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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What does it mean by saying that a number is 'asymptotic to ' another number?

The following comes from the book The Geometry of Numbers by C.D. Olds, Anneli Lax and Giuliana Davidoff. After the discussion of the geometry of numbers, its application, lattice-point packing is mentioned. Let $K$ be a convex set, or body,…
Nighty
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Given 3 circles with different center coordinates and radius, what is the maximum radius of the circle that can fit inside the 3 circle in 3D?

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 need to solve a geometry/linear algebra problem. Here is a figure representing the problem. There are…
myster
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Hyper-sphere packing in dimension 9

What is the best known lattice for sphere packing in dimension 9? The 'best' lattice is still unknown in dimension $d>8$ (except for the famous d=24!)
K. Sadri
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How goes the Refined Harmonic Bin packing algorithm?

I failed to find any paper that explains the algorithm in a simple manner. I understand Harmonic(M) which goes like this: Size(1/K - 1/(K-1)] -> Type K-1 -> Pack K-1 per bin at the end Size(0 - 1/K) -> Use Next Fit now i just need to understand what…
AngelicCore
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Correct best-fit algorithm for bin packing?

I have the following numbers 6,8,9,4,3,2,10,7,14,12,6,2,3,1,10,11,13,5 I wish to know the correct way to implement the best-fit 1D Bin packing algorithm for these. Because in this video http://www.youtube.com/watch?v=B2P1TzKKWOI&feature=related they…
AngelicCore
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If you had a number of cubes of different sizes, what would be the algorithm to figure out the smallest cube they could fit in?

Is this possible to do, if so how would you do it? "If you had a number of cubes of different sizes, what would be the algorithm to figure out the smallest cube they could fit in?"
Dean Leitersdorf
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Goldilocks Packing type problem

This is a resource allocation problem I am attempting to formulate myself, so bear with me this isn't from the 12th edition of some math book. A miner is selecting 'rocks' from amongst his mine to haul back to the top. There are three components,…
Mr. AM
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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 this wasn't so difficult. Nonetheless I was…
Marcus Mitchell
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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 smaller sphere touches the surface of the main…
feynman
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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 1 big circle) goes to inf, how to prove that the…
feynman
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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 and radii of the four other spheres. It seems that…
cosmogato
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Rectangles of certain Area fitting into larger rectangle of certain dimensions

I have an issue where I need to do the following: Smaller rectangles (e.g., any size, but must have Area = 100 sq. ft.) must fit within another, larger rectangle (e.g., dimensions = 400 x 600 feet) with 4 columns and 6 rows (thus, 24 smaller…
edsager
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An infinite packing in the plane with density $0$

Let $P$ be a packing in $\mathbb{R}^2$ consisting of infinitely many unit disks. Is it possible for $P$ to have density $0$? For clarification: A packing $P$ of $D \subset$ $\mathbb{R}^2$ is a set of unit disks with pairwise disjoint interior.…
3nondatur
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How is those cases trivial in packing L's in Tans

In packing problem "L's in Tans" as presented on https://erich-friedman.github.io/packing/Lintan/ . I don't understand how is some cases trivial and others are not. For example case $n=4$ isn't trivial but $n=8$ is trivial. Can someone give me prove…
Tutan Kamon
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What seating configuration respects Covid-19 distancing but provides maximum packing?

I am a musician who plays in Celtic sessions. Typically 5 to 20 of us crowd into a pub or living room, but due to Covid-19 physical distancing recommendations, we now have to play outside in a big field. What seating arrangement provides maximum…
Eric O
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