**6**

votes

**1**answer

317 views

### Triangle (constrained number, rather than shape) packing?

Are there any interesting results on optimal packings in the plane using a fixed number of triangles (without a fixed size or shape constraint)?
For instance, what's the maximum area packing of the ...

**6**

votes

**1**answer

159 views

### Integral straight-line embeddings of planar graphs

Wikipedia says (in the article on Fáry's theorem),
"Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The ...

**1**

vote

**0**answers

45 views

### Euclidean embedding of a graph based on 1-ring neighborhood distances only

Consider a graph $(V,E)$, $\vert V \vert = n$ and weights $\{l_{ij}\}$, where $l_{ij}>0$ iff there is an edge connecting vertices $v_i$ and $v_j$. Distances beyond the 1-ring neighborhood are not ...

**4**

votes

**2**answers

126 views

### Computational complexity of deciding isomorphism of rational polyhedral cones

Let $C,C'$ be rational polyhedral cones in $\mathbb R^n$ both with non-empty interior. Rational means they are generated by vectors with rational entries. One says that $C,C'$ are isomorphic if there ...

**2**

votes

**1**answer

114 views

### Omit each vertex in turn of convex polygon: Iterative limit?

Let $P=P_0$ be a convex polygon of $n$ vertices $v_k$.
Let $P_{i+1}$ be the convex polygon obtained by intersecting the halfplanes
determined by the lines through every other vertex.
Below, $P_0$ is ...

**7**

votes

**2**answers

375 views

### get a point in polygon (maximize the distance from borders)

I have several 2D polygons represented by lists of xy-coordinates of their vertices.
It is needed to get several points inside the polygon so that they lie possibly far from the polygon's borders ...

**6**

votes

**0**answers

165 views

### Linked circles in R3

Two circles in 3-D are linked iff each one passes through the interior of the other.
There are $N$ points in 3-D in general position (no four lie on a plane). Each triple of points defines a unique ...

**3**

votes

**1**answer

557 views

### The Stock Market Polytope: Explanation?

Ovidiu Racorean.
"Crossing Stocks and the Positive Grassmannian I: The Geometry behind Stock
Market."
(arXiv Abstract link)
Anyone care to offer a summary of what's going on here?
(The ...

**14**

votes

**2**answers

472 views

### Spearing rolling hula hoops

Or: Stabbing rolling disks.
Imagine there are $n$ unit-diameter disks rolling between $x=0$ and $x=d$,
reflecting off either end.
The disk centers start at a random location within $[\frac{1}{2}, ...

**1**

vote

**1**answer

355 views

### Study of convex polytopes via commutative algebra

Let $P \subset \mathbb{R}^d$ be any convex polytope with integral vertices, and let $M$ be the additive submonoid of $\mathbb{R}^{d+1}$ which is generated by $\{ (v,1) : v \in P \cap \mathbb{Z}^d \}$. ...

**18**

votes

**2**answers

370 views

### Trapping a convex body by a finite set of points

In $\mathbb{R}^n$, let $K$ be a convex body and $T$ a finite set of points disjoint from the interior of $K$. Say that $T$ traps $K$ if there is no continuous motion of $K$ carrying $K$ arbitrarily ...

**5**

votes

**2**answers

202 views

### Genus of Tutte-Coxeter Graph

What is the genus of the Tutte-Coxeter graph -- the incidence graph of the
GQ of order 2? Seems like it should be well known, since nearly every other
parameter for that graph is known, but I can ...

**2**

votes

**1**answer

58 views

### Integer point in a non-empty polytope

I have a high-dimensional, non-empty polytope $Ax\geq b$ sitting inside the cube ($0\leq x_i \leq 1$). Is there any general theory or technique to show that this polytope contains an integer point, ...

**14**

votes

**2**answers

769 views

### Randomly walking a leashed dog

Let a human $h(t)$ random walk on $\mathbb{Z}^2$ by taking a unit-length step at every
time step $t$. A dog $d(t)$ on a leash of length $\lambda$ follows $h(t)$, also
taking a unit-length step at ...

**4**

votes

**2**answers

109 views

### The maximal discrete parallelepiped in a convex body

Does the positive constant $c_d$, depending only from dimension, with the following property exist?
Property:
for every convex body $K\subset \mathbb R^d$ there exists parallelepiped $P\subset K$ ...

**11**

votes

**1**answer

296 views

### Wander distance of self-avoiding walk that backs out of culs-de-sac

Suppose a self-avoiding walk on $\mathbb{Z}^2$, with random steps each one unit long,
backs out of culs-de-sac, but retaining the lattice points on which it stepped
marked as unavailable for future ...

**4**

votes

**2**answers

157 views

### Bound on Minimal Length of Vectors in Lattice and its Dual Lattice

Let $\Lambda$ be a lattice in $\mathbb{R}^n$ and $\Lambda^\ast$ its dual lattice. Let $d=\min_{v\in\Lambda} (v,v)$ and $d^\ast =\min_{v\in\Lambda^\ast} (v,v)$ be the minimal squared lengths of vectors ...

**3**

votes

**1**answer

132 views

### Partition All $n$-bit Binaries into $n$ Parts

For what values of $n$, it is possible to partition $\mathbb{Z}_2^n$ into $n$ disjoint parts, say $A_1, ..., A_n$ such that every element in $\mathbb{Z}_2^n$ is at most one-edit away from each part, ...

**9**

votes

**2**answers

466 views

### Finite field Szemeredi-Trotter theorem with unequal number of points and lines

My question concerns the Szemerédi-Trotter theorem in $\mathbb{F}_q^2$. If we have $m$ points and $n$ lines in $\mathbb{F}_q^2$, then by Cauchy-Schwartz the number of point-line incidences is as most ...

**9**

votes

**3**answers

348 views

### Conjecture on NP-completeness of tesselation of Wang Tile up to finite size

Motivated by these following questions on tessellation:
coloring in lattice
Reference for Wang Tile
Computational approach deciding whether a set of Wang Tile could tile the space up to some size
...

**11**

votes

**2**answers

331 views

### The intersection of a circle and a rank 3 subgroup of the plane

Let $A$ be a rank 3 subgroup of the Euclidean plane, i.e. $A = \mathbb{Z} v_1 + \mathbb{Z} v_2 + \mathbb{Z} v_3$, where $v_1, v_2, v_3 \in \mathbb{R}^2$ are three $\mathbb{Q}$-linearly independent ...

**3**

votes

**1**answer

213 views

### 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

**3**answers

286 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**

votes

**2**answers

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**

votes

**0**answers

484 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**

votes

**3**answers

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

**0**answers

89 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

**1**answer

104 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

**3**answers

673 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 ...

**35**

votes

**3**answers

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 ...

**1**

vote

**0**answers

52 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

**1**answer

309 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**

votes

**0**answers

224 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

**1**answer

428 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

**1**answer

546 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

**1**answer

262 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**

votes

**0**answers

121 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 ...

**1**

vote

**2**answers

240 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

**3**answers

700 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

**1**answer

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

**2**answers

460 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

**2**answers

194 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 ...

**1**

vote

**0**answers

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

**1**answer

73 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

**2**answers

789 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**

votes

**2**answers

453 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

**1**answer

175 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

**1**answer

199 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

**0**answers

326 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

**2**answers

170 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 ...