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Questions tagged [packing-problem]

Questions on the packing of various (two- or three-dimensional) geometric objects.

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Packing tiny open sets into a large open set

The questions Let $n$ be a positive integer, let $\lambda$ be the Lebesgue measure of $\mathbb R^n$, let $B$ and $U$ be two open subsets of $\mathbb R^n$ such that $B$ is bounded, $\lambda(B)=1$, and $...
Pierre-Yves Gaillard's user avatar
1 vote
0 answers
19 views

How many vectors can be placed in $n$ dimensions given max cosine similarity? [duplicate]

In machine learning we usually use the concept of cosine similarity to compare things. Similar things should have embeddings with high cosine similarity and different things should have embeddings ...
F. Bruno Dias's user avatar
9 votes
0 answers
174 views

For what values of $n$ can coins of radius $\frac12,\frac13,\frac14,...,\frac1n$ be held rigidly in a circular tray of radius $1$?

For what values of $n$ can circular coins of radius $\frac12,\frac13,\frac14,...,\frac1n$ (at least one of each, and no other kind of coin) be held rigidly in a circular tray of radius $1$? By "...
Dan's user avatar
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70 votes
3 answers
7k views

Geometry question about a six-pack of beer

On a hot summer day like today, I like to put a six-pack of beer in my cooler and enjoy some cold ones outdoors. My cooler is in the shape of a cylinder. When I place the six-pack in the cooler ...
Dan's user avatar
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14 votes
3 answers
855 views

Sangaku problem involving eight circles

I made the following sangaku problem. $\dfrac{\text{Area of the orange circle}}{\text{Area of a blue circle}}=\space ?$ Description of diagram. In this question, circles of the same color are ...
Dan's user avatar
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1 vote
1 answer
34 views

Packing a sphere of each integer volume at most $N$ in $\mathbb R^3$ - Does the marginal radius ever approach zero?

Given $N \in \mathbb N$, let $S_i$ denote the sphere of volume $i$ for $i \in \{1, \cdots, N\}$. Now define $r \in \mathbb R$ as the minimal radius so that you can pack all of the spheres into a ...
Snared's user avatar
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1 vote
0 answers
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What is the maximum number of (non-overlapping) small squares that fit inside a larger square? And similar question for cubes.

I am having a hard time answering the following questions, despite them seeming elementary at first glance. Is it true that the maximum number of (non-overlapping) squares with side lengths $x$ cm ...
Adam Rubinson's user avatar
1 vote
0 answers
37 views

Recognize geometric pattern in natural form

UPDATED Although my question arises from biology, it’s about geometry. I’m interested in various natural structures: fractals, packing, hyperuniformity etc. Here is photo of pores of tinder fungus or ...
lesobrod's user avatar
  • 804
4 votes
1 answer
237 views

Combinatorial rectangle packing problem

Take the numbers 1, 2, and 3, and make a list of all possible unordered pairs (ie {1,1}, {1,2}, {1,3}, {2,2}, {2,3} and {3,3}). Interpreting these as the dimensions of rectangles, you get 6 rectangles ...
Elliott Price's user avatar
5 votes
0 answers
62 views

Can ellipsoids pack better than spheres?

It is known that same-sized spheres can be packed at a density of $\pi/3\surd2$. If we uniformly stretch or compress the packed configuration as a whole, in any direction, the packing density does not ...
John Bentin's user avatar
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3 votes
0 answers
64 views

Is there a collection of disks that can't cover a unit disk, but can cover every ring centered at the origin with multiplicity?

Say that a region $R$ is covered with multiplicity by some pieces $P_1,\ldots,P_n$ if $\sum_{i=1}^n\text{Area}(P_i\cap R)\ge \text{Area}(R)$ - ie, there's enough total overlap of $R$, it just isn't ...
RavenclawPrefect's user avatar
1 vote
0 answers
55 views

What is the minimum excess area when covering a unit square with $n$ circles?

Suppose you have a unit square and want to completely cover it with $1$ circle ($n=1$). If you want to minimize the excess area, the area inside of the circle and outside of the square, you would ...
Dylan Levine's user avatar
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1 vote
2 answers
60 views

Optimal "maze length" in plane tilings

Among the tesselations of the plane (the tiles employed are a finite number of shapes which are allowed to be translated and rotated at will), which one performs best at the following problem? To ...
5th decile's user avatar
  • 2,455
12 votes
2 answers
653 views

Placing kings on a 6x6 board - who wins?

Two players alternate placing kings on a $6\times6$ chessboard, such that no two kings are allowed to attack each other (not even two kings placed by the same player). The last person who can place a ...
Akiva Weinberger's user avatar
2 votes
0 answers
137 views

Circle packing in a circle

In this circle packing problem, the task is to determine the smallest circle for a given number of unit circles. However, in the case of $n=15$, it is not clear to me how the $$\ 1+ \sqrt{6+ {2\over \...
LorKris0128's user avatar
1 vote
0 answers
57 views

How to optimally pack random triangles with fixed perimeter

You have a machine that produces random triangles of perimeter $1$ in the following way. On a stick of length $1$, the machine chooses two independent uniformly random points. If breaking the stick at ...
Dan's user avatar
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1 vote
1 answer
61 views

For any $2-$d shape, can (at least) half its area be filled up with finitely many disjoint discs of the same size?

Here, the notation $B(x,r)$ means the ball with centre $x$ and radius $r$. So in $\mathbb{R}$ this represents an interval, in $\mathbb{R}^2,$ this represents a disc, etc. Whether the ball is open or ...
Adam Rubinson's user avatar
0 votes
0 answers
18 views

Packing problem in the 3D cube - generalisation of the Loomis-Whitney / Finner / Brascamp–Lieb / Holder inequality

I have the following geometrical problem: OBSERVATION Consider the unit cube in 3 dimensions, and N orthogonal parallelepiped of size 1/N x 1/N x 1 (hence each of volume $V=1/N^2$). It is easy to ...
MarcO's user avatar
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0 answers
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Show explicitly the rank-24 Leech lattice that is also symmetric unimodular matrix with integer entries?

A typical $E_8$ lattice is of the form of $E_8$ Cartan matrix: $$\begin{pmatrix} 2 &−1 &0 &0& 0& 0& 0& 0\\ −1& 2 &−1& 0& 0& 0& 0& 0\\ 0& −1&...
zeta's user avatar
  • 191
1 vote
1 answer
152 views

What's the minimum number of hexagons required to cover the whole circle?

Suppose I want to cover a whole circle of with hexagons. See the attached photo where coloured hexagons are those required to cover a whole circle. Given the length of the side of the hexagons and the ...
Albino Tiger Barb's user avatar
4 votes
0 answers
103 views

The 18 golden rational tetrahedra

In 2020, the 59 sporadic rational tetrahedra were identified. More recently, I found exact solutions for all of them. Most of them don't pair up well in terms of similar triangles that would allow ...
Ed Pegg's user avatar
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0 votes
1 answer
141 views

What is the max number of points I can put in an n-dimensional ball with radius 1, so that the distance between any two points is $\ge 1$?

I want calculate max number of points that I can put in an n-dimensional ball with radius 1, so that the distance between any two points is $\ge 1$. I want get the exact formula or lower bound for ...
Марат Медведев's user avatar
0 votes
1 answer
101 views

Can you help me solve this space optimization problem?

I am a high school student doing a mathematics research project and I am in dire need of help! My math project is "determining the crystals needed to bedazzle a top" and the topic might seem ...
aoife_er's user avatar
5 votes
0 answers
171 views

Configuration of $n$ points inside rectangle

Given a finite set $S$ of at least $2$ points in the Euclidean plane and $x\in S$ write $$ d_S(x)=\min_{y\in S,\, y\neq x}d(x,y) $$ and $$ d(S)=\sum_{x\in S}d_S(x) $$ Now given a rectangle (product of ...
I. Haage's user avatar
  • 253
3 votes
1 answer
88 views

Largest number of corner pieces in an $m \times n$ grid?

The other day the following combinatorics problem popped into my head: Given an $m \times n$ grid, how many corner pieces can fit in it without overlapping? A corner piece is defined as such: ...
David's user avatar
  • 31
0 votes
0 answers
39 views

How many balls to cover a hyperbolic or spherical disk?

Definitions: Let $D_k$ be the disk with constant curvature $k$ and with volume $1$. We call a finite subset $X\subset D_k$ a $\delta$-covering of $D_k$ if $D_k=\bigcup_{x\in X}B_\delta(x)$, where $B_\...
Alex's user avatar
  • 943
2 votes
1 answer
153 views

Find the 16 largest quadrilaterals that fit inside the rectangle

I have a rectangle of width $w$ and height $h$. Inside this rectangle, I want to draw quadrilaterals whose edges all form the same angle magnitude against the edges of the rectangle: $$|\phi_{top}| = |...
Harry's user avatar
  • 541
1 vote
1 answer
35 views

2D Binpacking efficiency on a bin-level

I want to minimize the amount of unused space in a binpacking problem. I keep seeing binpacking being described as using the least amount of bins possible. Example (emphasis mine): assign each item ...
Parrotmaster's user avatar
0 votes
0 answers
10 views

Total illumination from packed spherical balls

Imagine I have some spheres, all transparent and of equal radius $r$. Each sphere, at its center, has a light source of identical intensity $i$ at $r$. I want to take some number of spheres and ...
gomennathan's user avatar
2 votes
1 answer
78 views

For 4 balls having diameters $26,24,13$ and $9$ cm, find minimum length of a closed parallelepiped where the four balls can be placed.

I have 4 balls which diameters are $26, 24, 13$ and $9$ cm. What is the minimum length $L$ of a closed parallelepiped in which I can put the 4 balls? In the final answer please call $L, l$ and $W$ the ...
PierreR's user avatar
  • 21
2 votes
2 answers
254 views

Can you pack $53$ bricks of dimensions $1\times 1\times 4$ into a $6\times 6\times 6$ box?

Can you pack 53 bricks of 1×1×4 size into 6×6×6 box? Source: puzzledquant.com My approach: I looked at the solution and visualized the box as $3d$ checkerboard. So in total we have $27$ such $2\...
Charlie's user avatar
  • 305
0 votes
0 answers
461 views

Covering/packing number of a sphere instead of a ball

Let $\Theta$ be a subset of $\mathbb{R}^d$ with the Euclidean norm. Let the covering number $N_2(\Theta, \epsilon)$ denote the smallest $n$ such that there exists a covering $\{\theta_1, \ldots, \...
angryavian's user avatar
  • 91.2k
3 votes
0 answers
92 views

Maximum tiling by Y Hexomino

"Y Hexomino" has a shape as shown in the picture. What is the maximum number of Y Hexomino that can be placed on a $13\times 13$ chessboard, where each Hexomino does not overlap? From the ...
rack's user avatar
  • 183
0 votes
0 answers
37 views

Stacking bricks of various dimensions

In my line of work, I do a lot of stacking and packing cuboids of various proportions. Recently I was tasked with finding a stable arrangement of 5x6 units per layer using 2x3(x1) unit blocks, and ...
Marcus Mitchell's user avatar
0 votes
0 answers
61 views

Limit density of sphere packing on a spherical surface

Analagous to circle packing in a circle, let's consider sphere packing on a spherical surface. The little spheres all have the same radius. I think there could be 3 possibilities of packing: a. every ...
feynman's user avatar
  • 149
0 votes
0 answers
45 views

Limit density of circle packing in a circle

The circle packing in a circle can be found in https://en.wikipedia.org/wiki/Circle_packing_in_a_circle http://hydra.nat.uni-magdeburg.de/packing/cci/cci.html As the number of little circles (packing ...
feynman's user avatar
  • 149
3 votes
0 answers
76 views

Seeking a (simple) proof that the sphere packing density is always less than $1$ in $n \ge 2$ dimension.

For each natural number $n$, we may define the (optimal) sphere packing density in $\Bbb R^n$ to be the number $$ D_n = \limsup_{r\to\infty} D_n(r), $$ where $$\begin{align} D_n(r) = \sup \Big\{ \frac{...
BigbearZzz's user avatar
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1 vote
1 answer
111 views

Fractional covering and packing problems.

I found the fractional covering number of the graph in the figure. And as a result, I came across the result of 5/2. Next I wanted to show its equality with fractional packing number. I found the n-...
nortedor's user avatar
19 votes
2 answers
477 views

You have $n$ rectangles of area $1$ (and variable height). Pack as many as possible in a semicircle of area $n$. How many leftovers will there be?

You have $n$ rectangles of area $1$ (and variable height). Pack as many of these rectangles as possible in a semicircle of area $n$. How many leftover rectangles will there be, in terms of $n$? How to ...
Dan's user avatar
  • 25.8k
2 votes
5 answers
190 views

Do perfect squared rectangles with corners of sizes 10, 12 and 13 exist?

A squared rectangle is a rectangle dissected into squares. squared rectangles are called perfect if the squares in the tiling are all of different sizes and are positive integers. The smallest perfect ...
Stuart Anderson's user avatar
0 votes
1 answer
70 views

How many spaced squares fit in outer square

Say I have an outer square like this, which is 14m x 14m. Each inner square is separated by a space of 3m, and one square is 1m$^2$. I of course have the image up, but how would I mathematically ...
Marko's user avatar
  • 41
0 votes
0 answers
78 views

Find a sphere tangent to four other

I am working in an algorithm to order a bed of close-packed spheres. In the case where I have got four spheres, I understand that the fifth sphere position and radius is determined by the positions ...
cosmogato's user avatar
1 vote
0 answers
70 views

What does the + sign mean at the end of a calculation result?

I discovered yesterday the Square packing in a square problem, and I ended up on this page that shows the proven answers for the first 89 values of $n$. But at the end of some answers, we can see a ...
hacb's user avatar
  • 11
1 vote
1 answer
468 views

Using algebra to figure out how many boxes fit in an area without over estimating

I'm trying to figure out how many boxes fit in a specific area without over-estimating. The problem is the width of the space isn't a whole number multiple of the width of the boxes, the depth isn't ...
James's user avatar
  • 189
0 votes
2 answers
52 views

Sum of fractions in the form $\frac{1}{n}$ from $\frac{1}{2^n}$ to $\frac{1}{3*2^{(n-1)} - 1}$ less than $\frac{1}{2}$?

https://mathoverflow.net/a/278290/501460 I've been trying to figure out why this works, and why the tiles don't go past the middle, considering all the squares together have an infinite side length. ...
Faraz's user avatar
  • 3
4 votes
1 answer
462 views

Total placement number of battleship game

The board is 4x4 and there are three types of battleships: 3x1, 2x1, 1x1. One for each type. How many total placements are possible? Notice, the ships cannot overlap and we must use all ships. (there ...
Rieder's user avatar
  • 151
1 vote
0 answers
59 views

Fomin et al., Mathematical Circles Chapter 4- Pigeon Hole Principle Problem 12. Max. no. of kings that can be placed so no two put each other in check

I found this problem in Mathematical Circles in the Pigeon Hole Principle chapter: What is the largest number of kings which can be placed on a chessboard so that no two of them put each other in ...
D S's user avatar
  • 5,316
3 votes
1 answer
149 views

Does a (5,3,4)-code exist?

I am a bit confused on whether a binary (5,3,4)-code exists. As far as I am aware, this code exists if and only if a binary (4,3,3)-code exists according to Theorem 2.7 in Raymond Hill's book "A ...
am567's user avatar
  • 339
7 votes
0 answers
251 views

Marjorie Senechal (2011): "The question of packing tetrahedra is still unsolved" Is it still unsolved?

At about 15:49 in her 2011 talk Prof. Marjorie Senechal - "Quasicrystals Gifts to Mathematics": But Hilbert understood that groups aren't everything and maybe not even the main thing. And ...
uhoh's user avatar
  • 1,893
1 vote
1 answer
69 views

$1$-Lipschitz $f:[0,1]^k\to[0,1]^n$ With Maximally Dense Image

Is there anything known about which $1$-Lipschitz functions $$f:(X:=[0,1]^m)\to(Y:=[0,1]^n)$$ for $m$ < $n$ fill the codomain maximally dense, i.e. I want to minimize $\sup_{y\in Y}d(y, f(X))$ ...
fweth's user avatar
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