Mathematics

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.

831 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 triangles,…
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 multiple…
Oscar Cunningham
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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 smallest…
Herman Tulleken
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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 angled triangles $1:\sqrt{2}$ parallelograms The…
Kepler's Triangle
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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 successful in creating something useful. We have…
Mr Yve
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Are tetromino packings always 4-colorable?

Suppose that I have a $2n \times 2n$ grid which has had $n^2$ tetrominoes packed into it, as in the following illustration: Can I always assign the numbers 1,2,3,4 to the squares of the grid so that (1) each tetromino contains each number exactly…
Gregory J. Puleo
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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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Hexagons are best for tiling 2D space in terms of perimeter vs area. What's best for 3D space?

If you think of the bee-hive problem, you want to make 2D cells that divide the plane of honey into chunks of area while expending the least perimeter (since the perimeter of the cells is what takes up resources/effort). The solution ends up being…
chausies
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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…
user632577
36
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2 answers

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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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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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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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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