All Questions
7 questions
21
votes
0
answers
453
views
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.) One may ...
- 3,068
17
votes
1
answer
458
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 ...
- 151k
15
votes
2
answers
737
views
Tiling survey that updates "Tilings and patterns"?
Can anyone suggest a survey (or surveys) that provides an update to Tilings and patterns by Grunbaum and Shepard? If there's a more recent book, that would be fantastic, but I don't see one.
I am ...
- 881
11
votes
1
answer
406
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 ...
- 151k
6
votes
2
answers
424
views
A class of tilings with amazing visual qualities
For more examples please see my related question on MSE:
Interesting tiling with a lot of symmetrical shapes
This is achieved by rotation of square grid over itself by atan(3/4).
Resulting ...
- 161
4
votes
2
answers
207
views
Classification of symmetries of tilings in surfaces?
Is there a general study of the symmetries of tilings on surfaces?
Conway, Goodman-Strauss & Burgiel classified them on $\mathbb S^2, \mathbb R^2$ and $\mathbb H^2$, with their 'Magic Theorem'. ...
- 561
1
vote
2
answers
232
views
What does the extension theorem for tilings state?
I have seen several references to the so-called Extension Theorem in the context of tilings of Euclidean space.
E.g. in "The Local Theorem for Monotypic Tilings" one reads
The Extension Theorem [......
- 13.6k