All Questions
12 questions
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
16
votes
2
answers
1k
views
Are Penrose tilings universal? Do aperiodic universal tilings exist?
Consider a tiling of the plane using tiles of at least two types (e.g, a Penrose tiling such as that shown at the bottom of this question, which tiles the plane with two types of tiles). List the tile ...
- 4,922
12
votes
1
answer
373
views
A claim on partitioning a convex planar region into congruent pieces
Let us define a perfect congruent partition of a planar region $R$ as a partition of it with no portion left over into some finite number n of pieces that are all mutually congruent (ie any piece can ...
- 5,979
11
votes
1
answer
499
views
Tiling with incommensurate triangles
Say that two triangles are incommensurate if they do not
share an edge length or a vertex angle, and their areas differ.
Suppose you'd like to tile the plane with pairwise incommensurate triangles.
I ...
- 151k
7
votes
1
answer
186
views
Decidability of convex rearrangements of polygons
Triggered by the MO question,
"How many convex shapes can be made with the pieces of the Stomachion?," I would like to pose this question:
Q. Given $n$ polygons in a set $S$, say each with integer ...
- 151k
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
4
votes
1
answer
1k
views
What Islamic tiling patterns are constructible?
Eric Broug in his book Islamic Geometric Patterns gives
straightedge and compass construction of some simpler patterns.
It is clear his techniques will provide constructions for many
Islamic patterns.
...
3
votes
1
answer
152
views
Triangles that can be cut into mutually congruent and non-convex polygons
It is easy to note that an equilateral triangle can be cut into 3 mutually congruent and non-convex polygons (replace the 3 lines meeting at centroid and separating out the 3 congruent quadrilaterals ...
- 5,979
2
votes
1
answer
235
views
Tiling with one of each shape
Q. Is there a tiling of the plane by one each of simple polygons of $n$ vertices:
one triangle, one quadrilateral, one pentagon, $\ldots$ ,
one simple polygon of $n$ vertices, $\ldots$ ?
Here a ...
- 151k
2
votes
1
answer
84
views
What is the average component size of a coloring?
Supose each cell of a big (or infinite) grid is colored at random by one of $k$ colors. Then the connected monochromatic components (here components are not supposed to contain "wasp waists",...
- 13.4k
1
vote
0
answers
97
views
Tiling the plane with pair-wise non-congruent and mutually similar triangles
Question: Is it possible to tile the plane with triangles that are (1) mutually similar, (2) pairwise non-congruent and (3)non-right? No other constraints.
Note 1: Reg requirement 3 above: since any ...
- 5,979
0
votes
0
answers
36
views
Vertex configuration to tile repeat unit
I am working with elongated triangular tiling which has a vertex configuration of 3.3.3.4.4 and noticed that the representative symmetry is not the repeat unit. Is there a general formula to convert ...
- 1