper $1\times 6$

When I was too young one of my problems was in the list of problems of All-Russian Olympiad. The problem is the following: Problem. We have a surface of a cube $n\times n \times n$ such that each ...
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### Smallest tile to tessellate the hyperbolic plane

Is it known what the smallest tile (in terms of area) that can tessellate the hyperbolic plane is? In particular, it should tessellate the plane by itself. I think it will be a Triangle group, but I'...
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### "a shape that ... lies halfway between a square and a circle"

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 ...
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### Non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical?

Is there a non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical? It is a nice exercise with rational generating functions (or equivalently, ...
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### Monomer-Dimer tatami tilings need better relationships with other math. Summary of results

A monomer-dimer tiling of a rectangular grid with $r$ rows and $c$ columns satisfies the \emph{tatami} condition if no four tiles meet at any point. (or you can think of it as the removal of a ...
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### Does every 5-celled animal tile the plane?

An animal in the plane is a finite set of grid-aligned unit squares in $\mathbb{R}^2$. (The definition is the same as a polyomino, but where we relax the connectivity requirement.) One may ...
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### Rhombus tilings with more than three directions

The point of this question is to construct a list of references on the following subject: Fix vectors $v_1$, $v_2$, ..., $v_g$ in $\mathbb{R}^2$, all lying in a half plane in that cyclic order. I am ...
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### Can every tromino (including those with gaps) tile the plane?

I've generalized trominos to include "gaps", i.e. they are formed by removing all but $3$ squares from an $n$-omino where $n$ is finite. The generalized trominos pictured above can tile the plane ...
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### How hard is it to tell when a finite set tiles the integers?

Given a nonempty set $B$ of integers between 1 and $n$, we wish to determine whether or not $\mathbb{Z}$ can be tiled with translates of $B$ (that is, covered by disjoint translates of $B$). I know an ...
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### Higher dimensional generalization of: Any quadrilateral tiles the plane?

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., $\mathbb{R}^d$ ...
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### The sparsest planar net that captures every unit segment

Let $\cal C = \lbrace C_i \rbrace$ be a collection of rectifiable curves in the plane with the property that every unit-length segment meets at least one curve in at least one point. Call such a ...
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### ♢ ⧫ ⬠: the fourth kind of Penrose tiling?

It’s known that Penrose tilings have several implementations that are mutually locally derivable; but the sources (such as en.wikipedia) list no more than three essentially different variants. There ...
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### Tiling rectangle with trominoes - an invariant

There are two types of trominoes, straight shapes and L-shaped. Suppose a rectangle $R$ admits at least one tiling using trominoes, with an even number of L-trominoes. EDIT: we do not admit ALL ...
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### Mathematics of quasicrystals

I want to study quasicrystals from mathematical point of view, but I'm having hard time finding materials about it. If you could suggest me some books, articles or papers, I would be glad.
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### Decidability of tiling R^2

Does there exist a closed curve, with finite area and finite circumference, of which it is undecidable (in an axiomatic system where it is constructable) whether it can tile the plane? I know the ...
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### Tiling a rectangle with all simply connected polyominoes of fixed size

For which values of $n$ can we tile some rectangle with one copy of each free simply-connected $n$-omino (that is, each polyomino with $n$ squares that has no holes)? It appears that it is possible ...
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### Are Penrose tilings universal? Do aperiodic universal tilings exist?

Consider a tiling of the plane using tiles of at least two types (e.g, a Penrose tiling such as that shown at the bottom of this question, which tiles the plane with two types of tiles). List the tile ...
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### Bicoloring of $\mathbb{N}^2$, avoiding set of patterns, is the maximal limit density rational?

Consider a bi-coloring of $\mathbb{N}^2$, (black and white), where we wish to maximize the limit (limsup) of the density of black squares in $[n] \times [n]$ as $n \to \infty$. Here, we identify each ...
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### Arctic regions in higher dimensional zonotopes

Same way as the two dimensional tilings by rhombi come from minimal surfaces in a $D$ dimensional cubical lattice as mentioned in this answer, one can consider higher dimensional zonotopes tiled by ...
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### slick-proof-of-trick-for-counting-domino-tilings

The trick for rewriting the number of domino tilings of a simply-connected finite lattice region as the absolute value of the determinant of a matrix (due I believe to Kasteleyn and Percus, but if ...
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### Minimum number of distinct triangles for tesselating the sphere

Consider sequences of tesselations of the sphere. For instance, one such sequence might start with an icosahedron and proceed by subdividing each triangle face into 4 triangles and projecting the new ...
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### Relation Between Truncated Braid Groups and Regular Tilings of the Complex and Hyperbolic Plane

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question on math.SE. There exists a rather remarkable ...
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### Random walk on a Penrose tiling

Pólya proved that a random walk on $\mathbb{Z}^2$ almost surely returns to the origin, or, equivalently, returns to the origin infinitely often. It was subsequently established that in $\mathbb{Z}^3$, ...
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### Minimum tiling of a rectangle by squares

Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes). Is there an efficient way to calculate this?
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### Random Walk on Pentagonal Tiling

I’ve recently been looking at closed walks on tilings of the plane in which the “player” can move from one tile to any of its edge-adjacent neighbors. In particular, I’m trying to find asymptotic ...
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### Tiling survey that updates TIlings and Patterns?

Can anyone suggest a survey (or surveys) that provides an update to Tilings and Patterns by Grunbaum and Shepard? If there's a more recent book, that would be fantastic, but I don't see one. I am ...
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### Dividing a square into 5 equal squares

Can you divide one square paper into five equal squares? You have a scissor and glue. You can measure and cut and then attach as well. Only condition is You can't waste any paper.
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### What is the right way to think about / represent general tilings?

For periodic/symmetric tilings, it seems somewhat "obvious" to me that it just comes down to working out the right group of symmetries for each of the relevant shapes/tiles, but its not clear to me if ...
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### Growing a chain of unit-area triangles: Fills the plane?

Define a process to start with a unit-area equilateral triangle, and at each step glue on another unit-area triangle.                     $50$ ...
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### "Transcendental tilings": Do they exist?

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. Compute from $r$ a binary number $0 < b(r) < 1$ ...
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### Is this an instance of any existing convex pentagonal tilings?

Inspired by Wikipedia's article on pentagonal tiling, I made my own attempt. I believe this belongs to the 4-tile lattice category, because it's composed of pentagons pointing towards 4 different ...
This problem is a restatement of this question, first announced in MathStackExchange. We start with a triangle $T$ in the Euclidean plane and we define $A_n$ as the set of angles of the $6^n$ ...