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Results tagged with discrete-geometry
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user 20595
Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.
3
votes
0
answers
115
views
Infinite counter-examples to Borsuk conjecture
When I formulate the Borsuk conjecture in the form of a coloring problem, Aigner told me that the formulation is not quite correct: It only works if the set is finite. He's right.
All the constructi …
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9
votes
0
answers
290
views
Neighborly family of coins
Here is a puzzle:
Find 5 identical coins. Can you arrange them so that every coin is touching every other coin?
The solution is here. The hint is: use the third dimension.
My questions are ba …
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4
votes
0
answers
169
views
Graph drawing maximizing the volume of the convex hull
Given a graph $G=(V,E)$ and a length function $\ell:E\to\mathbb{R}_+$.
An embedding of the graph into the $d$-dimensional Euclidean space is a map $f:V\to\mathbb{R}^d$ such that $||f(u)-f(v)||=\ell(uv …
- 2,581
12
votes
1
answer
447
views
What can we learn from the newly discovered monohedral convex pentagonal tiling?
Wikipedia: https://en.wikipedia.org/wiki/Pentagonal_tiling#Stein_.281985.29_and_Mann.2FMcLoud.2FVon_Derau_.282015.29
Media coverage: http://www.theguardian.com/science/alexs-adventures-in-numberland/ …
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10
votes
Accepted
Coloring of the plane
Seems to be the polychromatic number of the plane.
According to my knowledge, the value is at least 4 (due to Raiskii) and at most 6 (due to Stechkin).
See Chap. 4 and 6 of The Mathematical Coloring …
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16
votes
Is {6,3,7} an 'ultrahyperbolic' Coxeter group?
Actually, every Coxeter system of rank four is either euclidean or hyperbolic. That is, the canonical bilinear form has at most one negative eigenvalue.
It follows from the paper Sphere packings and …
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5
votes
Three-dimensional Apollonian spirals
According to the discussion in Coxeter (1968), the tangent points lie asymptotically on a concho-spiral, so the distribution is not uniform on the sphere, but is uniform on a circle.
By the way, the …
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1
vote
Convex caps with prescribed edges and curvature
Given Gaussian curvatures at the vertices, there is a unique lift that realizes these curvatures, as you can see from Igor's note.
Given a graph, the set of liftings that projects to this graph form …
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