Deprecated; please use a more specific tag.

**5**

votes

**1**answer

636 views

### The Gauss circle problem on a hexagonal lattice

Take an infinite hexagonal lattice (or equivalently, an equilateral triangular lattice), with unit spacing between the closest lattice point pairs, and draw a disc of radius $r$ centered on a lattice ...

**0**

votes

**0**answers

78 views

### Building an invariant Sn structure from two invariant Zn structures

Take two mathematical structures with a $Z_n$ symmetry (cyclic symmetry). Which are the different ways, in "gluing" these structures, to obtain a mathematical structure with a $S_n$ symmetry ...

**4**

votes

**4**answers

755 views

### Proofs for doubly ruled surfaces

Hello,
I am interested in proofs for why the only irreducible doubly ruled surfaces in ${\mathbb R}^3$ are the one sheeted hyperboloid and the hyperbolic paraboloid. While many books and papers state ...

**1**

vote

**1**answer

158 views

### Overlaying two domino-like constructions such that all individual pairs of domino-like cells in the overlay have matching symbols

Imagine I have two $n$ x $m$ assemblies of $P = (n*m)$ unit square cells on the plane, $(c_{(a,1)}, ..., c_{(a,P)}) \in A$ and $(c_{(b,1)}, ..., c_{(b,P)}) \in B$, where every cell, $c_k$, in a ...

**2**

votes

**0**answers

142 views

### Honeycomb-type properties of the Delaunay triangulation and Voronoi diagram

Suppose I'd like to distribute a set of points $P=\lbrace p_1 ,\dots, p_n \rbrace$ in the unit square $S=[0,1]\times[0,1]$ to minimize a weighted sum of two things:
1) The average distance between a ...

**2**

votes

**2**answers

380 views

### Dilogarithm, tetrahedrons, and hyperbolic space

The Bloch-Wigner function $D(z)$, gives the volume of a ideal tetrahedron in hyperbolic space $\mathbb H3$, where z is the cross-ratio (z1,z2,z3,z4) parametrizing the tetrahedron in $\mathbb CP1$.
...

**2**

votes

**2**answers

236 views

### Surface of a Ideal Tetrahedron in Hyperbolic Space H3

The hyperbolic space $\mathbb{H}^3$, has a boundary $\mathbb{CP}^1$.
A ideal tetrahedron in $\mathbb{H}^3$, is a tetrahedron, where the four vertices are on the boundary $\mathbb{CP}^1$.
The four ...

**1**

vote

**1**answer

349 views

### H-representation versus V-representation

The $H$-representation of a convex polytope $S$, is just a set of linear inequalities corresponding to the intersection of halfspaces:
$S = ( x | Ax\leq b )$.
One could also represent a convex ...

**1**

vote

**1**answer

121 views

### Planar eucliean bipartite matching with squared distances

This is probably a really stupid question, but suppose I have two sets of points in the plane $X$ and $Y$ each with cardinality $|X| = |Y| = n$. For any bipartite matching $M$ between $X$ and $Y$, ...

**6**

votes

**3**answers

332 views

### Herringbone partitions of regions and surfaces

Let $R \subset \mathbb{R}^2$ be a region of the plane bounded
by a Jordan curve. The boundary $\partial R$ could be a polygon,
or a smooth curve—there are variations depending upon boundary ...

**8**

votes

**1**answer

527 views

### Approximating a convex function by a piecewise linear function

Suppose I have a Lipschitz-continuous convex function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. I wish to approximate it on the unit ball by a piecewise-linear function $g:\mathbb{R}^n\rightarrow ...

**8**

votes

**2**answers

463 views

### Three half circles on the plane may not meet nicely

Let $H$ denote the union of the northen hemisphere of the unit circle $S^{1}$ with the interval $[-1,1]$ on the $x$-axis. That is, $H=\{(x,\sqrt{1-x^{2}}):-1\le x\le 1\}\cup\{(x,0):-1\le x\le 1\}$
...

**1**

vote

**1**answer

137 views

### Covering a $d$-dimensional integer lattice by repeating a minimal set of deterministic moves

Imagine I place a turtle on some desired vertex, $v_i$, of a bounded $d$-dimensional integer lattice, $Z^d$, with dimensions $(l_1, ..., l_d)$. The turtle is able to travel from vertex to vertex ...

**0**

votes

**0**answers

143 views

### How do Hodge classes for Calabi-Yau 4-folds compare with the classes for tori?

Let $X$ be a Calabi-Yau 4-fold, i.e., a connected 4-dimensional compact Kahler manifold with $K_{X} \cong \mathscr{O}_{X}$ and $h^{i} (X,O_{X} )= 0$ for $0 \lt i \lt 4$.
Given a general ...

**4**

votes

**5**answers

603 views

### Collisions between rooks taking random flights on an $N$ by $M$ chessboard

I randomly place $k$ rooks on an (arbitrarily sized) $N$ by $M$ chessboard. Until only one rook remains, for each of $P$ time intervals we move the pieces as follows:
(1) We choose one of the $k$ ...

**3**

votes

**1**answer

263 views

### Can an ellipsoid be moved freely inside another ellipsoid?

An origin centric ellipsoid is defined by any positive semi-definite $n$ by $n$ matrix $X$, by taking all vectors $v$ such that $v^tXv\leq1$. Call two origin centric ellipsoid equivalent if one can be ...

**7**

votes

**4**answers

590 views

### Surfaces that can be rolled by a ball

Let $S$ be a smooth solid body in $\mathbb{R}^3$,
and $B$ a ball of radius $r$.
Say that $B$ is in contact with $S$ if
(1) they share a point $x$
that is on the surface of each,
$x \in \partial S$ ...

**2**

votes

**2**answers

99 views

### Worst-case nearest-neighbor distances between regions

Suppose that $S_1,\dots,S_n$ is a collection of disjoint shapes in the plane, and let $\mathcal{X}$ denote the set of all $n$-tuples of points $\lbrace x_1,\dots,x_n\rbrace$ such that $x_i\in S_i$ for ...

**0**

votes

**0**answers

112 views

### What is the intrinsic geometry of a feasible set?

In constraint optimization problem, one is often confronted with the following problem:
$min$ $f(x)$ , $x \in R^n$
given
$g_i(x) = c_i$ where $i = 1,...m$
$h_j(x) < c_j$ where $j = 1,...p$
...

**6**

votes

**3**answers

483 views

### Cylinders dividing $\mathbb{R}^{3}$

Consider $n$ affine copies of a compact cylinder, say $S^{1}\times [-3,3]$ with top and botom, sitting inside $\mathbb{R}^{3}$.
For each $n$ we may ask ourselves how to arrange the $n$ cylinders so ...

**5**

votes

**0**answers

102 views

### Simultaneous Strong Law of Large Number classes?

Say that $C$ is a SSLLN class of subsets of some topological space $V$ provided that for every sequence of i.i.d.r.v.s $X_1,X_2,...$ with values in $V$, we almost surely have: For every $A$ in $C$, ...

**4**

votes

**2**answers

285 views

### Realization spaces for regular convex polytopes

Q1.
Are there convex polytopes combinatorially equivalent to each of the regular polytopes
that are realized with integer vertex coordinates?
...

**11**

votes

**1**answer

345 views

### Can all convex polytopes be realized with vertices on surface of convex body?

The following question was asked by me on Mathematics.SE. Unfortunately, no one answered it so I thought I might give it a try one level higher. Below the line you can find the slightly edited ...

**4**

votes

**2**answers

339 views

### Product of random diagonals on the unit circle

Let $P_1, P_2, ..., P_n$ be points randomly placed on a unit circle from a uniform distribution. Consider the product $D$ of all pairwise distances:
$D=\displaystyle \prod_{1\leq i < j \leq n} ...

**11**

votes

**1**answer

320 views

### Detecting a hidden convex body with line probes

Imagine that, somewhere inside an origin-centered, unit-radius sphere
$S$ in $\mathbb{R}^3$,
sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$,
with $\alpha < 1$ the fraction of ...

**5**

votes

**2**answers

351 views

### In which geometries do triangles have a Euler-line ?

In Euclidean-geometry the centroid, orthocenter and circumcenter of a triangle lie on a line.
In which other geometries does this hold ?

**1**

vote

**1**answer

259 views

### The probability a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice takes more than $N$ steps before trapping itself

What is the probability that a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice is able to take more than $N$ steps, i.e. able to take more than $N$ steps before trapping itself ...

**12**

votes

**4**answers

529 views

### Partitions of $\mathbb{R}^d$ by implicit polynomial equations

Given a polynomial
$p(x_1,x_2,\ldots,x_d)$
in $d$ variables, with maximum degree $k$,
what is the maximum number of
components of $\mathbb{R}^d$ minus $p(\ldots)=0$?
In other words, into how many ...

**5**

votes

**5**answers

504 views

### Finding an axis-aligned ellipsoid of minimal volume which contains a given ellipsoid

A friend asked me to post the following question. He's not an MO user and felt it would be better received if asked by someone who was already known to the community. This is not my area, but I'll do ...

**5**

votes

**1**answer

284 views

### Can a birational morphism between smooth varieties be dominated by smooth blow ups sequences

Suppose $f:X\rightarrow Y$ be an birational morphism between smooth varieties and D is a snc divisor on $Y$. Can we find a smooth blow ups sequence on $Y$ which dominates $f$ such that the preimage of ...

**8**

votes

**2**answers

483 views

### Hilbert style axioms for Euclidean and/or hyperbolic geometry without reference to congruence?

Hilbert's axioms from Grundlagen der Geometrie involve notions of incidence, between-ness, segment congruence and angle congruence.
Consider the sub-theories of either Euclidean or hyperbolic ...

**1**

vote

**1**answer

179 views

### Polar duality and -1 [closed]

We define the polar dual of a polytope $P$ as the set
$$\{x\in \mathbb{R}^n: x \cdot a\geq -1 \text{ for all } a\in P\}$$
Why do we require $-1$ instead of $-2$ or any other constant?

**5**

votes

**0**answers

196 views

### Rigid-body matching algorithm and clustering algorithm with groups of lines in 3D

I've been struggling with this problem for weeks, and couldn't find an appropriate algorithm to solve it. Could you guys please give me some advices or suggestions in addressing this question. Or if ...

**1**

vote

**1**answer

324 views

### Why is the physical space equivalent to $\mathbb{R}^3$ [closed]

I am trying to understand what would be the logical reasons behind our assumption that our physical space is equivalent to $\mathbb{R}^3$ or 'physical straight line' is equivalent to $\mathbb{R}$.
...

**6**

votes

**1**answer

2k views

### Get Largest Inscribed Rectangle of a Concave Polygon

I'm looking for an algorithm to find a set of largest inscribed rectangles of a concave polygon where each rectangle must be collinear with one of the edges of the polygon.
In other words, I want to ...

**2**

votes

**1**answer

165 views

### The distance between the centroid of $P$ points and the centroid of a subset of the points

Imagine I have an $(p_1, ..., p_N) \in P$ points, on a two-dimensional plane, patterned in a rectangular or hexagonal lattice arrangement in a circle of radius $R_c$, with a spacing between the points ...

**0**

votes

**0**answers

189 views

### Mapping from $\text{PSL}(2,\mathbb{R})$ to transformations of the hyperboloid

Let $F$ map points in $\mathbb{R}^3$ to points on the unit disk, $\Delta$, in the $xz$-plane (identified with $\mathbb{C}$) by projecting through $\Delta$ along lines that intersect at $(0,-1,0)$. ...

**2**

votes

**1**answer

255 views

### The equilibrium position for the body of a spider-like spring system after randomly perturbing the anchor positions of its legs

Take $N$ springs, $(s_1, ..., s_N) \in S$ of length $(l_1, ..., l_N)$, and for each spring, label one end "A" and one end "B". Connect the "A" ends of the $N$ springs to a point-like particle on a ...

**7**

votes

**2**answers

2k views

### Covering a Polygon with Rectangles

I am tyring to cover a simple concave polygon with a minimum rectangles. My rectangles can be any length, but they have maximum widths, and the polygon will never have an acute angle.
I thought about ...

**2**

votes

**2**answers

328 views

### Infinite knot composed of parallel helices

I am wondering if there is a developed theory of "infinite knots" that could capture
this object, and tell me something of its knot properties.
Imagine vertical helices in $\mathbb{R}^3$, each ...

**3**

votes

**2**answers

245 views

### Empty lattice simplex or White's theorem

White has proved (White, G. K. Lattice tetrahedra -- Canad. J. Math. 16 1964 389–396.) the following theorem:
If $T$ is a closed tetrahedron and $\Lambda$ is a lattice which contains the vertices of ...

**9**

votes

**3**answers

446 views

### Circle-arc number of a knot

I would like to build knots in $\mathbb{R}^3$ from arcs
of unit-radius (planar) circles, joined together at points where
the tangents match. Thus the knot will have
curvature $1$ at all but the ...

**2**

votes

**2**answers

461 views

### Euclidean triangulation of the plane with degree 7 at each vertex.

Hyperbolic plane has a beautiful triangulation by congruent hyperbolic triangles where all the vertices of the triangulation have degree 7, this is of course not possible in the euclidean plane, even ...

**7**

votes

**1**answer

598 views

### A remark by Gromov on 4-manifolds

Gromov remarks in a a survey on manifolds (p.12) that "it is hard to imagine that there are infinitely many non-diffeomorphic, but mutually homeomorphic, quotients of the hyperbolic 4-space by ...

**4**

votes

**1**answer

227 views

### When is the hull of a space curve composed of developable patches?

Let $C$ be a smooth curve in $\mathbb{R}^3$ that lies entirely on its convex hull,
$\cal{H}(C)$.
Under what conditions on $C$ is $\cal{H}(C)$ the union of developable surface patches?
I believe ...

**2**

votes

**0**answers

155 views

### Variations of the mean curvature

Good evening everyone,
I am facing a technical problem, maybe one of you can help.
Given a spacelike surface $S$ with mean curvature 0 in a lorentzian $3$-manifold with constant sectionnal curvature ...

**3**

votes

**1**answer

374 views

### How to partition a graph into N groups with M elements nearest?

Hi.
This problem probably already here but I could not find the right words to find it.
I have a list with 1700 points (geographic coordinates) and a need to separate into 17 groups with 100 ...

**9**

votes

**5**answers

485 views

### Minimal blocking objects with shadows like a cube

This is a more geometric version of the previous question,
"Lattice-cube minimal blocking sets". I will first specialize to $\mathbb{R}^3$, $d=3$.
View an $n \times n \times n$ cube $C_3(n)$ as ...

**3**

votes

**1**answer

167 views

### Are faces of a compact, convex body “opposed” iff their extreme points are pairwise “opposed”?

Let $P$ be a compact, convex subset of $\mathbb{R}^n$ (infinite-dimensional generalisations welcome, but not necessary). Let's say that disjoint subsets $W_1$, $W_2$ $\subset P$ are opposed if there ...

**5**

votes

**2**answers

240 views

### Lattice-cube minimal blocking sets

Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with
each of its $d$ coorindates in $\lbrace 1,2,\ldots,n \rbrace$.
Define a blocking set for a lattice cube to be a set of points
in ...