All Questions
Tagged with tiling euclidean-geometry
9 questions
15
votes
1
answer
528
views
Dividing a polyhedron into two similar copies
The paper Dividing a polygon into two similar polygons proves that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original).
Right ...
6
votes
1
answer
435
views
On the aperiodic monotile
One of the more mind-boggling aspects of the Penrose tiles is that there are uncountably many distinct tilings of the plane, but every tiling contains every finite region that appears in another ...
- 4,898
22
votes
1
answer
1k
views
Aperiodic monotile without reflections?
The recently discovered amazing aperiodic monotile (or "einstein") of David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss tiles the plane only if reflections of the ...
- 82.6k
16
votes
0
answers
391
views
Is "Escherian metamorphosis" always possible?
$\DeclareMathOperator\int{int}\DeclareMathOperator\diam{diam}\DeclareMathOperator\area{area}\DeclareMathOperator\cl{cl}\DeclareMathOperator\ran{ran}\DeclareMathOperator\dom{dom}$This is a tweaked ...
- 21.1k
1
vote
0
answers
54
views
How to tile a plane such that moving from one tile to the next in any of the 8 cardinal directions is the same length?
When tiling the euclidean plane with squares (like most board games), moving diagonally to another tile is longer than moving vertically or horizontally. Is there a tiling such that moving in any of ...
- 11
5
votes
0
answers
119
views
What (if anything) is the connection between the Feit-Higman Theorem and the regular plane tilings?
Here are two facts that are superficially similar.
Tiling Theorem: The only regular tilings of $\mathbb{R}^2$ are achieved by triangles, squares, and hexagons.
Feit-Higman Theorem: The only finite ...
- 1,389
10
votes
1
answer
329
views
Is there a triangle which makes dense set of angles by drawing medians?
This problem is a restatement of this question, first announced in MathStackExchange.
We start with a triangle $T$ in the Euclidean plane and we define $A_n$ as the set of angles of the $6^n$ ...
- 125
8
votes
1
answer
265
views
Penrose tiling substitution is bijective
Let $\mathcal{P}$ a Penrose tiling built by a substitution $\omega$ with two triangles.
It is claimed, for instance, in the article of Anderson and Putnam "Topological invariants for substitution ...
- 81
10
votes
1
answer
231
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2-layer tilings with a center-of-gravity constraint
I've encountered a tiling problem with a physical constraint that
might place it outside the literature on tiling.
"Tiling" is a bit of a misnomer; it is a special type of cover.
All the tiles are ...
- 151k