4
votes
1answer
140 views

Thinnest 2-fold coverings of the plane by congruent convex shapes

It is an unsolved problem to determine the "thinnest" $2$-fold covering of the plane by disks. The $2$-fold coverage problem by disks is to find the minimum number of congruent (unit-radius) disks ...
9
votes
1answer
285 views

Are there irregular tilings by L-polyominoes?

I wonder if one can tile the plane with an order-$n$ L-polyomino in a fundamentally irregular manner. I seek help in defining what should constitute "irregular." An L-polyomino of order $n \ge 2$ is ...
9
votes
3answers
340 views

Conjecture on NP-completeness of tesselation of Wang Tile up to finite size

Motivated by these following questions on tessellation: coloring in lattice Reference for Wang Tile Computational approach deciding whether a set of Wang Tile could tile the space up to some size ...
3
votes
1answer
206 views

Computational approach deciding whether a set of Wang Tile could tile the space up to some size

As an applied person, I'm facing one practical problem deciding whether a set of Wang tile could tile the plane periodically or aperiodically. Although both problems seem undecidable, but I'm on a ...
22
votes
3answers
2k views

Can a unit square be cut into rectangles that tile a rectangle with irrational sides?

For arbitrary positive integers $m$ and $n$, if we dissect a unit square into an $m\times n$ rectangular grid of $1/m\times 1/n$ rectangles, we can reassemble these $mn$ rectangles into an $n/m\times ...
28
votes
1answer
540 views

Can the sphere be partitioned into small congruent cells?

On the unit $2$-sphere ${\mathbb S}^2$ furnished with the geodesic distance, a subset homeomorphic to a planar disk is called a cell. A finite family of cells is a tiling if their interiors are ...
4
votes
4answers
440 views

Consecutive Integer Squared Square

Is it possible to construct a squared square out of consecutive integer squares? Be it 1,2,3,...n or k,k+1,k+2,...n.
12
votes
1answer
297 views

The sparsest planar net that captures every unit segment

Let $\cal C = \lbrace C_i \rbrace$ be a collection of rectifiable curves in the plane with the property that every unit-length segment meets at least one curve in at least one point. Call such a ...
55
votes
5answers
2k views

Does every polyomino tile R^n for some n?

This is a question posed by Adam Chalcraft. I am posting it here because I think it deserves wider circulation, and because maybe someone already knows the answer. A polyomino is usually defined to ...
8
votes
3answers
9k views

Dividing a square into 5 equal squares

Can you divide one square paper into five equal squares? You have a scissor and glue. You can measure and cut and then attach as well. Only condition is You can't waste any paper.
8
votes
4answers
411 views

What is the right way to think about / represent general tilings?

For periodic/symmetric tilings, it seems somewhat "obvious" to me that it just comes down to working out the right group of symmetries for each of the relevant shapes/tiles, but its not clear to me if ...