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Questions tagged [discrete-geometry]

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

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0answers
103 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 $...
1
vote
2answers
120 views

Is a vertex- and edge-transitive polytope already a uniform polytope?

I want to consider (convex) polytopes $P=\mathrm{conv}\{p_1,...,p_n\}\subset\Bbb R^d$ which are both, vertex- and edge-transitive (or maybe stronger: 1-flag-transitive). Question: Is every such ...
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votes
0answers
17 views

Complexity of Calculating a Generalization of Planar Convex Hulls to Weighted $K_n$

Question: what is the complexity of finding a unique longest subtour in a $K_n$ with weighted edges, that is 2-optimal, i.e. its length can't be reduced by exchanging a pair of non-...
29
votes
5answers
8k views

Six yolks in a bowl: Why not optimal circle packing? [closed]

Making soufflé tonight, I wondered if the six yolks took on the optimal circle packing configuration. They do not. It is only with seven congruent circles that the optimal packing places one in the ...
3
votes
1answer
99 views

Incidences between points and circles in the plane

Suppose we have $n$ points $P$ and $m$ circles $C$ in the plane. Let $I(P,C)=\{(p,c), p \in P, c \in C, p \in c\}.$ Then what do we know about $\max_{m,n} |I(P,C)|$? Any references?
3
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0answers
121 views

Can bellows make loops?

Can flexible polyhedron (hyperbolic or euclidean) have non-simply connected configuration space not containing singular polyhedra?
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0answers
67 views

Are there half-transitive convex polytopes?

I only consider convex polytopes, i.e. convex hulls of finitely many points. The (edge-)graph of a polytope $P\subseteq\Bbb R^d$ is the graph consisting of the polytope's vertices, two are adjacent if ...
3
votes
1answer
71 views

Generalized digraph homomorphisms and graph cores

Given any digraphs $G$ and $H$ we say a surjection $f:V(G)\to V(H)$ reduces $G$ to $H$ if and only if it satisfies $(u,v)\in E(G)\iff (f(u),f(v))\in E(H)$. Where if there exists at least one ...
1
vote
0answers
45 views

Barycentric weighting of a point in a mesh

I know there is a way to get a barycentric weighting of a point $p$ inside a convex polygon; for weighting $\omega_n$ of component $q_n$ with adjacent angles $\gamma_n$ and $\delta_n$: $$\omega_n = \...
8
votes
1answer
255 views

Integer points avoiding three on a line, four on a circle

A century ago, Dudeney asked to place $16$ pawns on a chessboard with no three on a line:           As described by David Eppstein,1 the maximum number $g_3(n)$ points that ...
22
votes
1answer
413 views

Is there a short proof of the decidability of Kepler's Conjecture?

I've believed that the answer is "yes" for years, as suggested in various sources with reference to Tóth's work. For example, the Wikipedia article for Kepler Conjecture says: The next step toward ...
2
votes
0answers
73 views

Existence of a “generic enough” lattice point interior to a lattice triangle

Let $T$ be a lattice triangle in $\Bbb R^2$ (i.e. the convex hull of three noncolinear points in $\Bbb Z^2$), and assume it has at least one interior lattice point. Is it always possible to find a ...
14
votes
1answer
585 views

Holes in double-tileable polynominoes

This question was communicated to me by Evgeniy Romanov. Consider a connected polyomino $P$ that can be completely tiled in two different ways: with disjoint $2 \times 2$ square tetraminoes, and with ...
1
vote
1answer
73 views

How do I find the correct distance spacing for distributing equidistant points on a sphere of a given diameter?

How do I find the correct distance spacing for distributing equidistant points on a sphere of a given diameter? I don't need to fill the sphere with equidistant points. I just need less than a ...
1
vote
0answers
36 views

Shortest Lattice Vector with restricted $x$

Let $\Lambda$ be a lattice with basis, $B$ consisting of vectors $b_i$, so that the elements of $\Lambda$ are of form, $y\in \Lambda \iff y=Bx=\sum_i b_ix_i$ for some $x_i\in\mathbb{Z}$. My questions ...
10
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0answers
176 views

Fundamental circuit characterization of matroid independence complexes

I have the following characterization of independence complexes of matroids, which I think is standard but I can't find a reference. Here it goes: A pure simplicial complex $\Delta$ is the ...
1
vote
0answers
37 views

Compute the edge-skeleton of a polytope given by its vertices

Let $P$ be a polytope given by a vertex description, i.e., $P=conv(\{x_1,\ldots,x_m\})\subset\mathbb{R}^n$. Is there an efficient (i.e., not relying on Linear Programming) algorithm to compute the ...
1
vote
0answers
160 views

Acute triangles from 100 points

Given $n$ points in general position in the plane, let $P_n$ be the maximum proportion of the $\binom{n}{3}$ triangles with three acute angles. What is the limit $\lim\limits_{n \rightarrow \infty} ...
5
votes
0answers
43 views

Universal point sets for 1-plane graphs

It is a notorious open problem to find a smallest set of $N$ points that permit any $n$-vertex planar graph to be drawn in the plane without crossings, using only those $N$ points as vertices, and ...
8
votes
2answers
359 views

Simplicial set are to cubical sets what simplicial complexes are to …?

Simplicial sets and cubical sets (with or without connections) are defined as presheaves over some indexing categories. There is a full subcategory of simplicial sets that we can identify with the ...
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0answers
30 views

Defining a notion of “volume of its lattice” for non-rational subspaces

Let $V\subseteq \Bbb R^n$ be a vector subspace. If $V$ is rational, i.e. has a basis consisting of elements in $\Bbb Z^n$, then there’s a well-defined notion of the “volume of the lattice of V”: $$\...
8
votes
3answers
202 views

Random reflections unexpectedly produce banded distributions

Start with $p_1$ a random point on the origin-centered unit circle $C$. At step $i$, select a random point $q_i$ on $C$, and a random mirror line $M_i$ through $q_i$, and reflect $p_i$ in $M_i$ to ...
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vote
0answers
60 views

On Topological Tverberg Theorem

Let $r$ be a prime power. The Topological Tverberg Theorem says that any continuous map from a $(r-1)(d+1)$-simplex to $\mathbb{R}^d$ identifies points from $r$ disjoint faces. It is not hard to see ...
6
votes
0answers
103 views

Does the problem of recognizing 3DORG-graphs have polynomial complexity?

A 2DORG is the intersection graph of a finite family of rays directed $\to$ or $\uparrow$ in the plane. Such graphs can be recognized effectively (Felsner et al.). A 3DORG is the intersection graph of ...
9
votes
0answers
266 views

Why does Loday call the permutohedra “zylchgons”?

Today I was reading Jean-Louis Loday's classic paper, "Realization of the Stasheff polytope", in which he produces a simple and very pretty realization of the associahedra as convex polytopes. He ...
5
votes
4answers
215 views

Lattice points in a square pairwise-separated by integer distances

Let $S_n$ be an $n \times n$ square of lattice points in $\mathbb{Z}^2$. Q1. What is the largest subset $A(n)$ of lattice points in $S_n$ that have the property that every pair of points in $A(n)$...
7
votes
0answers
28 views

Inefficient covering by translates

While trying to answer this question, I arrived at another question: How many translates of $\{0,1\}^n$ does it take to cover $\mathbb F_3^n$? The broader context is: consider a set $S$ and a ...
2
votes
0answers
23 views

condition on rational polyhedral cone to guarantee dual cone is homogeneous

Let $\sigma\subseteq \Bbb R^d$ be a full-dimensional rational polyhedral cone which is strongly convex (i.e. $\sigma\cap-\sigma=0$). Definition. The cone $\sigma$ is homogeneous if there are ...
5
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0answers
79 views

Random walks in arrangements of lines in the plane

Let $\cal{A}_n$ be a simple arrangement of $n$ lines in $\mathbb{R}^2$. (Simple: each pair of lines meet in a distinct point, i.e., no three lines pass through the same point.) Start a random walk at ...
5
votes
0answers
145 views

Can we represent partitions by mutually parallel lines in the plane?

Lately I have become interested in the following idea: Suppose $n$ is a positive integer and $[n]=\{1,2,3,...,n\}$. Suppose we have 3 distinct partitions $b$, $g$, and $r$ of $[n]$. Assume that the ...
14
votes
1answer
279 views

Can the graph of a symmetric polytope have more symmetries than the polytope itself?

I consider convex polytopes $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-...
19
votes
2answers
979 views

Is it possible that both a graph and its complement have small connectivity?

Let $G=(V,E)$ be a simple graph with $n$ vertices. The isoperimetric constant of $G$ is defined as $$ i(G) := \min_{A \subset V,|A| \leq \frac n2} \frac{|\partial A|}{|A|} $$ where $\partial A$ is ...
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0answers
43 views

Upper bounds on $\epsilon$-covers of arbitrary compact manifolds

Let $M \subset \mathbb{R}^d$ be a compact $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$-cover $P$ of $M$, that is for every ...
8
votes
3answers
214 views

monochromatic subset

Suppose we have $n^2$ red points and $n(n-1)$ blue points in the plane in general position. Is it possible to find a subset $S$ of red points such that the convex hull of $S$ does not contain any blue ...
2
votes
1answer
81 views

Combinatorial problem about binary arrays with certain mutual distinctions

If there are m binary arrays (with 0 and 1) of length n, and between any two of these m arrays, there are k and only k same numbers (with the same site index in two different arrays). For example, if ...
2
votes
0answers
57 views

8-partition of a planar convex body by 4 concurrent lines

It is known1 that any convex body $K$ in the plane can be partitioned into $6$ equal-area pieces by $3$ concurrent lines which meet at a point in $K$. Call this a $6$-partition. This result cannot be ...
1
vote
1answer
165 views

A possible characterization of the cube?

Let $P$ be the $1$-skeleton of a convex polyhedron fixed in $\mathbb{R}^3$, and $|P|$ the sum of the Euclidean lengths of the edges of $P$. Let $P_1, P_2, P_3$ be the perpendicular projections of $P$ ...
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votes
0answers
101 views

Generating a Penrose tessellation around a given tile

Given a starting Penrose tile, I need to build a "spiraling" tessellation around it. The following picture illustrates the request: In this example, the starting tile is a "thin rhombus" (the pink ...
3
votes
2answers
215 views

Minimum weight triangulation of lattice points in a circle

Let $r$ be a natural number, and consider the $\mathbb{Z}^2$ lattice points $S$ inside or on the circle $C$ of radius $r$ centered on the origin. Let $P$ be the convex hull of $S$; so $P$ is inscribed ...
0
votes
0answers
24 views

Integer Distances in Pointsets Generated from Simplices with Integer Sidelengths

Let $\mathbb{S}_\mathbb{N}^n$ denote the set of all $n$-simplices in $n$-dimensional euclidean space $E^n$. Call an $n$-simplex aligned, if the set $C$ of its corners satisfies $\exists c_k=\left(x^...
2
votes
0answers
57 views

Can projecting a simplex onto orthogonal subspaces exposes the same vertices and edges?

Given the regular $n$-dimensional simplex $S\subset\Bbb R^n$ with $n\ge 4$, as well as two orthogonal subspaces $V,W\subset\Bbb R^n$ of dimension $\ge2$ (not necessarily of same dimension, not ...
5
votes
1answer
196 views

Can two non-equivalent polytopes of same dimension have the same graph?

By a polytope I mean the convex hull of finitely many points. The graph of a polytope is the graph isomorphic to its 1-skeleton. By equivalence of polytopes I mean combinatorial equivalence, i.e. ...
13
votes
0answers
254 views

Simple disproof of Danzer — Grünbaum conjecture

I asked this question on the MSE, but I did not get an answer. I hope that one of the experienced participants will check the correctness of the proof or the truth of the statement (and, perhaps, will ...
3
votes
0answers
65 views

Reconstructing plane graphs from degree- and face-sequences

Let $G$ be a plane $3$-connected graph; so it partitions the plane into regions bounded by faces. Let $\mathrm{deg}_v$ be the sequence of vertex degrees of $G$, and $\mathrm{deg}_f$ be the sequence of ...
5
votes
2answers
270 views

Is every polytope combinatorially equivalent to the intersection of a simplex and a linear subspace?

I wonder whether such a result is known, and if so, whether the proof is trivial. By polytope I mean the convex hull of finitely many points in $\Bbb R^n$. Assume the simplex to be symmetric and ...
1
vote
0answers
55 views

Can sufficiently symmetric polytopes be uniquely reconstructed from their 1-skeleton?

General convex polytopes can not be uniquely reconstructed from their 1-skeleton1, as explained here. Not even the dimension is known from the skeleton, as e.g. the complete graph $K_n,n\ge 5$ is the ...
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votes
0answers
33 views

How many points are still contained in a common (hyper-)ellipsoid?

It is known that $${d+2\choose 2}-1$$ points uniquely determine a quadric in $\Bbb R^d$. However, I want my points not on an arbitrary quadric, but on a centered hyperellipsoid in $\Bbb R^d$, or ...
5
votes
0answers
117 views

Polychromatic number of plane

Let $\chi$ be the least size of a partition of plane into pieces each of which omits unit distance. Let $\chi_p$ be the least size of a partition of plane into pieces each of which omits some distance....
2
votes
1answer
188 views

Name and Algorithms for a Sparsest Circle Packing

The ordinary circle packing problem in the variant with equal radii asks for the largest radius $r_{max}$ that allows placing $n$ non-overlapping circles with radius $r_{max}$ e.g. in the unit square, ...