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Results tagged with discrete-geometry
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user 39521
Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.
7
votes
A variant of Nelson-Hadwiger Problem on the chromatic number of the plane
It's apparently an open problem as to whether it's finite:
Olga Kosheleva & Vladik Kreinovich, “On chromatic numbers of space-times: open problems” (UTEP Technical Report UTEP-CS-08-42)
- 12.4k
13
votes
Accepted
Are there irregular tilings by L-polyominoes?
It is possible to not only avoid periodic monotonic staircases, but to avoid any infinite monotonic staircases whatsoever. This tiling of L-tetrominoes is periodic and features neither infinite monoto …
- 12.4k
4
votes
Accepted
Can $n$ circles on a plane generate $m$ intersection points where at least $k$ circles inter...
Yes, we can. Consider the usual drawing of the Fano plane with 7 vertices, 6 lines, and a circle. Replace the circle with a line through two of the three vertices.
Now we have 7 lines with 6 triple i …
- 12.4k
7
votes
Accepted
Is there a 3d equivalent of this picture?
The restriction to conformal maps is a natural one, as it means that there is no affine distortion in the neighbourhood of a point. Specifically, the Voronoi cells of the points will not be oblated or …
- 12.4k
47
votes
Accepted
Can we find lattice polyhedra with faces of area 1,2,3,...?
I found a 32-face example with face areas $\{ 1, 2, \dots, 32 \}$:
It took a reasonable amount of experimentation to stop it from self-intersecting.
- 12.4k
24
votes
Accepted
Rational inscribed realization of the regular dodecahedron
An example
Yes, here is a list of rational coordinates lying on the unit sphere, the convex hull of which is combinatorially equivalent to a regular dodecahedron. This polyhedron is invariant under r …
- 12.4k
8
votes
Accepted
Is a polytope that has in-spheres for faces of all dimensions already regular?
This is true in all dimensions, and can be proved by induction (on $d$) applied to the following (slightly stronger) hypothesis:
Theorem: If $P$ is a convex $d$-polytope with $k$-in-spheres for all $ …
- 12.4k
5
votes
Do triple-linked graphs exist?
If you restrict to straight-line embeddings (where edges are line segments), then the answer is yes: using the result in Erdos-Szekeres in high dimensions there exists some $n$ such that if you have $ …
- 12.4k