1
vote
1answer
81 views

Repetitive tiling implies finite local complexity

My question probably needs to include the definitions of the terms in the title so I will first ask the question and then introduce the necessary definitions. The following Theorem is stated without ...
10
votes
1answer
343 views

Covering points on a sphere with a disk

Suppose $m$ points ("sites") are selected on the unit sphere $S^2$. For a given radius $r < \pi$, we can define a disk around any point on the sphere as the set of points at geodesic distance at ...
0
votes
1answer
87 views

Tiling the Cantor space

Given the Cantor space $2^\mathbb{N}$, and a set of disjoint open sets $D$, are there any non-trivial upper bounds on the number of further open sets needed to complete the tiling of the space?
3
votes
1answer
120 views

Planar kelvin problem

What is the minimal possible value of the maximal total sidelength shared by any two tiles in a tiling of the plane if all tiles have the same area A? total sidelength = Length-integral of the curve ...
4
votes
1answer
318 views

Decomposing a circle into similar pieces

Is it possible to decompose a circle into finitely many similar disjoint pieces, one of which contains the circle's center in its interior?
2
votes
2answers
168 views

Status of single aperiodic tile

What is known about the existence of a single tile, that tiles R^n only aperiodically? Has such a tile been found/proven to exist/not exist for any R^n?