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      Questions tagged [discrete-geometry]

      Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.

      2
      votes
      1answer
      73 views

      Algorithm to compute the convex hull of a set of $m$ possibly intersecting convex polygons in the plane

      I am trying to find an algorithm to compute the convex hull of a set of $m$ possibly intersecting convex polygons in the plane, with a total of $n$ vertices. Let $h$ denote the number of vertices on ...
      8
      votes
      0answers
      53 views

      Minimizing union of overlapping rectangles

      Believe it or not, this has something to do with making triangle-free graphs bipartite... We have a collection of $k$ axis parallel rectangles with side lengths $(a_i,b_i)$. We want to arrange them (...
      2
      votes
      0answers
      57 views

      Polygons such that $n^2 $ times magnification of a polygon could be covered by exactly $n^2$ original polygon

      While studying about covering problems in combinatorics, I got to a simple question: What polygons can be covered exactly, without any area that is covered twice or area that is outside the covered ...
      3
      votes
      0answers
      92 views

      Fitting $\frac1n\times\frac1{n+1}$ rectangles into the unit square [duplicate]

      Consider the set of rectangles $r_n | n \in \Bbb N$ such that rectangle $r_n$ has shape $\frac1n\times\frac1{n+1}$. The total area composed by one copy of each $r_n$ as $n$ ranges from $1$ to ...
      1
      vote
      0answers
      103 views

      Arithmetic that corresponds to combinatorial rectangles and cylinder intersections?

      Definable subsets of $\mathbb N$ in the language of Presburger arithmetic are exactly the eventually periodic sets. In communication complexity the interpretation is more on intersection and union of ...
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      votes
      0answers
      53 views

      Large finite subsets of Euclidean space with no isosceles (or approximately isosceles) triangles

      Here's a question in combinatorial geometry which feels very much like other questions I'm familiar with but which I can't see how to get a hold of. I'll actually propose two different questions on ...
      1
      vote
      1answer
      75 views

      Is a polytope with vertices on a sphere and all edges of same length already rigid?

      Let's say $P\subset\Bbb R^d$ is some convex polytope with the following two properties: all vertices are on a common sphere. all edges are of the same length. I suspect that such a polytope is ...
      26
      votes
      3answers
      659 views

      How many random walk steps until the path self-intersects?

      Take a random walk in the plane from the origin, each step of unit length in a uniformly random direction. Q. How many steps on average until the path self-intersects? My simulations suggest ~$8....
      2
      votes
      0answers
      138 views

      How to calculate the positions of the vertices of a deformed cube with different local and global symmetry?

      Going over degrees of freedom, it appears that the following construction works, but I have no idea how to calculate exact positions of the vertices, or even a practical approach to approximating them ...
      6
      votes
      2answers
      472 views

      Tiling a square with rectangles whose areas or perimeters are 1, 2, 3, …, N

      For which positive integers N does there exist a square that can be completely tiled with N rectangles of integer sides whose areas or perimeters are precisely 1, 2, 3, ..., N?
      17
      votes
      0answers
      192 views

      Can 4-space be partitioned into Klein bottles?

      It is known that $\mathbb{R}^3$ can be partitioned into disjoint circles, or into disjoint unit circles, or into congruent copies of a real-analytic curve (Is it possible to partition $\mathbb R^3$ ...
      2
      votes
      1answer
      67 views

      A questions concerning Laguerre/Voronoi tessellations

      Fix $n>1$ distinguished points $x_1,\ldots, x_n\in \mathbb R^d$, the Voronoi tessellations are the subsets $V_1,\ldots V_n\subset\mathbb R^d$ defined by $$V_k~~ := ~~ \big\{x\in\mathbb R^d:\quad |...
      0
      votes
      0answers
      38 views

      Periodicity of oscillators in Langton's Ant and powers of $2$

      This question based on previous one by me. As Christopher Purcell noticed in his comment, there exist conjecture (which has a lot of counterexamples) that if you take a pair of ants $(n,n+1)$ apart (...
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      votes
      0answers
      140 views

      Absolute oscillator in Langton's Ant

      We have a simple (or single) block of Langton's Ants colony which includes two ants looking in the same direction. Their positions can be interpreted as knight's walk. The distances between each next ...
      1
      vote
      0answers
      30 views

      Algorithm for Calculating Spheric Convex Hulls of Finite Pointsets

      Let the Spheric Convex Hull ($\mathrm{CH}_S$) denote the intersection of all closed spheres that contain a compact $\Sigma\subset\mathbb{R}^n$ and on their boundary at least $n+1$ distinct points of $...

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