**14**

votes

**3**answers

237 views

### Covering a hexagon

For $\epsilon > 0$ sufficiently small, can a regular hexagon with sides of length $1 + \epsilon$ be covered by seven equilateral triangles with sides of length $1$?
Motivation: Conway and Soifer ...

**7**

votes

**0**answers

68 views

### Is the equidissection spectrum closed under addition?

If a polygon can be cut into $m$ as well as into $n$ triangular pieces of equal area, can it also be cut into $m+n$ triangles of equal area?
(I'm editing after realizing that my conjecture that a ...

**11**

votes

**3**answers

593 views

### What fraction of n-point sets in the unit ball have diameter smaller than 1?

This question is inspired by a recent talk by Matt Kahle on random geometric complexes.
Some simple notation: let $\mathcal{B} \subset \mathbb{R}^d$ be the unit ball in $d$-dimensional Euclidean ...

**2**

votes

**1**answer

62 views

### Covering the annulus of d-cube

Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C+x$ = {$~\lambda c+x|c\in C$} for some $x\in R^d$ is called homothetic copy of $C$. The number
...

**3**

votes

**1**answer

102 views

### Covering a convex body with its smaller homothetic copies

Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C + x = \{ \lambda c + x \mid c\in C \}$ for some $x\in R^d$ is called a homothetic copy of $C$. The ...

**3**

votes

**2**answers

291 views

### Consecutive Integer Squared Square

Is it possible to construct a squared square out of consecutive integer squares?
Be it 1,2,3,...n or k,k+1,k+2,...n.

**45**

votes

**5**answers

2k views

### If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?

Update: The answer to the title question is not necessarily, as pointed out by Tapio and Willie. I would be more interested in lower bounds.
Monsky's famous and amazingly tricky proof says that if we ...

**1**

vote

**1**answer

148 views

### Helly's number from biconvex functions

Helly's Theorem states the following. Suppose $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...

**0**

votes

**0**answers

36 views

### Covering the annulus of symmetric convex body

Consider a symmetric convex body $A$ in $R^d$. Now, we draw another object, $A'$, concentric and translated with respect to A and having radius slightly greater than twice to the radius of $A$.
Now ...

**4**

votes

**1**answer

262 views

### Algorithms for covering a rectilinear polygon using the same multiple rectangles

Sorry for the crossing-posting: original post is here
All angles of the polygon (representing a room) are right. It may be convex or concave. Use rectangles of the same size (representing a sensor ...

**5**

votes

**0**answers

383 views

### N-balls covering n-balls

This question is a follow-on question from:
Covering a unit ball with balls half the radius
The questions are these:
Given an arbitrary dimension d, and a unit n-ball in d-dimensional Euclidean ...

**7**

votes

**0**answers

327 views

### Rectangology and squareology

I thought that rectangles were simple, and squares even simpler. Until my research has led me to several questions about rectangles and squares, which I can't solve.
I started by posting this ...

**2**

votes

**1**answer

137 views

### Helly's Theorem for Biconvex Sets

Helly's Theorem states the following. Suppose that $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...

**9**

votes

**1**answer

298 views

### Needle probing for a convex body

Suppose there is an unknown closed convex body $K$ of
volume vol$(K) = V$ inside the
unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$.
You are permitted to probe with a (one-dimensional)
...

**8**

votes

**1**answer

287 views

### Question about tetrahedron decomposition

Are there tetrahedra which can be subdivided into three parts similar to the original? I believe this would require splitting one face into three parts. I know some types of tetrahedron for which this ...

**2**

votes

**1**answer

136 views

### Is this cube packing possible?

I know how to pack $5$ unit squares in a square of side length $2+\frac{\sqrt{2}}{2}$. Is there an $\varepsilon>0$ such that there exists a packing of $9$ unit cubes in a cube of side length ...

**2**

votes

**1**answer

217 views

### Regularity of Delaunay triangulation of a hypercube

First using a three dimensional unit cube as an example for the term "regularity", we can have two possible triangulations:
(A)
(B)
We say the lower triangulation is more "regular" than upper ...

**3**

votes

**1**answer

148 views

### Simplex with edges of length at least s having smallest circumradius

Is it true that of all $n$-simplices with edge lengths greater than or equal to some parameter $s$, the regular simplex with edge lengths $s$ has the smallest circumradius? It seems obvious, but I ...

**9**

votes

**1**answer

265 views

### Maximum number of Vertices of Hypercube covered by Ball of radius R

Let $R>0$ be given and let $H^n$ be the unit hypercube in $\mathbb{R}^n$. The problem I am facing is to find the maximum number of vertices of $H^n$ which can be covered by a closed $n$-dimensional ...

**0**

votes

**1**answer

150 views

### Counting integer points in a Minkowski sum

We have known from Ehrhart theory that if $P$ is a $d$-dimensional polytope of $\mathbb R^n$ which has integer vertices then the number of integer points in $nP$ is a polynomial of degree $d$. We also ...

**3**

votes

**1**answer

113 views

### The discrete theory of compressible fluids dynamics

I am working on the discrete theory of compressible fluids dynamics, i.e., numerically solving and simulating the compressible fluids , we are interested in the way using discrete exterior calculus, ...

**11**

votes

**0**answers

333 views

### Drawings of complete graphs with $Z(n)$ crossings

Hill conjectured that the minimum number of crossings in a drawing of the complete graph $K_n$ in the plane is exactly
$$Z(n) = \frac{1}{4} \bigg\lfloor\frac{n}{2}\bigg\rfloor ...

**27**

votes

**4**answers

734 views

### Can every $\mathbb{Z}^2$ disk be pinball-reached?

Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$,
except leave the origin $(0,0)$ unoccupied by a disk.
Q. Is it the case that every disk can be ...

**2**

votes

**0**answers

86 views

### Symmetric dominance regions surrounding a Gaussian prime

Let $z=a + b i$ be a complex number which is a Gaussian prime,
on neither the $x$- nor the $y$-axis.
So $a^2+b^2$ is a prime.
Construct a region $D(z)$ surrounding $z$ which is the
largest ...

**2**

votes

**1**answer

169 views

### Maintaining boundary of unit circle arrangement

I have a process which in each step creates a new unit circle and I am interested in maintaining the boundary of the resulting arrangement in linear time.
Is there anything known about computing this ...

**3**

votes

**0**answers

116 views

### Graph drawing maximizing the volume of the convex hull

Given a graph $G=(V,E)$ and a length function $\ell:E\to\mathbb{R}_+$.
An embedding of the graph into the $d$-dimensional Euclidean space is a map $f:V\to\mathbb{R}^d$ such that ...

**3**

votes

**2**answers

247 views

### Questions on Discrete Exterior Calculus in numerial computing

I have several questions about the Discrete Exterior Calculus (DEC) in the numerical method for solving partial differential equation in physics:
(Discrete Exterious Calculus is the newly developed ...

**2**

votes

**2**answers

185 views

### Primitive orthogonal vectors/Unimodular matrices

Primitive sets of vectors are very important in the theory of point lattices, since they constitute the sets of vectors that are part of a basis for the lattice.
A set of integer vectors ...

**6**

votes

**2**answers

169 views

### Untangling entwined rigid chains in 3-space

I am interested in exploring the degree of "tangledness"
of two rigid chains in space.
A polygonal chain is a simple (non-self-intersecting) path
of segments in
$\mathbb{R}^3$, viewed as a rigid body. ...

**3**

votes

**1**answer

141 views

### Triangulation of the surface determined by sampling two of its cross-sections

I have a data set that essentially looks like the picture below, i.e., it's given by sets of points in $\mathbb{R}^3$ that sample the cross-sections of a certain surface that in principle I do not ...

**7**

votes

**0**answers

140 views

### Fractional Helly for more than one piercing

Fractional Helly Theorem says the following:
For every $0<\alpha\leq 1$ there exists $\beta = \beta(d, \alpha)$ with the following property. Let $C_1 , C_2 , ..., C_n$ be convex sets in $R^d$, $n ...

**1**

vote

**1**answer

222 views

### Finding a point farthest away from $k$ points in a polygon

There are $k$ points placed inside a polygon and I am interested in finding a point inside the polygon (not necessarily on its boundary) who's minimum distance to any of the $k$ points is maximized.
...

**11**

votes

**0**answers

380 views

### Does every connected set that is not a line segment cross some dyadic square?

A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...

**8**

votes

**0**answers

110 views

### Diameter of simplicial complex mirrored in property of Stanley-Reisner ring?

Consider a pure finite abstract simplicial complex $\Delta$. Define its diameter as the maximal distance between any two facets, i.e., between any two faces of maximal dimension $d-1$. The distance ...

**17**

votes

**3**answers

1k views

### Research trends in geometry of numbers?

Geometry of numbers was initiated by Hermann Minkowski roughly a hundred years ago. At its heart is the relation between lattices (the group, not the poset) and convex bodies. One of its fundamental ...

**2**

votes

**1**answer

295 views

### Apollonian gasket and the degree of convergence

Let $r_1,r_2\dots$ be the radii of Apollonian gasket.
I would like to know for which values $\alpha$ we have
$$\sum_{n=1}^\infty r_n^\alpha<\infty.$$
I know that if three circles $A$, $B$ and ...

**1**

vote

**0**answers

115 views

### Spectrum of Combinatorial Laplacian

The spectrum of the combinatorial laplacian is well understood for a square lattice. What about for other lattices?
In particular:
Let $ f: \mathbb{Z}^2 \rightarrow \mathbb{R} $. The usual ...

**23**

votes

**2**answers

706 views

### The kissing number of a square, cube, hypercube?

How many nonoverlapping unit squares can (nonoverlappingly) touch one unit square?
By "nonoverlapping" I mean: not sharing an interior point.
By "touch" I mean: sharing a boundary point.
...

**2**

votes

**2**answers

245 views

### packing disks tightly in the plane

Given a discrete point set $S$ in ${\bf R}^2$ with a specified base-point $p_0 \in S$, label the remaining points as $p_1, p_2, \dots$ in order of increasing distance from $p_0$ (with ties resolved ...

**3**

votes

**1**answer

393 views

### Combinatorial geodesics

[There has been a flaw in my definition - as Sergei and Andreas pointed out. I hope I could fix it.]
I want to understand how the concepts of directions, straight (or shortest) lines, and geodesics ...

**8**

votes

**1**answer

359 views

### Polyhedra that combinatorially shadow a sequence

Let $P$ be a polyhedron in $\mathbb{R}^3$.
Say that $P$ combinatorially shadows a sequence of natural numbers $S$ if
there is a continuous rotation of $P$ such that its orthogonal-projection
shadows ...

**0**

votes

**1**answer

214 views

### Is this bounded?

May be better to ask for help here. Let $v_{1}$, $v_{2}$, $\ldots$, $v_{m}$ be the vertices of a
convex polygon in the plane and $v_{m+1}$ be a vertex in the interior
of the convex polygon. Connect ...

**13**

votes

**1**answer

324 views

### The optimal constant in Vitali covering lemma

Let me restate Vitali covering lemma.
Let $\{B_i\}_{i\in F}$ be a finite collection of balls in the $\mathbb{R}^n$. Then there is $S\subset F$ such that the balls $\{B_i\}_{i\in S}$ are disjoint and
...

**14**

votes

**2**answers

682 views

### Particles chasing one another around a circle

Two particles start out at random positions on a unit-circumference circle.
Each has a random speed (distance per unit time) moving counterclockwise uniformly distributed
within $[0,1]$. How long ...

**1**

vote

**0**answers

149 views

### Lattice-point enumeration question involving linear combinations of matrices

I would like to know some references to learn more about an answer to this question, if there are any references:
Let $A_1, \dots , A_m$ and $B$ be $n\times n$ symmetric matrices. Let $$S = \{(x_1, ...

**1**

vote

**0**answers

91 views

### Rational viewing points in a polygon

We refer to the question posed in Seeing the vertices of a polygon with rational angles, but now ask for constructions or for the existence of rational viewing points. We'll call a point $p$ inside ...

**5**

votes

**1**answer

189 views

### Seeing the vertices of a polygon with rational angles

Given any convex polygon in the plane, is it always possible to find a point $p$ in its interior such that when we draw the line segments from $p$ to each of its vertices, the angles formed at $p$ are ...

**12**

votes

**1**answer

287 views

### The sparsest planar net that captures every unit segment

Let $\cal C = \lbrace C_i \rbrace$ be a collection
of rectifiable curves in the plane with the property that
every unit-length segment meets at least one curve
in at least one point.
Call such a ...

**12**

votes

**1**answer

557 views

### The geometry of crinkled aluminum foil

I wonder if the geometry of crinkled aluminum foil has been studied?
The above is a photo of foil I flattened to reuse.
It might be ...

**4**

votes

**1**answer

193 views

### Best upper bound on rate for q-ary codes

Among the many upper bounds for families of codes in $\mathbb F _2 ^n$, the best known bound is the one by McEliece, Rodemich, Rumsey and Welch which states that the rate $R(\delta)$ corresponding to ...