# Questions tagged [packing-problem]

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

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### What will be the upper bound of covering number of $r B_{n}^2$

I am very curious to know The covering numbers of the r-radius Euclidean ball $B_{n}^2$ for any $r$ > 0: so $r B_{n}^2$ is a Euclidean ball with radius $r$ Have read this but unable to find. Can ...
1 vote
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### Finding the smallest box that would fit different sets or stacks of other boxes

I am trying to find an efficient way of solving this. I have 3 sets of boxes. Each set of boxes goes together. I want to find a one size bigger Box that can hold each of these sets. By this I mean one ...
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### Approximation for maximum space 2D irregular bin packing

So usually bin packing algorithms compute the tightest packed solution. I want to calculate the opposite, in my case the solution with the most space between the packed objects is needed. I tried ...
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### Which shape is the worst to pack in $\mathbb{R}^n$?

Yesterday I bought a box of chocolates and remarked with a friend how they had just put enough in to get above a certain transparent window to the inside but everything else was empty space. I added ...
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### Articles from the Journal of Recreational Mathematics from the 1970s

When reading some articles about packing puzzles I regularly stumble accros references from the "Journal of Recreational Mathematics" from the 1970s. I did not found any way of accessing ...
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### Where do the "classical lattices" come from?

I am trying to understand sphere packing problems, and it seems well-known that lattices can be only of certain kinds: the classical types (e.g. $A_n, B_n$, etc.) or the exceptional types (e.g. $E_8$)....
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### How to fill a $d$-dimensional rectangle of shape $n_1 \times n_2 \times \dots \times n_d$ with $k$ points homogeneously?

Given a $d$-dimensional rectangle ("hyper-rectangle"?) $$A = [0,n_1]\times[0,n_2]\times\dots\times[0,n_d]$$ with $n_1,\dots,n_d\in\mathbb{R}^+$ How to fill it with $k$ points homogeneously? ...
336 views

### Number of plates that can be placed on the table so that they neither overlap each other nor the edge of the table?

Table in my room is round in shape and its radius is 15 times the radius of our plates, which are also round in shape. Find the number of plates that can be placed on the table so that they neither ...
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### How to prove that a square packaging problem has an optimal solution?

In a mathematics book "Which way did the bicycle go" were given a problem: What is the smallest square into which you are able to fit 11 unit squares without overlapping. There were no ...
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### Understanding the Honeybee Conjecture vs. Sphere Packing Conjecture

I am trying to understand the differences between the Honeybee Conjecture and the Sphere Packing Conjecture (also called the Kepler Conjecture). As a quick overview: "The honeycomb conjecture ...
1 vote
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### How much wire will fit on an industrial reel?

I am trying to figure out how much cable would fit on an industrial reel. I have found a few calculators but nothing that will help me calculate this myself. If you have a reel with dimensions of 3m ...
1 vote
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### algorithm for fitting different sized circles into a square

I have the following question. I want to stack differently sized circles into a square/rectangle in a pyramid pattern and write code that can calculate the amount of circles in it. Kind of like this. ...
1 vote
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### Find $m$ (approx) maximally distant unit vectors in a unit $n$-sphere, where $m > n$

I want to find $m$ unit vectors on the surface of a $n$-sphere, where $m > n$ and the cosine angle amongst the $m$ vectors is maximal. There are a few posts on this particular problem (ref 1, ref 2,...
1 vote
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### If two lattices have the same kissing number and center density, then are they similar?

Two lattices $\Lambda$ and $\Omega$ (the $\mathbb{Z}$-span of a linearly independent set $B\subset \mathbb{R}^n$) are said to be similar if there exist a real orthogonal $n \times n$ matrix $A$ and a ...
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### 2D packing problem - how to optimise/maximise area of a a set of irregular convex polygons within a polygon?

I'm interested in a particular case of this problem: fitting odd-shaped polygons/shapes within the bounds of a rectangle. Say you have 10 sticker designs and you want to fit them all on a sheet of A4 ...
106 views

### What is this branch of mathematics?

What is the branch and topic of mathematics concerned with volume optimization of objects that can be morphed/manipulated? I.e. finding the most efficient way to position 3D objects, where each object ...
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### If you fill space with spheres and inflate them

Say that you were to fill space with spheres and inflate them what shape would the bounds make? We can assume the spheres are packed optimally, just like you would stack cannon balls, what kind of ...
1 vote
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### Circle Packing - Average distance between a random point in a circle and the second, third, fourth, ... closest circle centre

I am currently working on the following problem (see also the example drawing): Suppose I have, for example, a hexagonal circle packing or a square arrangement. And we randomly place a point in a ...
59 views

### How to calculate the number of smaller rectangles that fit into a larger rectangle with minimum wastage.

I have a larger rectangle of 2438mm width x 1219mm height. I have smaller rectangles that I need to cut out from the larger rectangle. The ideal dimensions of the smaller rectangles are 200mm width x ...
63 views

### Packing spheres in a sphere

For dimension $d$, let $N(r,d)$ = number of spheres of radius $r$ which can be packed in a sphere of radius 1. For dimension 1, $N(r,1) = \lfloor 1/r \rfloor$. From the articles below, I infer that $N$...
1 vote
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### Packing maximum number of identical rectangles in a polygon

Given a 2D polygon, convex or nonconvex, maybe some holes (also small polygons) inside. How can we pack an (approximately) maximum number of identical rectangles in the polygon (the rectangle can not ...
26 views

### How can I solve the circle packing problem for 2 unitary radius circles using KKT?

I am trying to solve the packing circle problem for two circles with unitary radius using Karush Kuhn Tucker KKT conditions. But I am stuck with the following equations: Is it possible to solve the ...
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### Why can't Pascal's Simplex with n terms be used to describe close-packing of equal n-spheres?

Pascal's simplex is a generalization of Pascal's triangle into n dimensions, just as multinomial theorem is a generalization of binomial theorem. In Pascal's triangle, binomial coefficients are ...
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### Are there semidefinite programming bounds for the number of kissing spheres in the upper hemispace with respect to the central sphere?

For dimensions n=5-7 there are good known lower bounds for the number of n-dimensional spheres that that can touch (kiss) the central sphere, but the maximum number of spheres that can do so is not ...
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### Optimal distribution of the playing cards on print templates

I have a mathematical Problem: I plan to have about 1200 cards printed for the open source game INCANTATA from ZeroNet. In doing so, I now come across the problem of distributing the cards on the ...
1 vote
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### How many rectangles of different sizes can fit in a large rectangle, until you need to exchange the large rectangle for a bigger rectangle

I'm trying to figure out how to plan something without actually going out and buying the stuff. the title is pretty much the problem, but let me give an example let's say you have 2 rectangles of ...