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

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2
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76 views

Computing Voronoi poles in $\mathbb{R}^d$ (the farthest points within each cell)

Say I have a Voronoi diagram of some points $p_1,\dots,p_n\in\mathbb{R}^d$, which tesselates $\mathbb{R}^d$ into cells $V_1,\dots,V_n$. Within each cell $V_i$, the pole is defined as the vertex of ...
5
votes
1answer
123 views

Upperbounding the number of regions induced by a set of unit disks

Given a set $D$ of $n$ same radius disks, embedded in the plane, their arrangement induces a number $k$ of connected regions in $\mathbb{R}^2 \setminus \cup_{d \in D}$ . I am interested in an upper ...
0
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0answers
36 views

Lattice isotopy type of uniform hyperplane arrangements

I am working on a problem related to the isotopy type of a certain class of hyperplane arrangements in $\mathbb{C}^{d}.$ For more references, compare Randell's work "Lattice-isotopic arrangements are ...
9
votes
0answers
220 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 ...
1
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1answer
22 views

number of affine pieces of linear interpolation of convex functions in high dimension

Consider a convex function $f$ defined on a $d$-dimensional hypercube $[0,1]^d$. Now for fixed $m \in \mathbb{N}$, consider the grids $\mathcal{G}_m=\{(i_1/m,\cdots,i_d/m)\}$ where ...
3
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3answers
317 views

Aperiodic graphs

The concepts of being non-periodic and aperiodic for tilings have obvious versions for connected graphs with a countable set of vertices and a finite number of edges meeting at each vertex. A graph ...
3
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0answers
84 views

Modeling bubble rafts

If you go to images.google.com and search on "bubble rafts", you'll see various pictures of disk packings that in large patches look like the six-around-one dense packing of the plane by equal-sized ...
3
votes
1answer
280 views

Balancing real numbers in one dimension

Given numbers $0 \leq d_i \leq 1$ for $i=1,\ldots,m$, it is easy to see that you can always find signs $\varepsilon_i \in \{-1,1\}$ such that the partial sums $\sum_{i=1}^k \varepsilon_i d_i/2$, for ...
3
votes
2answers
107 views

regular polyhedra (and polytopes) in hyperbolic geometry, and generalisations

While there exist regular tesselations of the hyperbolic plane with arbitrary regular polygons, there are no new regular polyhedra in hyperbolic (3D) space. This being quite trivial, it is probably ...
4
votes
3answers
256 views

Perimeter/Neighborhood of a graph on grid

Hello, I have a $\sqrt{n}\times\sqrt{n}$ lattice graph $G=(V,E)$ i.e. vertices on said 2-dim integer lattice, and two vertices have an edge if their $L_1$ distance is one. Now I want to claim ...
3
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1answer
235 views

Min Bend Orthogonal Knots

I am seeking literature on 3D orthogonal drawings of knots, especially minimum bend drawings. An orthogonal drawing employs segments parallel to the axes of a Cartesian coordinate system. A bend is a ...
13
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1answer
339 views

Is the Ford disk packing optimal?

Given two unit-diameter disks tangent to a given line and to each other, determining a region bounded by two circular arcs and a line segment, is the Ford disk packing of that region the unique ...
9
votes
1answer
143 views

Optimal shape for stabbing balls in $\mathbb{R}^3$

I have radius $r < \frac{1}{2}$ congruent balls with centers randomly distributed uniformly within a region, say, within a unit-radius sphere $S$. I shoot a ray/path through $S$, hoping to ...
2
votes
0answers
70 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 ...
7
votes
0answers
81 views

Points on $k$ Circles

Let $k$ be a fixed positive integer. We want to find the minimum number $f(k)$, such that for a set of finite points in the plane, if any $f(k)$ of them are on $k$ circles, then all of them are on $k$ ...
12
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0answers
246 views

Grothendieck on polyhedra over finite fields

In Grothendieck's Sketch of a Programme he spends a few pages discussing polyhedra over arbitrary rings and concludes with some intriguing remarks on specializing polyhedra over their "most singular ...
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0answers
66 views

degenerate abelian surfaces

I am wondering if the family of degenerate abelian surfaces constructed by K. Hulek, C. Kahn and S.H. Weintraub in "Moduli spaces of Abelian Surfaces: Compactification, Degenarations, and Theta ...
3
votes
1answer
132 views

Packing bounds for sumsets, or, very discrete balls

Let $D\subset \mathbb{F}_2^n$ with $D=-D$ and $0\in D$. Write $k D$ for the set of all sums of $k$ (not necessarily distinct) elements of $D$. (This is the "ball" in the title.) Now let $d(g,h)$ be ...
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0answers
74 views

An exact fraction of a matrix

Let $A$ be a $n \times m$ real matrix with $n<<m$ and of rank $r<n$. It is known that $A$ has exactly two distinct non-zero singular values: $\sigma_{\max}$ and $\sigma_{2}$, and also that ...
2
votes
1answer
75 views

Complex lines arrangements from a given wiring diagram

I'm working on some presentation of the fundamental group of the complement of a complex lines arrangement in $\mathbb{C}^{2}.$ In particular, in Arvola's article "The fundamental group of the ...
6
votes
2answers
163 views

Geometric dominating set: NP-complete?

Let $G=(V,E)$ be a geometric graph, a graph embedded in the plane whose edge lengths are the Euclidean distance between its endpoint vertices. Say that a set of vertices $D \subseteq V$ is a geometric ...
4
votes
0answers
79 views

Can we replace 2-fold cover by n rectangles with 1-fold cover by n rectangles?

Suppose that $n$ rectangles cover every point of their union exactly twice (except for points on their boundaries). Can we partition this union into at most $n$ rectangles? I think it's pretty ...
3
votes
1answer
130 views

Braid wiring diagrams and matroids

recently I started reading some articles about the presentation of the fundamental group of lines arrangements in $\mathbb{C}^{2}$ via Wiring diagrams. I also found some relation with matroid theory. ...
10
votes
1answer
280 views

Soft question: mathematics about truchet tiles

It seems that this is the first question on Truchet tiles on MO. Shown above is a picture of a random tile, which you can see the resulting configuration is much like many membranes of cells. I ...
3
votes
2answers
302 views

How to flip one triangulation on a surface into another

Let $S$ be a compact orientable surface and $p_1,\dots, p_n\in S$ be distinct points. We consider all triangulations on $S$ with vertices $p_1,\dots, p_n$. Is there an algorithm which takes two ...
7
votes
2answers
680 views

Which (semi)regular polyhedra are combinations of two others?

The convex combination of convex polytopes is a convex polytope. An example in $\mathbb{R}^2$ is that a regular octagon can be obtained as $\frac{1}{2} S + \frac{1}{2} S'$, where $S$ is a square and ...
3
votes
2answers
80 views

Average vertex degree in finite Delaunay triangulations in high dimensions

In $\mathbb{R}^2$ it's known that with a "random" point configuration, the average degree of a vertex in its Delaunay triangulation is 6. Does anyone know of a similar result in higher dimension? I ...
0
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0answers
43 views

Number of faces of polytope projecting to lower dimensional polyhedron

Denote $K=\mathrm{conv}(v_1, \ldots, v_n)\subsetneq\Bbb R^m$ to be convex set spanned by vectors $v_i\in\Bbb R^m$ with $m\leq n$ then what technique could be useful to upper bound minimum number of ...
16
votes
1answer
451 views

Longest of random worm-like paths in $\mathbb{Z}^2$

Imagine at each lattice point of $\mathbb{Z}^2$ within $[1,3n]^2$, with coordinates $\equiv 2 \bmod 3$, we place, with equal probability, one of these six patterns:       The result ...
3
votes
1answer
164 views

n-dimensional Delaunay Triangulation of Lattices

I have several questions concerning the Delaunay triangulation of a high dimensional lattice. Given an $n$-dimensional lattice $L$ and its Delaunay triangulation (partition of $R^n$ into simplices ...
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0answers
32 views

Covering number of the range of a function

I have come across the need to know a bound on a certain curious quantity: the covering number of the range of a continuous function $f: D \rightarrow \mathbb{R}^n$, where $D \subseteq \mathbb{R}^m$. ...
2
votes
1answer
89 views

mean length of the non-crossing graphs on n points

My original question is rather vague so I'll start with a precise example and then indicate possible generalisations. Given a n-tuple $x=(x_1,\dots,x_n)$ in, say, a square with side-length $1$ in the ...
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0answers
142 views

covering high dimensional hypercube by balls

suppose we are given the $d$-dimensional hypercube $H^d$ defined as $$ H^d:=\left\{\sum_{i=1}^d\epsilon_ie_i:\ \epsilon_i\in \{0,1\}\mbox{ for }i=1,\dots , d\right\} $$ and $(e_i)_{i=1}^d$ the ...
8
votes
1answer
111 views

Tilting the $d$-cube to vertically separate its vertices

Let $C_d$ be a unit edge-length cube in $d$ dimensions. I would like to orient it ("tilt" it) so that the vertical (last) coordinates of its $2^d$ vertices are maximally separated, in the sense that ...
2
votes
1answer
216 views

Given a set of 2D vertices, how to create a minimum-area polygon which contains all the given vertices?

Not sure whether this question belongs here or math.stackexchange. You can assume that all the vertices are unique. The given vertices can be the vertices of the polygon, thus they do NOT have to be ...
11
votes
1answer
285 views

Are all well behaved “mean” functions on $\mathbb{R}^+$ equivalent?

Given a set $S$, a function $M: S\times S \rightarrow S$ is a mean if it satisfies the properties: $M(a,a)=a\qquad$ (identity) $M(a,b)=M(b,a)\qquad$ (commutativity). and possibly ...
4
votes
1answer
212 views

convex polyhedron in the unit cube

Let $P$ be a given finite set of points within the $n$-dimensional unit cube. A finite set $Q$ of points within the $n$-dimensional unit cube covers $P$ if $\operatorname{conv}(Q) \supseteq P$ where ...
36
votes
3answers
2k views

Is the “Napkin conjecture” open? (origami)

The falsity of the following conjecture would be a nice counter-intuitive fact. Given a square sheet of perimeter $P$, when folding it along origami moves, you end up with some polygonal flat figure ...
13
votes
2answers
979 views

Are the Platonic solids shadows of 4-polytopes?

Say that a 3D shadow of a 4-polytope is a parallel projection to 3-space, not necessarily orthogonal to that 3-space (that would make it an orthogonal projection). I am wondering if each of the five ...
2
votes
1answer
75 views

Measuring the Randomness and Statistics of Convex Polygons

How can I tell, how likely it is, that a given convex polygon with a sufficiently high number of edges is random and, if so, what kind of randomness it is (e.g. white noise)? What is known about ...
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vote
2answers
166 views

Generalization of Bracketing (or one of its many equivalences)

I asked the following question on MathStackExchange, but I have not received any answers after almost 3 days. Although it may not be a research level question, I thought I could ask it here. *"Is ...
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0answers
55 views
3
votes
1answer
79 views

What is Known About the Complexity of Calculating Minimal Surface Polyhedra?

I am currently ruminating about ways of generalizing Minimum Spanning Trees to Minimum Spanning "Hypertrees", where the cost is associated with simplex volumes and, where certain topological ...
7
votes
0answers
180 views

Minimal “basis” in $n$ dimensional unit cube

Let's $$ B^n=\{\bar\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_n)|\alpha_i\in \{0,1\}\};~~~~n=1,2,\ldots $$ and let's $$ C\subseteq B^n, $$ $$ S(C)=\{\bar\alpha\oplus\bar\beta\ | \bar\alpha,\bar\beta\in ...
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0answers
50 views

Can any Delone set be approximated by a model set?

Let $\Lambda \subset \mathbb{R}^d$ be a Delone set (uniformly discrete and relatively dense). I would like to know whether $\Lambda$ can be approximated by a model set in the Hausdorff distance. ...
25
votes
5answers
2k views

Nonconvex manhole covers

One common reason given for the circularity of manhole covers is that they can't fall through the manhole. For convex manhole covers, this property is equivalent to having constant width — if ...
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vote
0answers
161 views

Find m most distant points from a set of n points [closed]

I would like to find the $m$ (where $n$ $\geq$ $m$ > 1) maximally distant subset of points from a collection of $n$ $d$-dimensional points. Maximally distant means the sum of the pairwise distances ...
17
votes
0answers
267 views

Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to (0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space $X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point set ...
6
votes
0answers
107 views

How big can a family of pairwise intesecting affine spaces be?

I apologize if this question might seem to be a bit too elementary. Let $\mathbb{P}^n$ be the projective space over $k$ - an algebraically closed field of characteristic 0. Let $1\leq l\leq n-1$, and ...