The packing tag has no wiki summary.

**4**

votes

**1**answer

109 views

### Equitably distributed curve on a sphere

Let $\gamma=\gamma(L)$ be a
simple (non-self-intersecting) closed curve of length $L$
on the unit-radius sphere $S$.
So if $L=2\pi$, $\gamma$ could be a great circle.
I am seeking the most equitably ...

**8**

votes

**0**answers

117 views

### Randomly placing nonoverlapping unit cuboids

Suppose one places unit cuboids of dimension $d$ with min-corners
uniformly distributed to lie in $[0,n]^d$, but with cuboid (strict) overlap forbidden.
At some point, the region is "saturated," ...

**31**

votes

**1**answer

459 views

### What measurable quantity can constrain the number of odors human can discriminate?

This is not a very typical MO question, but I hope you bear with me. It concerns a recent disagreement in the biology literature about how many different odors humans can discriminate. The authors of ...

**3**

votes

**0**answers

97 views

### Is the kissing number in $n$ dimensions always divisible by $n$? And what is the base of exponential growth of the kissing number?

And why are the kissing numbers for 1, 2, 3, 4 and 8 dimensions all highly composite numbers?

**6**

votes

**1**answer

106 views

### Translative packing constant strictly larger than lattice packing constant

Simply put, my question is this: what is the smallest dimension, if any,
where we can know for sure that a convex body exists whose translative
packing constant is strictly larger than its lattice ...

**4**

votes

**0**answers

164 views

### Packing space by cones: Translates best?

Let $C$ be a right circular cone, the convex hull of a unit-radius disk
and a point directly above the disk center at height $h$.
Is the ...

**6**

votes

**1**answer

317 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 ...

**9**

votes

**0**answers

147 views

### Characterizing the norms on $\mathbb{R}^3$ coming from Platonic solids

Recall that any sufficiently nice compact centrally symmetric convex body in $B \subset \mathbb{R}^3$ gives rise to a Banach norm on $\mathbb{R}^3$ which has $B$ as its unit ball.
Is there a nice ...

**-2**

votes

**1**answer

114 views

### Combination of 2 × 1 × 1 cubes inside a 3 × 3 × n cube [closed]

You have a block with a width of 3, depth of 3 and a height n
Given n, in how many ways can you fill this block with smaller blocks of 2 x 1 x 1?
if n is uneven, one 1x1x1 block will be unused. This ...

**4**

votes

**1**answer

168 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.

**7**

votes

**0**answers

333 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

181 views

### Is there an “accepted” jamming limit for hard spheres placed in the unit cube by random sequential adsorption?

I have a unit cube, and operating in the continuum limit (i.e. not on a lattice), I sequentially place spheres of some radius $r$ inside the cube until a filled volume "jamming limit" ...

**2**

votes

**1**answer

141 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 ...

**1**

vote

**0**answers

257 views

### 2d bin packing problem, with opportunity to optimize the size of the bin

I have been tasked with optimizing a manufacturing process. It is a non-trivial, NP hard problem. The problem is similar to the 2d bin packing problem, but we are trying to optimize the size of the ...

**0**

votes

**2**answers

133 views

### Disks Packing Variant

Usually disk packing problems require that no two disks of the packing intersect.
Does anybody know if the problem has been studied when disks may intersect but they are not allowed to contain the ...

**6**

votes

**4**answers

432 views

### Packing and isoperimetrics

Suppose a manufacturer bottles small units of liquid and ships them via very large trucks.
If the transportation cost nothing, spherical bottles would minimize the packaging cost (isoperimetric ...

**11**

votes

**2**answers

484 views

### What is the largest possible thirteenth kissing sphere?

It is well-known that it is impossible to arrange 13 spheres of unit radius all tangent to another unit sphere without their interiors intersecting. This was apparently the subject of disagreement ...

**26**

votes

**2**answers

615 views

### Careless packing

The sequence $\frac{1}{2}, \frac{1}{2\cdot 2}, \frac{1}{3\cdot 4}, \frac{1}{4\cdot 6}, \frac{1}{5\cdot 8}, \frac{1}{6\cdot 10},\ldots$ has a curious property, as follows:
a) the series with these ...

**13**

votes

**2**answers

2k views

### How many vertices/edges/faces at most for a convex polyhedron that tiles space?

I wonder if this problem has already been examined before:
Consider a convex polyhedron that tiles $\mathbb R^3$. What is the maximum of vertices/edges/faces that such a polyhedron can have?
...

**12**

votes

**2**answers

693 views

### Average degree of contact graph for balls in a box

Imagine you dump congruent, hard, frictionless balls in a box,
letting gravity compress the balls into a stable configuration
(I believe such configurations are called
jammed.)
Assume the box ...

**3**

votes

**2**answers

503 views

### Torus in $\mathcal{R}^3$

Hi
I'm interested in packing the 3 space as dense as possible using equally sized tori whose major radius is much bigger than their minor radius in.
Do you have any idea how to attack this problem?
...

**5**

votes

**3**answers

733 views

### Packing density of randomly deposited circles on a plane

Let's say that I have a rectangular two-dimensional surface of bounded dimensions, $[0,A]$ and $[0,B]$:
Under "no overlap" constraints, I sequentially deposit circles of radii $r_c$ on this ...

**13**

votes

**0**answers

473 views

### $\epsilon$-nets with respect to the cut norm

The cut norm $||A||\_C$ of a real matrix $A = (a_{i,j}) \in \mathcal{R}^{n\times n}$ is the maximum over all $I \subseteq [n], J \subseteq [n]$ of the quantity $\left|\sum_{i \in I, j \in ...

**4**

votes

**1**answer

375 views

### Are there non-tiling polyhedra that pack arbitrarily well?

The fact that an upper bound on the packing density $< 1$ has only recently been exhibited for regular tetrahedra in $\mathbb{R}^3$ (see this question) suggests that proving concrete bounds of ...

**1**

vote

**1**answer

739 views

### Sphere packing in a sphere

Let $S_a^d$ be the $(d-1)$-dimensional sphere of radius $a$ in $\mathbb{R}^d$. Let $r>0$ be a constant and $R=\nu r$ where $\nu>1$ (some constant). Are there any known upper bounds on the number ...

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votes

**0**answers

182 views

### A polynomial counting some packings in $\mathbb Z/N\mathbb Z$

Given two integers $n$ and $N$ such that $N>{n+1\choose 2}$, we denote by $\alpha_n(N)$
the number of elements $(x_1,\dots,x_n)$ in $(\mathbb Z/N\mathbb Z)^n$ such that the
$2n$ elements ...

**11**

votes

**1**answer

787 views

### Cobounded ⇒ cocompact?

Assume $\Gamma$ acts by isometries on a separable Hilbert space $H$, and
$$\operatorname{diam} H/\Gamma\le 1.$$
Is it true that $H/\Gamma$ is compact?
Stupid example. Assume the action of ...

**9**

votes

**1**answer

398 views

### Are packing-homogeneous spaces homogeneous?

Given a metric space (M,d) define the packing function P(x,R,r) to be the maximum number of non-intersecting balls of radius r with centers in the ball B(x,R). Let’s call M packing-homogeneous if the ...

**2**

votes

**1**answer

441 views

### Compute number vertex disjoint cycles in graph surrounding a face

Hi all,
If anyone has insight into the following variant of the classic problem of packing vertex-disjoint cycle into graphs I would be interested.
Given a finite undirected graph $G$ embedded in ...

**0**

votes

**0**answers

172 views

### Packing Icons Onto A screen

You are trying to pack icons onto a screen that is divided into n horizontal rows of uniformly varying size. The rows narrow by a fixed ratio as one goes up the screen from the bottom. Since the icons ...

**8**

votes

**0**answers

155 views

### Asymptotics of packing

Define $m(n,k,l)$ as the maximal size of a family $k$-element subsets of $[n]$ having the property that the intersection of every two sets is less than $l$.
As stated on wikipedia, in 1985 Rödl ...

**13**

votes

**1**answer

2k views

### A circle packing conjecture

Consider $n$ circles with variable radii $r_1,\ldots, r_n$ that pack inside a fixed circle of unit radius. In other words, all $n$ variable-radius circles are contained in the unit radius circle and ...

**13**

votes

**4**answers

797 views

### Pennies on a carpet problem

I recently read the following "open problem" titled "Pennies on a carpet" in "An Introduction To Probability and Random Processes" by Baclawski and Rota (page viii of book, page 10 of following ...

**9**

votes

**1**answer

551 views

### Upper bound for tetrahedron packing?

There have been several recent advances on packing regular tetrahedra in $\mathbb{R}^3$. All the results I've seen have been lower bounds -- first John Conway and Sal Torquato showed that there ...

**29**

votes

**3**answers

4k 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 } ...

**4**

votes

**7**answers

1k views

### How to generate a net on a 8-dimensional sphere

Using Matlab, how to generate a net of 3^10 points that are evenly located (or distributed) on the 8-dimensional unit sphere?
Thanks for any helpful answers!

**7**

votes

**2**answers

481 views

### Coiling Rope in a Box: Decidable?

Is the problem Coiling Rope in a Box decidable? To be specific, is this decidable?
Given $L > 0$ and $r \in (0,\frac{1}{2})$,
both rational,
can a rope of length $L$ and radius $r$
fit ...

**17**

votes

**1**answer

1k views

### Coiling Rope in a Box

What is the longest rope length L of radius r that can fit into a box?
The rope is a smooth curve with a tubular
neighborhood of radius r, such that the rope does not
self-penetrate. For an open ...

**9**

votes

**1**answer

626 views

### Packing twelve spherical caps to maximize tangencies

Suppose that $v_i$, for $i \in \{1, 2, \ldots 11, 12\}$, are twelve unit length vectors
based at the origin in $R^3$. Suppose that $|v_i - v_j| \geq 1$ for all $i
\neq j$. What arrangement of ...

**16**

votes

**3**answers

2k views

### How many unit squares can you pack into a rectangle with nearly integer side lengths?

Earlier today, somebody asked what looks like a homework problem, but admits the following reading which I think is interesting:
Suppose $a_1,\dots, a_n$ are positive integers, and $\varepsilon$ ...