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3
votes
0answers
128 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 ...
5
votes
0answers
142 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
0answers
143 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
1answer
113 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
1answer
157 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
0answers
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
1answer
170 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
1answer
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 ...
1
vote
0answers
174 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
2answers
130 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
4answers
379 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
2answers
466 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
2answers
603 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
2answers
1k 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
2answers
659 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
2answers
497 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
3answers
663 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
0answers
461 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
1answer
354 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
1answer
727 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 ...
6
votes
0answers
180 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
1answer
776 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
1answer
387 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
1answer
435 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
0answers
170 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
0answers
152 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
1answer
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
4answers
785 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
1answer
542 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 ...
27
votes
3answers
3k 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
7answers
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
2answers
476 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
1answer
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
1answer
616 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
3answers
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 $\varepsilon$ ...