# Tagged Questions

**8**

votes

**1**answer

297 views

### Billiard dynamics with angle of reflection a fraction of angle of incidence

Suppose that a billiard ball bouncing in a unit square (or a lightray reflecting
in a mirrored square) has the property that the angle of reflection is a fraction
of the angle of incidence, rather ...

**9**

votes

**1**answer

360 views

### Are there irregular tilings by L-polyominoes?

I wonder if one can tile the plane with an order-$n$ L-polyomino
in a fundamentally irregular manner.
I seek help in defining what should constitute "irregular."
An L-polyomino of order $n \ge 2$ is ...

**2**

votes

**1**answer

90 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 ...

**3**

votes

**1**answer

106 views

### Polygonal Venn diagrams

Suppose that the interiors of $n$ $m$-sided planar simple closed polygons generate a $\sigma$-algebra $A$.
How many atoms can $A$ possess, at the most?
Failing an exact answer, how about good ...

**1**

vote

**0**answers

55 views

### Approximating Unit covering of d-dimensional points

Given a $d$-dimensional disk of radius $2$ in $\mathbb{R}^d$, how many disks of radius $1$ suffice to cover it. Of course, it's fine if the smaller disks overlap. What matters is to specify a finite ...

**7**

votes

**2**answers

164 views

### Uniqueness of an equilateral triangle decomposition into three similar polygons, exactly two congruent

A favorite fun problem I give to students and (even non-mathematical) friends is the following:
Find a decomposition of an equilateral triangle into three similar polygons, exactly two of which are ...

**1**

vote

**1**answer

73 views

### Unit covering of $d$-dimensional points

Given a set of points in $X$ axis, we want to cover them with minimum number of unit intervals.
For this problem we can assume that each interval in the optimal solution is starting or ending in one ...

**2**

votes

**1**answer

118 views

### Bound on maximum distance between points on a unit N-Sphere

I want to select M points on the N-sphere such that $min_{i\neq j,i,j\in \{1..M\}} ||x_i - x_j||$ is maximized.
Are there good upper bounds for this max-min distance?

**7**

votes

**2**answers

277 views

### Wait time to grid network disconnection with failing edges

Let $G_n$ be an $n \times n$ planar toroidal grid graph, with each node
connected to its four neighbors, with the top row connected to the bottom,
and the right column connected to the left.
Suppose ...

**2**

votes

**1**answer

123 views

### The Mahler conjecture and non-zonoidal 3-polytopes (4-polytopes)

I have been working on the Mahler conjecture for over a year now and have made some progress for certain classes of convex polytopes and I'm now attempting to write up my results specified to ...

**5**

votes

**1**answer

320 views

### A result from Peter McMullen's thesis

The classical definition of regular polytopes is recursive. It says that a polytope is regular if its facets and vertex figures (both smaller-dimensional polytopes) are regular.
The modern definition ...

**14**

votes

**0**answers

300 views

### Knight's tours in higher dimensions

I wonder if Knight's Tours have been explored in higher dimensions,
using the following definition of a knight move.
In dimension $d=2$, the knight moves left/right and forward/back
one step and two ...

**1**

vote

**1**answer

35 views

### Maximum crossings of curvature-constrained curve

Let $C$ be a curve in the plane whose curvature is everywhere $\le 1$.
If $C$ has length $L$, what is the largest number of proper self-crossings
of $C$ as a function of $L$?
For example, the curve ...

**1**

vote

**1**answer

145 views

### Calculate the discrete set of points B which are in the convex hull of the set of points A

This problem is likely best described with the following picture:
Given the discrete set of points $A$ (shown in blue), I wish to calculate the discrete set of points that are contained within the ...

**12**

votes

**3**answers

626 views

### Is {6,3,7} an 'ultrahyperbolic' Coxeter group?

These pictures, drawn by Roice Nelson, are attempts to visualize a geometry having as symmetries the {6,3,7} Coxeter group, by which I mean the one coming from the Coxeter diagram
...

**3**

votes

**3**answers

100 views

### Minimal area of non-planar lattice curves

Consider a $\mathbb{Z}^d$ lattice whose edges connect nearest-neighbor sites only, i.e. a $d$-dimensional hypercubic grid. Let $C$ be a closed curve along such edges. In general, for $d>2$ such ...

**28**

votes

**1**answer

998 views

### Can we find lattice polyhedra with faces of area 1,2,3,…?

I asked this question two months ago on MSE, where it earned the rare
Tumbleweed badge for garnering zero votes, zero answers, and 25 views over 61 days.
Perhaps justifiably so! Here I repeat it with ...

**2**

votes

**0**answers

90 views

### What is the projective dual of a planar graph?

Everybody learns the usual definition of the dual of a planar graph when edges are preserved and faces are mapped to vertices. Everybody learns the projective duality. What if we apply it to a ...

**7**

votes

**1**answer

362 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

171 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

52 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

140 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

115 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 ...

**8**

votes

**2**answers

595 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

174 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 ...

**0**

votes

**0**answers

111 views

### Generalized Sphere Kissing Problem

I am currently researching discrete geometry and I am in need of an upper bound on a generalized kissing number in 3-dimensions dependent upon a parameter $\eta$ which is the radii of spheres touching ...

**2**

votes

**1**answer

595 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

478 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

362 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

378 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

225 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

107 views

### Relation between non-integral polytopes, integrally closed polytopes and polynomial Erhart quasi-polynomials

By lattice points, I will always mean points in $\mathbb{Z}^n$ and all polytopes here are convex rational polytopes.
If $P$ is an integral polytope, the counting function for the number of lattice ...

**2**

votes

**1**answer

61 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

791 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

117 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

319 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

162 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

133 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

524 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

372 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

332 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

220 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

310 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?
...

**4**

votes

**5**answers

444 views

### Lattice points in dilated polytopes and sumsets

Let $P$ be an integral polytope, that is, the convex hull of some points in $\mathbb{N}^d$.
Let $p_1,\dots,p_m$ be all lattice points in $P$.
Question: What is the condition on $P$ that guarantees ...

**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 ...

**1**

vote

**0**answers

73 views

### Separating unit disks by circles

This is inspired by the recent question about separating unit disks by lines, which I will refer to as the "line case". Replacing "line" by "circle" adds one degree of freedom, and I'm wondering if ...

**7**

votes

**1**answer

240 views

### Separating unit disks by lines

Given $n\ge 2$. For a real $d>2$, consider a constellation $C$ of $2n$ disks of radius $1$ in the plane such that $h(C)$, the minimal distance between any two of their centers, is equal to $d$. Let ...

**1**

vote

**0**answers

101 views

### On 'Very Movable' Geometric Configurations (Configurations with a large degree of freedom)

Let $C$ be an $(n_r, b_k)$ combinatorial configuration that admits a geometric realization in the plane. I'm interested in the maximum number of points/lines $M$ of $C$ we can place in general ...

**33**

votes

**0**answers

514 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 ...