Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [packing-problem]

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

Packing is distinct from tiling in that the given shapes may have gaps between them; the goal is often to minimise the relative area of those gaps, or maximise the density. For example, the best packing of equal circles in the plane is $\pi/\sqrt{12}=0.907$, and that of equal spheres $\pi/(3\sqrt2)=0.740$ (the content of Hales's theorem). Packing within a bounded region poses very different challenges due to the boundaries and is an active research topic. is often paired with this tag.

494 questions
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Gardening problem - mass planting in a circular area

First of all I am not a mathematician, forgive me if this is a stupid question. A circular area is given. How to place n plants within the area so that the minimal distance between any of two plants is maximized? I would truly appreciate some visual…
Aleksandar M
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Given 4 spheres with different center coordinates and radius, what is the maximum radius of the sphere that can fit inside the 4 sphere?

In the context of determining pore volumes in adsorbing materials, I'm trying to find the pores that a gas molecule can go through. To do so, I have so solve a complex geometry/linear algebra problem. Here is a 2D representation of the problem,…
myster
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Records of densest sphere packings

Last year it was a big news that Maryna Viazovska solved the densest sphere packing problem in dimension 8. As you know, the proof for dimension 24 soon followed. I would like to find the current records of densest sphere packings. I thought that…
P.-S. Park
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Close packing of eggs

I recently ate in a restaurant where you could see part of the kitchen, and they had a plastic bin full of chicken eggs. This prompted me to wonder about the close packing properties of egg-shaped solids, analogously to the close packing of spheres.…
mgold
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Circle packing with a twist

A quick look at Wikipedia makes it quite clear that circle packing is an open question in mathematics, with only n<20 having efficient packings and many of those are merely conjectured. My question is slightly different, but related and therefore…
Wasabi
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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 $N$ does not have a closed form for higher…
LenB
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Packing spheres in a spherical space

Given a large sphere with radius $R$, and given a set of smaller spheres with radius $r$, how many smaller spheres can fit in the larger sphere. Ignoring the boundaries between spheres, the max number is $(R/r)^3$, but how much space do the gaps…
Imran Q
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What is the least dense rigid congruent sphere packing?

I was wondering what the least dense rigid uniform packing of congruent spheres was. The lowest density packing of circles is the truncated hexagonal packing.
B H
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Packing problem cube and cuboids

Is it possible to fill a box with dimensions $10\times10\times10$ using bricks of dimensions $1\times1\times4$? If yes, how? I think De Bruijn's theorem on harmonic bricks could be helpful, but I don't know if this theorem can be applied when 2…
wnvl
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How to create successive layers of Hexagonal Closed Packing?

How do I create a HCP using the bottom A layer? In other words, if I take the first layer and make a second layer, how much do I shift it vertically (z direction) and how much do I shift it backwards and forwards (along y axis) each time I make a…
user477818
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Distant packing of spheres INTO a sphere

I would like to find the N points inside a given sphere that maximises the minimum distance between any two. In other words, how can I position N equal (unit) spheres inside a larger sphere, as they stay as distant as possible? Could anybody give…
tSirmen
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Packing of n-balls

Much has been written about the packing of circles and spheres, but I was wondering what the most efficient way there was to pack n-balls in an n-dimensional box. I saw that the most dense packing of circles is approximately .91 and the packing of…
Cheese
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Height of a Hexagonal Closing Packing Unit Cell

According to my book, the dimensions of a HCP unit cell is $2r$,$2r$, $2.83r$. How in the world is the height $2.83r$? The length and width are obviously $2r$ because there the base is a rhombus and the atoms at each corner of the rhombus are…
Nova
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Packing Problem in cuboids

I can't seem to comment on this question Packing problem cube and cuboids but it is related. I just want to know what is the specific method used in the answer so I can try to replicate it for my own question. Specifically, how do you get…
winter
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What is the side length of the smallest square containing $n$ dominoes with short side lengths $1,2,\dots,n$?

Erich Friedman has collected solutions and notes: All of these are probably optimal, except for possibly n=20. But by adding the domino areas, one gets: $$A=\frac{20(20+1)(2\cdot20+1)}3=5740$$ So the side length must be at least $\lceil\sqrt…
Ivan
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