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

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4
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1answer
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Computational approach deciding whether a set of Wang Tile could tile the space up to some size

As an applied person, I'm facing one practical problem deciding whether a set of Wang tile could tile the plane periodically or aperiodically. Although both problems seem undecidable, but I'm on a ...
5
votes
3answers
260 views

How hard is it to determine if a weighted graph can be isometrically embedded in R^3?

Consider a graph $G$ with nonnegative edge weights. Question: In $\mathbb{R}^3$, how hard is it to assign coordinates to vertices such that the Euclidean length of each edge is equal to its weight? ...
42
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2answers
1k views

How many unit cylinders can touch a unit ball?

What is the maximum number $k$ of unit-radius cylinders with mutually disjoint interiors that can touch a unit ball? By a cylinder I mean a set congruent to the Cartesian product of a line and a ...
32
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0answers
457 views

Extending a line-arrangement so that the bounded components of its complement are triangles

Given a finite collection of lines $L_1,\dots,L_m$ in ${\bf{R}}^2$, let $R_1,\dots,R_n$ be the connected components of ${\bf{R}}^2 \setminus (L_1 \cup \dots \cup L_m)$, and say that {$L_1,\dots,L_m$} ...
22
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3answers
2k views

Can a unit square be cut into rectangles that tile a rectangle with irrational sides?

For arbitrary positive integers $m$ and $n$, if we dissect a unit square into an $m\times n$ rectangular grid of $1/m\times 1/n$ rectangles, we can reassemble these $mn$ rectangles into an $n/m\times ...
3
votes
0answers
87 views

Lattices achieving best density

Let $\Lambda \subset \mathbb{R}^n$ be an Euclidean lattice with generator matrix $B$. Define the center density $\delta(\Lambda)$ in the usual way as $\delta(\Lambda) = \rho^n/|\det{B}|$, where $\rho$ ...
4
votes
1answer
99 views

Local rigidity of square disk packing

Is the packing of the plane by disks of radius 1/2 centered at the points of ${\bf Z} \times {\bf Z}$ "locally rigid" in the sense that no finite subcollection of the disks admits any joint ...
13
votes
3answers
644 views

Are infinite planar graphs still 4-colorable?

Imagine you have a finite number of "sites" $S$ in the positive quadrant of the integer lattice $\mathbb{Z}^2$, and from each site $s \in S$, one connects $s$ to every lattice point to which it has a ...
34
votes
3answers
1k views

What fraction of the integer lattice can be seen from the origin?

Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$. Say that a point $(x,y)$ of $Q$ is visible from the origin if the segment from $(0,0)$ to $(x,y) \in Q$ passes ...
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vote
0answers
51 views

Finding special vectors generated by a matrix

Let $G\in \Bbb Z^{n\times n}$ be a unimodular matrix. Are there any efficient algorithms to find the maximum norm of a vector $v$ that satisfies $\langle\Delta(v),v\rangle=0$ over all vectors $v\in ...
3
votes
1answer
252 views

Geodesic convex hulls in a graph; and their properties

This question asks for an analog of the convex hull in a graph that parallels (as far as possible) convex sets in Euclidean space. Let $G$ be a simple, undirected graph, and let $S \subseteq V$ be a ...
4
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0answers
209 views

Unique Domino Tiling

Question: how does one enumerate all star-convex $2n$-vertex sublattices of the plane that have the unique domino-tiling property? Definitions: A subset S of the xy-plane is star-convex if there is ...
6
votes
1answer
394 views

Higher dimensional Rubik's cube group

Since "cubes" with higher dimension than three exist I think it's natural to ask for higher dimensional Rubik's cubes. These so called hypercubes don't seem to have been described from a group ...
28
votes
1answer
528 views

Can the sphere be partitioned into small congruent cells?

On the unit $2$-sphere ${\mathbb S}^2$ furnished with the geodesic distance, a subset homeomorphic to a planar disk is called a cell. A finite family of cells is a tiling if their interiors are ...
19
votes
1answer
253 views

Hidden points in polygons

Let $h(n)$ be the largest number of mutually invisible points that can be located in a polygon $P$ of $n$ vertices. Two points $x$ and $y$ are mutually invisible if the segment $xy$ contains a point ...
3
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0answers
110 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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vote
2answers
227 views

Discrete Laplace operator and its eigenvalues

I wonder is there any geometric interpretation of the eigenvalues ​​of the discrete Laplace operator on graphs? Maybe there is a relationship between the eigenvalues ​​and combinatorial properties of ...
22
votes
3answers
684 views

Tetrahedron insphere iteration

I know that iterating the following incircle construction approaches an equilateral triangle in the limit:       Starting with any triangle $T$, one forms $T'$ by connecting ...
0
votes
1answer
55 views

Optimal radiating $(d{-}1)$-flats within a sphere

Permit me to revisit an earlier unresolved MO question, "Chord arrangement that avoids confining small or large disks" with a (very!) specific version, inspired by radiation therapy. The main idea is ...
5
votes
2answers
444 views

Is there a 3d equivalent of this picture?

This question arises apropos of an earlier question I asked that was (VERY!!!) helpfully answered by Anton Petrunin: Fitting a mesh to a density function The picture below is the image of a regular ...
5
votes
2answers
177 views

Embedding points in 2D based on distance estimates?

Suppose we have a collection of exactly $N$ points (say $N=1000$), with each point belonging to 2-dimensional Euclidean space $\mathbb{R}^2$, but we don't know the coordinates of the points. Suppose ...
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0answers
74 views

Group actions on polytopes in indefinite integer lattices

Is anything at all known about polytopes in indefinite integer lattices? I'm interested in lattice automorphisms which preserve certain polytopes of "high regularity" (e.g. cones). As a first step, ...
3
votes
1answer
67 views

Points in general position on a small grid

A point set $P$ is said to be embedded in $\mathbf{Z}^2$ in general position, if no three points lie on a common line. Assume that $|P|=n$, I am interested in the smallest $k \times k$ integer grid in ...
19
votes
2answers
768 views

Erdős, Harary, Tutte's “dimension of graph”: Progress in last 48 yrs?

I just ran across this delightful paper by an amazing triumvirate: Paul Erdős, Frank Harary, and William Tutte. "On the dimension of a graph." Mathematika 12.118-122 (1965): 20. (Cambridge link) ...
12
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2answers
429 views

Sets of evenly distributed points in the Euclidean plane

Is there a set $P \subset \mathbb{R}^2$ of points in the Euclidean plane whose intersection with every convex subset of $\mathbb{R}^2$ of area $1$ is nonempty but finite? If the answer is yes, can ...
7
votes
1answer
162 views

Can a tangle of arcs interlock in plane?

This is a variation of the question Can a tangle of arcs interlock?, asked by Joseph O'Rourke, and solved. I reproduce the question here: Can a (finite) collection of disjoint circle arcs in ...
4
votes
1answer
194 views

n-simplex in an intersection of n balls

Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ in the simplex. The question: ...
2
votes
0answers
313 views

Partitioning the Projective Plane

Throughout this post, by projective plane I mean the set of all lines through the origin in $\mathbb{R}^3$. Side Note: If there are more standard definitions for any of the ideas presented here, ...
2
votes
2answers
166 views

Three questions concerning lattice points on sphere surfaces

Pardon my ignorance of this topic. Q1. In which dimensions $d$ is it the case that, for every natural number $n$, there exists a sphere having exactly $n$ lattice points on it ...
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vote
1answer
185 views

Request for some references exploring the connections of Riemann surfaces with medical imaging

I'd like to know some references for a beginner who has basic background in Riemann surfaces and differential geometry, and would like to start learning/working on more applied areas, medical ...
4
votes
1answer
159 views

Cover of a n-simplex with balls

Consider a n-simplex. For each edge (i,j), consider a n-ball, such that vertices i and j are antipodal on this ball. Is the simplex covered by the union of these balls? Thank you.
16
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4answers
703 views

Non-chaotic bouncing-ball curves

I was surprised to learn from two Mathematica Demos by Enrique Zeleny that an elastic ball bouncing in a V or in a sinusoidal channel exhibits choatic behavior:     (The Poincaré map ...
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0answers
29 views

Bounds on number of linear top-$k$ orderings of lattice point sets

Given a set of $n$ lattice points $A = \{a_i\} \subset {\mathbb Z}^d$ I call a $k$-tuple $(a_{i_1},\ldots,a_{i_k})$ from $A$ a (linear) top-$k$ ordering if there is a vector $c$ such that $$c\cdot ...
13
votes
1answer
197 views

Smallest regular simplex containing the unit cube in $R^n$

What is the length $e_n$ of the edge of the smallest $n$-dimensional regular simplex $S_n$ containing the $n$-dimensional unit cube $Q_n$? In particular, is there $n$ such that ...
12
votes
7answers
457 views

Can a tangle of arcs of ellipses interlock

This is a variation on an earlier question resolved by user35353: Can a tangle of arcs interlock? In that question, the arcs were restricted to circular arcs, and user35353's proof that one arc can be ...
11
votes
3answers
405 views

Can a tangle of arcs interlock?

Can a (finite) collection of disjoint circle arcs in $\mathbb{R}^3$ be interlocked in the sense in that they cannot be separated, i.e. each moved arbitrarily far from one another while remaining ...
5
votes
2answers
144 views

Functionals on oriented matroids

Oriented matroids are abstractions of hyperplane arrangements, or equivalently vector configurations. Let me recall the definition in terms of covectors. Let $R=\lbrace 0,+,-\rbrace$ with the monoid ...
8
votes
3answers
599 views

On maximal regular polyhedra inscribed in a regular polyhedron

Let T, C, O, D, or I be regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively. Suppose that the outer polyhedron have edge-length 1. For example, it's easy to prove that ...
9
votes
2answers
207 views

Do maximal polyhedra have algebraic volume?

Is it possible to prove that for every $n > 3$ the maximal possible volume of a convex polyhedron having $n$ vertices inscribed in a sphere of unit radius is an algebraic number? Update: What ...
5
votes
2answers
252 views

Covering convex polygons with inscribed disks

The following problem came up when discussing mapping software (e.g., Google maps) with computer scientists. By $B(c,r)$ I mean the planar disk (open or closed, it doesn't matter) of radius $r$ around ...
3
votes
0answers
113 views

On understanding Discrete-Valued Stochastic Processes( time series, panel data )

It seems to me that a significant proportion of work in probability theory, statistics and machine learning are on understanding continuous-valued, relatively weakly dependent, or linear dependent ...
11
votes
1answer
499 views

Tiling the square with rectangles of small diagonals

For a given integer $k\ge3$, tile the unit square with $k$ rectangles so that the longest of the rectangles' diagonals be as short as possible. Call such a tiling optimal. The solutions are obvious in ...
2
votes
0answers
72 views

Criteria to decide whether a subset of the boundary complex of a polyhedron is a manifold?

Let $\mathcal{P} \subseteq \mathbb{R}^d$ be a convex polyhedron. Let $K$ be a subset of the boundary complex of $\mathcal{P}$. (Perhaps $K$ could be defined in terms of a system of linear ...
2
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0answers
123 views

A question on discrete numerical simulation on fluids mechanics

I read the paper "Stable, circulation-preseving simplicial fuids" by Elcott, et al: http://www.cs.jhu.edu/~misha/Fall09/Elcott07.pdf. It gives a structure preseving discretization of fluids. I have ...
3
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1answer
181 views

cover and hide with squares

I am studying two numbers, related to squares, that can characterize a polygon P: MinCoverNumber = the minimum number of axis-aligned squares required to exactly cover P (the covering squares may ...
5
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1answer
398 views

Interesting behaviour of Brion's formula under a degenerate change of variables

This is, probably, a question for those knowledgeable on the subject of Brion's theorem and its applications. Lately, I've been dealing with situations of the following sort. Suppose we are given a ...
8
votes
1answer
208 views

Fold-and-cut problem in three dimensions

The fold-and-cut theory states that "Any shape with straight sides can be cut from a single (idealized) sheet of paper by folding it flat and making a single straight complete cut. Such shapes include ...
8
votes
1answer
268 views

Complexity of the union of randomly rotated unit cubes

It is a remarkable fact that the union of congrent cubes has only at most near-quadratic combinatorial complexity, $O^*(n^2)$ for $n$ cubes, known to be almost tight. This contrasts with the union of ...
6
votes
2answers
179 views

Volumes of convex vs non-convex polyhedra with prescribed facets areas

It is known that given a set of Areas $A_f$ and normals $\vec{n}_f$ if $\sum_f A_f \vec{n}_f=0$ exist a unique convex polyhedron with given face areas and normals. (Minkowski theorem - See Alexandrov ...
6
votes
1answer
285 views

Pick's Theorem for rational points of bounded height

I wonder if the various lattice-point theorems, such as Pick's Theorem or Minkowski's Lattice Theorem, have been generalized to the collection of points with rational coordinates no more than height ...