80
votes

Accepted

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

It seems that one can color a 15-15-15-30 trapezoid with the given tiles. Here is a picture (sorry about adjacent figures that are the same color, I used random colors so hopefully there are no ...

49
votes

Accepted

### Polyomino that can cover an arbitrarily large square but not the entire plane

Suppose you have a sequence $S^0 = (s_1, s_2, \ldots)$ of partial tilings, where $s_i$ covers a square of side length $2i-1$ centered at the origin.
Now let's consider two of these partial tilings ...

40
votes

Accepted

### Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?

Answer to Q1: All of the 261.
I looked at this question because of a video of Matt Parker and wrote an algorithm to find solutions. See here for an example of how a solution would look like. I dumped ...

38
votes

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

Since nobody has posted it, here's the smallest triangle tilable by the 'straight pentiamond', ie a side-30 triangle. Simple backtracking program, takes 0.5 seconds to show no tilings of the side-20 ...

34
votes

Accepted

### Can we three-color a tiling of the plane with Smith, Meyers, Kaplan, and Goodman-Strauss's einstein?

No, you cannot three-color that tiling.
Here's a finite part of the tiling from page 10 top left of the article, the tile numbered 2 here is the one darkened on that figure. This part cannot be three-...

32
votes

### Smallest tile to tessellate the hyperbolic plane

Binary Tiling
In fact, one can tile the hyperbolic plane with arbitrarily small tiles. There is a tiling of the hyperbolic plane (apparently due to Boroczky) by pentagons.
The horizontal edges are ...

28
votes

Accepted

### Tiling the plane with pairwise non-congruent rational triangles

First answer: The plane can be tiled as requested. First, we tile the plane with equilateral triangles with side lengths $1$. Now each such triangle can be tiled into two rational-sided triangles in ...

22
votes

Accepted

### (non-)existence of the aperiodic monotile

This recent preprint claims to find such a tile.
David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, “An aperiodic monotile”, (2023-03-20) arXiv:2303.10798
A longstanding open ...

Community wiki

20
votes

### Tiling a rectangle with all simply connected polyominoes of fixed size

A tiling is not possible for $n=77$. Consider the polyomino
of order 77 here.
The two $4\times 19$ missing rectangles can be filled with only one
tile, so this tile must be repeated. It should be ...

20
votes

### Polyomino that can tile itself

As John S. Adair commented, the relevant keyword is rep-tile. Wikipedia provides a partial answer to your second question (shapes other than polyominoes); it cites a paper by Viorel Niţică, "Rep-...

20
votes

Accepted

### Aperiodic monotile without reflections?

The same authors have just released a preprint claiming a positive answer to this question.
EDIT: Here is a picture of the reflection-free aperiodic monotile:
More visualizations and other data are ...

19
votes

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

There was a conference in July 2007 at the University of Minnesota—Duluth in honor of Joseph Gallian's 65th birthday. At that conference, Michael Reid gave a talk about tilings, and among other ...

19
votes

Accepted

### Smallest tile to tessellate the hyperbolic plane

The tilings mentioned by Ian Agol are related to an action of a Baumslag-Solitar group $\{ a,b \bigg| b^{-1}a^2b=a \}$ on the hyperbolic plane. They have arbitrarily small area, but diameter ...

19
votes

Accepted

### An aperiodic hexagonal tile?

This seems to admit periodic tilings, so there is probably some problem. . Numbers are rotation amounts.

18
votes

### Can the sphere be partitioned into small congruent cells?

Here is a partial result that says that if the sphere can be partitioned into small congruent polygons, then the polygons must be "thin" in some sense. Specifically:
Theorem Suppose a ...

17
votes

### How can one construct a four-coloring of a tiling of the plane with Smith, Myers, Kaplan, and Goodman-Strauss's aperiodic monotile?

I tried to find a colouring where one of the colours comprised exactly the 'flipped' tiles. So I coloured all of the flipped tiles blue and then found three of the remaining tiles which were all ...

16
votes

### Terrible tilers for covering the plane

Here is a better (possibly best) way of covering the plane with congruent regular pentagons:
The density of this covering is ${\sqrt5}/2 = 1.1180...$.
This covering is generated by the ...

16
votes

### Tiling a square with rectangles

The Nick Baxter solution is actually Blanche's Dissection, published in 1971.
I've outlined a general solution method at my Commuunity post Blanche Dissections. As a proof of concept, here are 16 ...

16
votes

Accepted

### Tiling with similar tiles

Non-convex solutions to Question 1
Consider the following polygon (the outward angle on the right is the same as the inward angle at the top)
Since I didn't know any better way to show it does not ...

16
votes

Accepted

### A claim on partitioning a convex planar region into congruent pieces

This picture looks like a counterexample with $N=2$ and $R$ a convex pentagon:
This should work more generally starting from an $n \times (n+1)$ rectangles
for any integer $n>1$, removing two ...

16
votes

Accepted

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

Here is at least a heuristic argument for why we should expect fewer vertical dominoes than horizontal dominoes. As we increase the length of the strip (to the right, say), let us think about how the ...

15
votes

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

I do not know whether the triangular region with size a multiple of 5 is tileable in general, but I can address the question in the last paragraph:
I'd also be interested in learning what kind of ...

15
votes

Accepted

### How many positions of a tiling polygon can occur simultaneousy?

Infinitely many, even for a triangle.
Let $T$ be the "Pythagorean" triangle with sides of length $3,4,5$. First, tile the $3\times 4$ rectangle $R$ by two copies of $T$. Tile the $60\times 60$ ...

15
votes

Accepted

### Is there a triangle which makes dense set of angles by drawing medians?

The answer to the second question is yes, for any non-flat triangle $T$, the set of angles $A$ is dense in $(0, \pi)$. This follows from a stronger result of Barany et al. [Theorem 1, 1]:
...

15
votes

### Tiling a rectangle with all simply connected polyominoes of fixed size

Continuing Richard Stanley's thought, here are two tiles that show that it is not possible for $n=25$ and $n=20$. The 25-omino is hostile (pun intended), in the sense that its complement does not ...

14
votes

### Polyomino that can cover an arbitrarily large square but not the entire plane

@Magma gives a beautiful self-contained answer — but to put it into a bigger picture, it’s worth also mentioning the word compactness, the term for a lot of phenomena like this one, and which gives ...

13
votes

### Tiling a square with rectangles whose areas or perimeters are 1, 2, 3, ..., N

Here are solutions for $N=10$ in a square of side $7$:
Here is a solution for $N=52$ in a square of side $47$:
At user44191's suggestion, here is the list of areas and perimeters of the rectangles ...

13
votes

### Fair cutting of the plane with lines

This is not a full answer to the question, but it is perhaps a start, and too long for a comment anyway.
If $n$ distinct lines intersect at a single point, let's say the intersection is regular if the ...

13
votes

### Dividing a polyhedron into two similar copies

Definitely not tetrahedra, see this paper. The proof generalizes to other families of polytopes, actually. For example, by Sydler's theorem these cannot be polytopes with nonzero Dehn invariant ($\...

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