# Questions tagged [circle-packing]

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### Is the maximal packing density of identical circles in a circle always an algebraic number?

There is a lot of interest in the maximal density of equal circle packing in a circle. And I thought that knowing whether or not the solution is always algebraic or not would be useful. My original ...
1k views

### Does greedy circle packing exhaust the measure of every bounded open set in the plane?

The greedy circle packing of a bounded region in the plane is the result of placing at each stage the largest possible disk into the region that remains uncovered. The greedy circle packing of a ...
79 views

### Another variant of the Malfatti problem

We try to add to A Variant of the Malfatti Problem As stated in the Wikipedia entry on Malfatti circles, it is an open problem to decide, given a number $n$ and any triangle, whether a greedy method ...
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### The problem of finding the smallest number of copies of a certain shape that can be placed into a space to make fitting another copy impossible

Packing problems often ask for the largest number of some identical shape that can fit in a given space without overlapping, if they are placed optimally. I'm interested in the opposite question: Q. ...
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### Can a convex frame hold all circles of radius $1/n$ immobile?

Here is a frame that holds circles of radius $1, \frac{1}{2}, \frac13, ..., \frac17$ immobile. By "immobile", I mean no circle can move without overlapping other circles or the frame, ...
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### Are there general principles that allow us to easily determine whether coins in simple arrangements in a frame can move?

Circular coins in a frame may all be stuck in their positions; for example: Another possibility is that they can all move simultaneously; I claim the following examples: It is not always obvious ...
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### Conjecture: If equal size circular coins are in a convex polygonal frame, with each coin touching exactly one edge, then all the coins can move

Earlier I conjectured that if circular coins of any sizes are in a convex polygonal frame, with each coin touching exactly one edge, then all the coins can move. A counter-example using coins of ...
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### Conjecture: If circular coins of any sizes are in a convex polygonal frame, with each coin touching exactly one edge, then all the coins can move

Suppose some circular coins (not necessarily the same size) are in a frame. The coins may be immobile, as in this example: On the other hand, they may be free to move, as in these examples (in which ...
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214 views

### Can a billiard rack be a square, for every number of balls?

A billiard rack is a rack, usually a triangle, that can hold a certain number of equal size billiard balls, such that the balls' centres cannot move within the rack. Can the rack be a square, for ...
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212 views

### Harmonic functions as limits of harmonic functions on graphs?

I have recently learned about Rodin and Sullivan's work that proved a conjecture of Thurston involving giving a construction for the map in the Riemann mapping theorem using circle packings and this ...
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### Can Chang and Wang's proof of Thue’s Theorem on circular packing be extended into other dimentions?

The simplicity of Chang and Wang's proof of Thue’s Theorem (link on arxiv) on circular packing took me by surprise. Have similar ideas been found helpful in other dimensions? For example, partition ...
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### Implications of combinatorial results towards discrete function theory on circle packings

Spurred primarily by a conjecture of Thurston in 1985, there was a series of developments in creating a "discrete analytic function" theory for maps between circle packings of complex ...
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1 vote
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### Densest safe disk packing

Inspired by current regulations regarding the minimal distance to be kept among people to prevent spreading of the COVID-19 virus and the maximal number of people in a group that is not subjected to ...
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768 views

### Packing circles with radii 1, 2, 3, ..., n in a rectangle

For each positive integer n, let $a_n$ be the area of the smallest rectangle whose area is a whole number, and inside which it is possible to pack all n circles of radii 1, 2, 3, ..., n respectively (...
119 views

### Do kissing numbers with distance $d$ always grow polynomially or exponentially in dimension?

Let $A_d(n)$ be the largest number of points that can be packed on the $n$-unit sphere, such that every point is at least $d$ apart. Compare with, for instance, https://arxiv.org/abs/1507.03631 When ...
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### Six yolks in a bowl: Why not optimal circle packing? [closed]

Making soufflé tonight, I wondered if the six yolks took on the optimal circle packing configuration. They do not. It is only with seven congruent circles that the optimal packing places one in the ...
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1 vote
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### Density of a saturated random packing of congruent circles

The problem of the expected density of a saturated random packing of unit circles in the plane can be described as follows. In a circular region $C$ of a large radius pick a point at random and draw ...
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1 vote