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.

# Questions tagged [tiling]

615 questions

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### What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?
The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif
Laczkovich gave a solution with many hundreds of…

Ed Pegg

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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've now also asked this question on mathoverflow.
Notes:
Obviously I'm ignoring the triangle of side $0$.
Because the area of the triangle has to be a…

Oscar Cunningham

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**60**

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### Can squares of infinite area always cover a unit square?

This is a claim one of my students made without justification on his exam. It definitely wasn't the right way to approach the problem, but now I've been nerdsniped into trying to figure out if it is true.
Let $a_i$ be a sequence of positive reals…

David E Speyer

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### Can any number of squares sum to a square?

Suppose
$$a^2 = \sum_{i=1}^k b_i^2$$
where $a, b_i \in \mathbb{Z}$, $a>0, b_i > 0$ (and $b_i$ are not necessarily distinct).
Can any positive integer be the value of $k$?
The reason I am interested in this: in a irreptile tiling where the…

Herman Tulleken

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**36**

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### How few $(42^\circ,60^\circ,78^\circ)$ triangles can an equilateral triangle be divided into?

This is the parallel question to this other post with many answers already, in the sense that the $(42^\circ,60^\circ,78^\circ)$-similar triangles form the only non-trivial rational-angle tiling of the equilateral triangle (and the regular hexagon),…

user632577

**36**

votes

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### The Scutoid, a new shape

The scutoid (Nature, Gizmodo, New Scientist, eurekalert) is a newly defined shape found in epithelial cells. It's a 5-prism with a truncated vertex. The g6 format of the graph is KsP`?_HCoW?T .
They are apparently a building block for living…

Ed Pegg

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### How few disks are needed to cover a square efficiently?

A unit square can be covered by a single disk of area $\pi/2$. Let us call the ratio of the square's area to that of the covering disks (i.e. the sum of the areas of the disks) the efficiency of the covering, so that in the base case with one disk…

John Bentin

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### Dividing a square into equal-area rectangles

How many ways are there to tile an $n\times n$ square with exactly $n$ rectangles, each of which has integer sides and area $n$?
The sequence $C(n)$ begins 1, 2, 2, 9, 2, 46, 2, 250, 37. Clearly $C(p) = 2$ for prime $p$. The value $C(8) = 250$ was…

MJD

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### Algorithm to get the maximum size of n squares that fit into a rectangle with a given width and height

I am looking for an algorithm that can return the number of size of n squares that fit into a a rectangle of a given width and height, maximizing the use of space (thus, leaving the least amount of leftover space for squares that do not fit).…

Anton

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### On existence of boards that be covered by every free tetromino

There is a board which can be covered by each of five free tetrominoes:
However, it's not simply-connected (has a hole). I wonder if there is a simply-connected board with the same property.

G. Strukov

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### How many "prime" rectangle tilings are there?

Given two tilings of a rectangle by other rectangles, say that they are equivalent if there is a bijection from the edges, vertices, and faces of the tilings which preserves inclusion. For instance, the following two tilings are equivalent (some…

RavenclawPrefect

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### how to generate tessellation cells using the Poincare disk model?

I'm a computer programmer, and while I like math, this is an area where my understanding of math falls short of what I need in order to apply it successfully.
I've been looking at M.C. Escher's "Circle Limit" drawings, which use a Poincare disk…

LarsH

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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.)
In this investigation of tilings of rectangles by 1D animals, it is…

RavenclawPrefect

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### Tiling a square with rectangles

Consider the set of all the rectangles with dimensions $2^a\times 2^b\,a,b\in \mathbb{Z}^{\ge 0}$. We want to tile an $n\times n$ square by rectangles from this set (you can use a rectangle several times). What is the minimum number of rectangles we…

ali

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### What's wrong with this Penrose pattern?

I programmed the Penrose tiling by projecting a portion of 5D lattice to 2D space, by the "cut and project" method described in
Quasicrystals: projections of 5-D lattice into 2 and 3 dimensions, H. Au-Yang and J. Perk.
Generalised 2D Penrose…

whitegreen

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