# Tagged Questions

**3**

votes

**0**answers

1k views

### How many combinations does Android pattern have? [closed]

Rules-
1) At-least 4 and at-max 9 dots must be connected.
2) There can be no jumps
3) Once a dot is crossed, you can jump over it.

**2**

votes

**0**answers

41 views

### How many regions are created by the set of all hyperplanes defined by a set of points?

If we have a set of points X in d-dimensional euclidean space, and we look at the set of all hyperplanes that are defined by any subset Y of X (in the sense of being the unique hyperplane containing ...

**1**

vote

**0**answers

109 views

### intersection points and lines, part two

Two years ago I asked a question about finding couples of integers (p,l) such that one may draw l lines on the euclidean plane, creating exactly p intersection points, and I got very interesting ...

**3**

votes

**0**answers

92 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

108 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

715 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

2k 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 ...

**5**

votes

**1**answer

239 views

### Regular lattice polygons

Suppose I want to construct an $N$-gon in the plane whose vertices are integer lattice points, and which is close to a regular $N$-gon (which means, the ratio of longest to the shortest side is within ...

**1**

vote

**0**answers

53 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 ...

**6**

votes

**1**answer

123 views

### How “accidental” are equalities between parts of Ehrhart quasi-polynomials? When do they persist to Euler-Maclaurin?

Background
What I think of Ehrhart theory (http://en.wikipedia.org/wiki/Ehrhart_polynomial) asserts that if we take a lattice polytope $P$, and count the number of lattice points in the $t$th ...

**0**

votes

**1**answer

169 views

### Expected length of the shortest polygonal chain connecting N random points in the unit square

N points are selected uniformly at random in the unit square. Let L(N) be the expected length of the shortest (possibly self-intersecting) polygonal chain connecting all the points. It can be proved ...

**6**

votes

**0**answers

112 views

### Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?

After reading these very interesting questions, I came up with another one:
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...

**6**

votes

**0**answers

78 views

### Constructing a polyhedron of maximal possible volume from given bounds on areas of its faces

Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the ...

**2**

votes

**1**answer

365 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

251 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

469 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

587 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

271 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

**1**answer

165 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

271 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

736 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 ...

**2**

votes

**2**answers

300 views

### Sphere - Symmetry and Triangulation [closed]

The sphere is symmetric with respect to any rotation. However, it loses this property as soon as it is triangulated. Are there sequences of triangulations that possess particular large symmetry groups ...

**0**

votes

**1**answer

56 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 ...

**2**

votes

**1**answer

303 views

### Number of 3-tuple partitions of a multiple of three which follow the triangle inequality

Given n=3t, t$\in \mathbb N$; let $\mathbb L_3$ be set of all distinct integer partitions of n having 3 parts; say $\lambda_1,\lambda_2,\lambda_3$ .
If I chose any one partition randomly from ...

**1**

vote

**1**answer

83 views

### Softwares to draw euclidean polyhedral surfaces

Let $S$ be an abstract euclidean polyhedral surface. By this, I mean a orientable compact 2D topological surface obtained by gluing together some (convex) euclidean polygons (arbitrary genus).
...

**5**

votes

**2**answers

497 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

**1**answer

337 views

### What are the essential properties of algebraic closure on an arbitrary structure?

Define the "model theoretic" notion of a closure function as follows:
Definition (1): Let $D$ be a non-empty set. A function $cl:P(D)\longrightarrow P(D)$ called a closure function iff it has the ...

**6**

votes

**2**answers

236 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 ...

**6**

votes

**1**answer

168 views

### Curvature flows for PL closed curves in the plane?

I'm curious to what extent people have studied "curvature flows" on PL closed curves in the plane.
There's a paper by Gage and Hamilton from 1986 that describes the long-term behaviour of smooth ...

**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

**0**answers

245 views

### Tiling a rectangle with weighted cells (min-max problem)

I have been struggling with a research problem. The problem can be formalized as follows:
Given a $n\times m$ matrix $A$ containing cells with non-negative integer values, partition it in $J$ ...

**3**

votes

**1**answer

79 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 ...

**8**

votes

**1**answer

238 views

### Largest convex hull of a unit length path

What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?

**2**

votes

**1**answer

93 views

### Estimates on the number of vertices of reflexive polytopes

Suppose $M \cong \mathbb{Z}^n$ is a rank $n$ lattice, with dual lattice $N$. Suppose $\Delta$ is a full dimensional lattice polytope (i.e. convex hull of finite lattice points) in $M$. Then $\Delta$ ...

**1**

vote

**1**answer

239 views

### Rational points of non-rational curves

An algebraic curve (in this question) is the zero set $C = f^{-1}(X\ Y)$ of any polynomial $f\in\mathbb R[X\ Y]$; we say then that $f$ represents $C$. ...

**1**

vote

**0**answers

227 views

### A problem in Galois Geometry

Given a prime $p$, out of $N$ vectors of length $p^k$ over $\Bbb F_2$ of Hamming weight $w^{k}$ that are chosen, how many vectors can there be with pairwise Hamming distance at least $2w^{k}$ given ...

**13**

votes

**0**answers

775 views

### Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories (In the 80s).
Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following:
(a) Trivial ...

**11**

votes

**3**answers

412 views

### Is there a nice way to “unravel” a great dodecahedron?

The great dodecahedron is a non-convex regular polyhedron bounded by 12 pentagonal faces, crossing each other, arranged in a star-shaped manner around each of its 12 vertices (see the Wikipedia page ...

**19**

votes

**2**answers

836 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

495 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

184 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

215 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

340 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

178 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 ...

**1**

vote

**1**answer

197 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

**1**answer

175 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.

**17**

votes

**4**answers

848 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 ...

**14**

votes

**1**answer

247 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

**7**answers

500 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

**3**answers

490 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 ...