**48**

votes

**3**answers

5k views

### What is the status of the Gauss Circle Problem?

For $r > 0$, let $L(r) = \# \{ (x,y) \in \mathbb{Z}^2 \ | \ x^2 + y^2 \leq r^2\}$ be the number of lattice points lying on or inside the standard circle of radius $r$. It is easy to see that $L(r) ...

**36**

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 ...

**24**

votes

**3**answers

713 views

### Largest possible volume of the convex hull of a curve of unit length

What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$?

**5**

votes

**1**answer

472 views

### Difference Sets

Suppose
$$
P \subseteq \{1,2,\dots,N\},\quad |P| = K
$$
We calculate the differences as: $$d=p_i-p_j\mod N,\quad i\ne j$$
Now let $a_d$ denote the number of occurrence of $d$ (for $d = 1, 2, \dots , N ...

**4**

votes

**1**answer

225 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 ...

**59**

votes

**6**answers

2k views

### Does every polyomino tile R^n for some n?

This is a question posed by Adam Chalcraft. I am posting it here because I think it deserves wider circulation, and because maybe someone already knows the answer.
A polyomino is usually defined to ...

**40**

votes

**12**answers

2k views

### Can a discrete set of the plane of uniform density intersect all large triangles?

Let S be a discrete subset of the Euclidean plane such that the number of points in a large disc is approximately equal to the area of the disc. Does the complement of S necessarily contain triangles ...

**33**

votes

**6**answers

3k views

### Is it possible to partition $\mathbb R^3$ into unit circles?

Is it possible to partition $\mathbb R^3$ into unit circles?

**44**

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 ...

**13**

votes

**1**answer

348 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

**3**answers

519 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 ...

**17**

votes

**4**answers

1k views

### What is the minimum N for which there exist N points in the plane that cannot be covered by any number of non-overlapping closed unit discs?

This problem was posed in March 2010 at G4G9 in a talk by the Japanese mathematician Hirokazu "Iwahiro" Iwasawa. He claims there is a simple proof that N > 10, though he did not share it with the ...

**10**

votes

**1**answer

391 views

### Chord arrangement that avoids confining small or large disks

This question is
These two questions are two-dimensional variations on this recent MO question,
"Threading pinholes in the wall of cylinder to pass through an internal coordinate."
Noam Elkies ...

**20**

votes

**3**answers

880 views

### Visibility of vertices in polyhedra

Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ is an interior point of $P$. Is it true that $X$ can see at least one vertex of $P$? More ...

**4**

votes

**4**answers

1k views

### Delaunay triangulations and convex hulls

This is a reference request.
I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you ...

**14**

votes

**1**answer

332 views

### Are all well behaved “mean” functions on $\mathbb{R}^+$ equivalent?

Given a set $S$, a function $M: S\times S \rightarrow S$ is a mean if it satisfies the properties:
$M(a,a)=a\qquad$ (identity)
$M(a,b)=M(b,a)\qquad$ (commutativity).
and possibly
...

**9**

votes

**3**answers

393 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
...

**3**

votes

**1**answer

240 views

### Min Bend Orthogonal Knots

I am seeking literature on 3D orthogonal drawings of knots,
especially minimum bend drawings.
An orthogonal drawing employs segments parallel to the axes of
a Cartesian coordinate system.
A bend is a ...

**25**

votes

**6**answers

3k views

### Covering a unit ball with balls half the radius

This is a direct (and obvious) generalization of the recent MO question, "Covering disks with smaller disks":
How many balls of radius $\frac{1}{2}$ are needed to cover completely a ball of radius ...

**9**

votes

**2**answers

958 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 ...

**3**

votes

**1**answer

238 views

### cover and hide with squares

I am studying two numbers, related to squares, that can characterize a polygon P:
MinCoverNumber = the minimum number of axis-aligned squares required to exactly cover P (the covering squares may ...

**92**

votes

**5**answers

5k views

### Gaussian prime spirals

Imagine a particle in the complex plane, starting at $c_0$, a Gaussian integer,
moving initially $\pm$ in the horizontal
or vertical directions. When it hits a Gaussian prime, it turns left ...

**68**

votes

**5**answers

6k views

### Is there a dense subset of the real plane with all pairwise distances rational?

I heard the following two questions recently from Carl Mummert, who encouraged me to spread them around. Part of his motivation for the questions was to give the subject of computable model theory ...

**39**

votes

**3**answers

5k views

### Can we cover the unit square by these rectangles?

The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1.
It is easy to show that
$$\sum_{1 \leq k } ...

**47**

votes

**5**answers

3k views

### Can an arbitrary collection of circles of total area 1/2 fit into a circle of area 1?

Assume the circles are actually open disks, otherwise two circles each of area $\frac{1}{4}$ wouldn't fit into the circle of area 1.
This seems like it should be true, thinking about packing ...

**15**

votes

**1**answer

354 views

### Does every convex polyhedron have a combinatorially isomorphic counterpart whose all faces have rational areas?

Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas?
Does every convex polyhedron have a combinatorially isomorphic counterpart whose edges ...

**34**

votes

**1**answer

1k views

### Pach's “Animals”: What if the genus is positive?

Janos Pach asked a deep question 23 years ago (1988) that remains unsolved today:
Can every animal—a topological ball in $\mathbb{R^3}$ composed of unit cubes glued face-to-face—be ...

**20**

votes

**1**answer

497 views

### Voronoi cell of lattices with the same profile

Definition 1. Given a body $V$ in $\mathbb R^n$,
the function $p_V\colon \mathbb R_+\to \mathbb R_+$
$$p_V(r)=\mathop{\rm vol} [V\cap B_r(0)]$$
will be called profile of $V$.
Definition 2. Define ...

**19**

votes

**8**answers

2k views

### Higher-dimensional Catalan numbers?

One could imagine defining various notions of higher-dimensional Catalan numbers,
by generalizing objects they count.
For example, because the Catalan numbers count the triangulations of convex ...

**15**

votes

**3**answers

1k views

### Is Euler characteristic of a simplicial complex upper bounded by a polynomial in the number of its facets ?

What is the best upper bound known on the (absolute value of) the
Euler characteristic of a simplicial complex
in terms of the number of its facets ?
In particular, I am interested in proving or ...

**6**

votes

**0**answers

115 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 ...

**15**

votes

**5**answers

959 views

### Is a rhombus rigid on a sphere or torus? And generalizations.

If a rectangle is formed from rigid bars for edges and joints
at vertices, then it is flexible in the plane: it can flex
to a parallelogram.
On any smooth surface with a metric, one can define a ...

**12**

votes

**0**answers

442 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 ...

**7**

votes

**2**answers

383 views

### Two cubes in unit cube

A cube of side one contains two cubes of sides a and b having non-overlapping interiors. How to prove the inequality $a+b \le 1$? The same question in higher dimensions. It was asked, but not answered ...

**10**

votes

**1**answer

468 views

### Largest pair of homometric Golomb rulers?

A Golomb ruler is a set of $n$ integers that determines $\binom{n}{2}$ distinct differences.
Two sets are homometric if they determine the same (multiset) of differences.
For example,
...

**17**

votes

**1**answer

500 views

### Longest of random worm-like paths in $\mathbb{Z}^2$

Imagine at each lattice point of $\mathbb{Z}^2$ within $[1,3n]^2$,
with coordinates
$\equiv 2 \bmod 3$,
we place, with equal probability, one of these six patterns:
The result ...

**8**

votes

**1**answer

246 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?

**8**

votes

**3**answers

770 views

### On maximal regular polyhedra inscribed in a regular polyhedron

Let T, C, O, D, or I be regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively. Suppose that the outer polyhedron have edge-length 1.
For example, it's easy to prove that ...

**15**

votes

**2**answers

650 views

### Tiling the square with rectangles of small diagonals

For a given integer $k\ge3$, tile the unit square with $k$ rectangles so that the longest of the rectangles' diagonals be as short as possible. Call such a tiling optimal. The solutions are obvious in ...

**12**

votes

**7**answers

518 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 ...

**7**

votes

**1**answer

243 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 ...

**6**

votes

**5**answers

261 views

### Packing obtuse vectors in $\mathbb{R}^d$

I came across this attractive theorem:
Theorem. In $\mathbb{R}^d$, there can be at most $d+1$ vectors that
form an obtuse angle with one another.
This was proved1 as a corollary of a lemma about ...

**5**

votes

**1**answer

541 views

### Linear transformation of a polyhedron

Is there a simple proof that shows:
Linear transformation of a $\mathcal{H}$-polyhedron (i.e. the intersection of
finitely many closed half-spaces) is a $\mathcal{H}$-polyhedron.
Minkowski sum of ...

**4**

votes

**0**answers

260 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 ...

**14**

votes

**1**answer

272 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 ...

**9**

votes

**1**answer

2k views

### Numbers of intersection points and lines

Hello,
I don't know if this question has already been posted, I have made a little search with keywords and did not found it, sorry if I missed anything.
Is it possible to characterize the set of ...

**8**

votes

**1**answer

447 views

### Polyhedron not circumscribed about a sphere

Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere.
My teacher ...

**7**

votes

**1**answer

383 views

### Building a polyhedron from areas of its faces

Is there a known algorithm which, given a finite multiset (unordered list) of integers $A$, returns a yes/no answer for ...

**6**

votes

**1**answer

188 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 ...

**6**

votes

**1**answer

285 views

### When is a 0-1 matrix a one-intersection incidence matrix?

The following problem is what motivated my previous MO question.
It is easily seen that for any given 0-1 matrix $M$, one can always find
a set $\mathcal P$ of points, and a set $\mathcal C$ of ...