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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 ...
Linus Hamilton's user avatar
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 ...
Magma's user avatar
  • 1,016
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 ...
Moritz Firsching's user avatar
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 ...
theonetruepath's user avatar
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-...
Zsbán Ambrus's user avatar
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 ...
Ian Agol's user avatar
  • 67.8k
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 ...
Peter Mueller's user avatar
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 ...
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 ...
Richard Stanley's user avatar
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-...
Timothy Chow's user avatar
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 ...
Terry Tao's user avatar
  • 112k
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 ...
Timothy Chow's user avatar
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 ...
Douglas Zare's user avatar
  • 27.9k
19 votes
Accepted

An aperiodic hexagonal tile?

This seems to admit periodic tilings, so there is probably some problem. . Numbers are rotation amounts.
Ville Salo's user avatar
  • 6,457
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 ...
Terry Tao's user avatar
  • 112k
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 ...
Oscar Cunningham's user avatar
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 ...
Wlodek Kuperberg's user avatar
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 ...
Ed Pegg Jr's user avatar
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 ...
Florian Lehner's user avatar
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 ...
Noam D. Elkies's user avatar
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 ...
Timothy Chow's user avatar
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 ...
Victor Protsak's user avatar
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 $60\times 60$ ...
Jan Kyncl's user avatar
  • 6,046
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]: ...
Pietro Majer's user avatar
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 ...
Timothy Budd's user avatar
  • 3,555
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 ...
Peter LeFanu Lumsdaine's user avatar
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 ...
Freddy Barrera's user avatar
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 ...
Will Brian's user avatar
  • 17.8k
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 ($\...
Igor Pak's user avatar
  • 16.8k

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