# Questions tagged [tiling]

Use this tag for questions about (not necessarily periodic) tilings of metric spaces, their combinatorial, topological and dynamical properties, as well as basic definitions and concepts.

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### Covering a $4\times 4\times 4$ cube with $2\times 2\times 1$ squares

Covering a $4\times 4\times 4$ cube with $2\times 2\times 1$ squares. I am trying to solve the above problem, hoping it has a connection to the problem of finding the number of perfect matchings in ...
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### Combinatorial rectangle packing problem

Take the numbers 1, 2, and 3, and make a list of all possible unordered pairs (ie {1,1}, {1,2}, {1,3}, {2,2}, {2,3} and {3,3}). Interpreting these as the dimensions of rectangles, you get 6 rectangles ...
48 views

### Regular spherical quadrilateral tiling for a game board

Are there constructions for tiling a sphere with mostly regular spherical quadrilaterals, but with correction spherical polygons whose number and total area are minimized? In other words, a corrected ...
1k views

### Dividing a polyhedron into two similar copies of itself

The paper Dividing a polygon into two similar polygons provides that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original). Right ...
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### consistent vertex configurations for Archimedean tiling

I am trying to find if there is a simple rule for vertex configurations $n_1.n_2\ldots n_k$ that define Archimedean tilings (equivalently called uniform/semi-regular/vertex-transitive tilings). In ...
74 views

### Questions on aperiodicity in graphs

Background I am trying to understand different notions of aperiodicity in graphs, and how these concepts relate to aperiodic tilings. There is a different description of aperiodicity in graphs on this ...
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### Guaranteed graph labyrinth solving sequence

Starting from a vertex of an unknown, finite, strongly connected directed graph, we want to 'get out' (reach the vertex of the labyrinth called 'end'). Each vertex has two exits (edge which goes from ...
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### What are the possible surfaces that one can construct from a finite set ot triangles?

I am looking for references in discrete differential geometry for a concept I've been interested in. It is very common to approximate smooth surfaces using discrete triangulations. I am interested in ...
1 vote
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### Tilings closed under translation in only one direction

Background A tiling or tessellation of the plane is periodic if it is closed under at least two non parallel translations. Three examples of periodic tilings, including their corresponding translation ...
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### A game of magic Egyptian tilings

Background I've recently been formulating a game that incorporates elements from Egyptian fractions, magic squares, and tilings. It is a single-player game in which the objective is to tessellate a ...
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### Hyperbolic Reflection of polygon

I'm working on visualizing the reflections of a polygon in the Poincaré disk along each side of it using SageMath. The figures below show the reflections of a polygon (a 4-gon and a 3-gon, the ...
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1 vote
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### Can Langton's Ant draw a Penrose tiling?

Given the possibility for automata like Langton's Ant to lead to complex, intricate structures, I'm curious whether a Penrose tiling can be generated via this sort of local exploration. More formally: ...
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### Can we form a rectangle with integral lengths using an odd number of copies of this domino?

Question: This figure is made up of 6 unit cells. Can we form a rectangle with integral lengths using an odd number of copies of this domino? Rotating and flipping of the figure is allowed. This ...
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### How can I resolve this graphically derived identity?

This problem arose when looking into the area of a dignomonic tiling. I found an identity for an arbitrary number, call it $\Phi$, that is totally independent of the tiling itself. The result is given ...
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### Tiling Puzzle - ways to tile a 1xn board given coloring restrictions.

This problem has come up in a puzzle I've been trying to solve -- Given a 1xn board and tiles up to 1xn length how many ways can you construct the board. All of the tiles are the same color, let's ...
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1 vote
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### Question about the proof of the Ornstein-Weiss tiling lemma

In the original paper of Ornstein and Weiss "entropy and isomorphism theorems for actions of amenable groups" I have trouble understanding the proof of the quasi-tiling lemma for countable ...
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### Why are there no identities for the golden ratio?

Why are there no identities for the golden ratio? While there are a plethora of identities for Fibonacci and Lucas numbers [see, for example, here and here: S Vajda, Fibonacci and Lucas numbers, and ...
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### Can a dodecagon be cut into $n$ congruent pieces for any $n$ not of the form $1,2,3,4,6,8,12k^2,24k^2$?

Suppose I want to cut a regular dodecagon into $n$ congruent simply-connected pieces. For which $n$ is this possible? I can cut it into 24 right triangles, by cutting from the center to each vertex ...
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### Rigorous version of the Symmetry of Things

I’m reading through The Symmetry of Things and I love it, but some of the arguments are handwavey and it’s hard for me to know if I fully understand. I’m curious if there is a rigorous reference that ...
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1 vote
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### Optimal "maze length" in plane tilings

Among the tesselations of the plane (the tiles employed are a finite number of shapes which are allowed to be translated and rotated at will), which one performs best at the following problem? To ...
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### Are there any 2D aperiodic tilings that are the projections from a 3D lattice?

I need to give a short talk to some students to introduce a few ideas related to quasicrystals. It's not a proper lecture, more of an "ice-breaker" as I am not that well versed myself. It is ...
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### Covering a Square Floor with Square Rugs

You are given a finite collection of axis-aligned square rugs. (You do not choose the collection of rugs that you receive and the rugs are not necessarily all the same size.) Your objective is to move ...
129 views

### New Spectre tile moire pattern is very different from that of the Penrose tiling, why?

When you take two copies of the Penrose tiling, as Penrose himself demonstrated, they form a 5 fold symmetric Moire pattern which matches the 5 fold symmetry of their construction. When you "zoom ...
92 views

### Is it really impossible to use hexagons for mixed-resolution cover?

"To cover" and, more strictly, subpaving is a set of nonoverlapping "boxes" of R⁺. A subset X of R² can be approximated by two subpavings X⁻ and X⁺ such that  X⁻ ⊂ X ⊂ X⁺. ...
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1 vote
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### An one dimensional sampling version of Penrose tiles in 2D

I was initially interested in aperiodic sampling for signals to address the problem of aliasing in the frequency domain. A design like Penrose tiling in 2D (which is non-repeating) can be very ...
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1 vote
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### 2D tiling with regular pentagons (and generalizations)

We know that the only regular polygons that can tile the 2D plane are triangles, squares, and hexagons. One way of seeing this is that, if we try to place regular pentagons (for instance) around a ...
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### Navigating in Z-Order curve with different ordering

I want to navigate in a Z-order curve by moving up, down, left or right. The Wikipedia page has this section : This property can be used to offset a Z-value, for example in two dimensions the ...
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### Is every tiling pattern of $S$ connected by this simple flipping rule?

Let $S$ be a shape made out of a finite number of squares, equilateral triangles, and rhombi with angles $30^\circ, 150^\circ$, all having unit length sides. Often there are multiple ways to compose/...
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### How many Katamino solutions are there on a $5 \times 12$ board?

Katamino is the puzzle of placing twelve polygonal pieces so as to form a $5\times 12$ rectangular array. The pieces consist of all possible arrangements of five connected $1\times 1$ squares. For ...
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### Prove that the rest of the square can be tiled with $L$-trominos [duplicate]

A square of sidelength $2^n$ is divided into unit squares. One of the unit squares is deleted. Prove that the rest of the square can be tiled with $L$-trominos. I noticed that using L trominos, we ...
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### How many hexagons to fill a square tile

I am filling a square tile of width wTile with equal hexagons stacked flat side on top of each other at an angle I call colourAngle as shown in the diagram. I call the rows of hexagons "Perp Line&...
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### Notation for referring to specific graph colourings

Consider the following four graphs where $k$ represents the number of colours used to colour the vertices of each graph. Here cycle graphs are used to represent regular polygons, in this specific case ...
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### Partition a stable (Middle School Math)

Middle School Math Club Question : The stable, $6$ yards by $6$ yards with concrete walls, is divided by internal wooden partitions into stalls $1$ yard by $2$ yards. What could be the total length of ...
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1 vote
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### All the solutions to tile an 8 by 8 square with tetrominoes

I want to fully tile an 8x8 board using the 19 fixed Tetrominoes, allowing repeats of Tetrominoes. For example, this is a valid solution: Here are the 19 fixed Tetrominoes: I am looking for a list ...
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### Coloring a polyomino tiling so that no two pieces with the same color have a common point

How many colors are enough to color all polyomino tilings so that no two adjacent or touching polyominoes have the same color? In the following example 6 colors are required (each region has a common ...
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### How many domino tilings of a $2n\times 2n$ board are uniquely 3-colourable?

The closed-form formula for the number of domino tilings of a $2n\times 2n$ board is known to be $$\prod_{j=1}^{n}\prod_{k=1}^{n}\Big{(}4\cos^2\frac{\pi j}{2n+1} + 4\cos^2\frac{\pi k}{2n+1}\Big{)}.$$ (...
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### The 18 golden rational tetrahedra

In 2020, the 59 sporadic rational tetrahedra were identified. More recently, I found exact solutions for all of them. Most of them don't pair up well in terms of similar triangles that would allow ...
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### Wiping a plane clean with a rectangle (chalkboard erasing alogrithm?)

Let's say I use some green chalk to draw and fill a continous shape on a chalkboard. Let's assume the shape has no holes in it. I then use some red chalk to cover the area around my blob for a ...
• 71
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### Is it possible to assemble copies of this shape into a cube?

A couple of friends of mine were discussing a problem concerning this shape: Is it possible to assemble enough of these to form a cube? I have discovered a lot of impossible positions but was not ...
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### Reinhardt's Polyhedron as the first couterexample to the second part of Hilbert's 18th problem

In 1928 Karl Reinhardt published a first solution to the second part of Hilbert's 18th problem "Über die Zerlegung der euklidischen Räume in kongruente Polytope" in "Sitz. Ber. Preuß. ...
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### One X-pentomino and any number of I-trominoes cannot tile a rectangle?

Is it possible to tile a rectangle by one X-pentomino and any number of I-trominoes? Consider a $3m+1$ by $3n+2$ rectangle $R$ to be tiled by one X-pentomino and many I-trominoes. Using 3-coloring ...
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### Union of two disjoint congruent polygons is centrally symmetric. Must the polygons differ by a 180 degree rotation?

Let $P$ be a polygon with $180^\circ$ rotational symmetry. Let $O$ be the center of $P$ and suppose $P$ is dissected into congruent polygons $A$ and $B$. Must the $180^\circ$ rotation around $O$ ...