Questions tagged [packing-problem]
Questions on the packing of various (two- or three-dimensional) geometric objects.
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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 "...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 \...
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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 ...
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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 ...
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