All Questions
Tagged with combinatorial-geometry or discrete-geometry
1,992 questions
5
votes
2
answers
219
views
The smallest set of polygonal regions that can all together form 2 different convex polyhedrons
We add a little to On reconstructing convex polyhedrons from disconnected faces that are all mutually non congruent
We call a set of polygonal regions that all together form a convex polyhedron a ‘...
- 7,221
3
votes
0
answers
54
views
Properties of a set of infinitely stacked regular tetrahedra
If two regular tetrahedra $S_1$ and $S_2$ in $\mathbb{R}^3$ share a triangular face $f$, we will say that each of them is obtained from the other by stacking over $f$. Now, choose some regular ...
- 31
5
votes
1
answer
163
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On reconstructing convex polyhedrons from disconnected faces that are all mutually non congruent
Ref: https://arxiv.org/pdf/1307.3472
It is well known that given only a set of convex polygonal regions (call this set of polygons a 'face set') and no further information, one cannot uniquely ...
- 7,221
1
vote
0
answers
110
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To choose a set of $n$ triangles which together form the largest number of different triangular layouts
Ref: To choose a set of $n$ rectangles which together form largest number of rectangular layouts
We present a variant of above question:
General Question: given an integer $n$, how do we find $n$ ...
- 7,221
9
votes
1
answer
147
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Triangulating the cube with small Hamming distance
For a triangulation $T$ of an $n$-dimensional cube, whose vertices are the $2^n$ original vertices, let $d(T)$ be the largest Hamming-distance of two vertices that are in the same simplex.
How small ...
- 19.7k
0
votes
0
answers
55
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To build Heesch-like configurations of 'coronas' around a central triangle with triangles all of same area and perimeter
Ref 1: https://arxiv.org/pdf/1711.04504
Ref 2: On 'Walls' with 'non-tiles' - a variant of the Heesch problem
It is known that we cannot tile the plane with triangles that are pairwise ...
- 7,221
2
votes
0
answers
96
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On 'Walls' with 'non-tiles' - a variant of the Heesch problem
Ref: "Non-tiles and Walls - a variant on the Heesch problem" (https://arxiv.org/pdf/1605.09203)
Definitions (adapted from above doc): A non-tile is any polygon that does not tile the ...
- 7,221
3
votes
1
answer
225
views
Tiling the plane with pair-wise non-congruent and mutually similar quadrilaterals
We add a bit to Tiling the plane with pair-wise non-congruent and mutually similar triangles
Starting with 2 unit squares and with squares with sides 2,3,5,... (all Fibonacci numbers), one can form an ...
- 7,221
3
votes
0
answers
57
views
Changes to the Delaunay Triangulation after deleting a point inside the convex hull
Consider the Delaunay Triangulation $\mathcal{DT}(P)$ of a finite set $P$ of points in the euclidean plane.
let $CH\subset P$ be all points of $P$ that are on $P$'s convex hull, and not just the ...
- 13.9k
2
votes
0
answers
79
views
Relation of the most distant point-pair to the smallest enclosing circle
I am looking for a counter example to, resp. a proof of the correctness of, the following conjecture:
among all pairs points from a finite set in the euclidean plane, that are at maximal distance, ...
- 13.9k
6
votes
1
answer
284
views
Beardon's version of Poincaré's theorem for fundamental polygons
I am currently trying to understand Beardon's proof for Poincaré's Theorem, which can be found in his book The Geometry of Discrete Groups. The last condition in the theorem is giving me a headache to ...
- 181
3
votes
0
answers
166
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Are these two definitions of an NTA (nontangentially accessible) domain equivalent? If so, is the constant unchanged?
Jerison and Kenig give the following two conditions (alongside an exterior corkscrew condition, which is not relevant to this discussion) in the definition of an $(M,r_0)$ nontangentially accessible ...
- 221
0
votes
0
answers
104
views
On partitioning convex polygons into kites -2
We add a little to On partitioning convex polygons into kites
In his answer to above post, Tom Sirgedas has shown that a convex polygon having two successive angles acute or right is a sufficient ...
- 7,221
0
votes
0
answers
144
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4-Flower free set family of 3-uniform sets (AKA f(3, 4)) largest construction?
On Polymath, the table shows that the largest known 3-uniform 4-flower free set family has size 38, sourced by this paper. I don't have access to the paper though, and wanted to see the set family ...
- 11
9
votes
0
answers
254
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Twisted Rupert property
It has recently been proven that the 2017 conjecture that all
convex polyhedra $P$ are Rupert is false:
"A convex polyhedron without Rupert's property,"
Jakob Steininger, Sergey Yurkevich. ...
- 153k