Questions tagged [circle-packing]
The circle-packing tag has no usage guidance.
52
questions
-2
votes
0
answers
79
views
Expressing the cosine rule with the sine function [migrated]
I have a triangle with sides of length $(x+y)$, $(x+z)$ and $(y+z)$. We can get the angle near $x$ by the cosine rule: $\alpha(x,y,z) =\arccos \frac{(x+y)^2 + (x+z)^2 - (y+z)^2}{2(x+y)(x+z)}$. I found ...
5
votes
1
answer
251
views
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 ...
- 63
21
votes
1
answer
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 ...
- 230k
2
votes
0
answers
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 ...
- 5,827
2
votes
1
answer
100
views
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. ...
- 121
4
votes
0
answers
128
views
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, ...
- 2,997
16
votes
1
answer
748
views
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 ...
- 2,997
15
votes
1
answer
1k
views
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 ...
- 2,997
40
votes
2
answers
3k
views
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 ...
- 2,997
11
votes
0
answers
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 ...
- 2,997
4
votes
1
answer
211
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 ...
- 1,065
2
votes
0
answers
91
views
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 ...
- 161
6
votes
0
answers
77
views
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 ...
- 161
2
votes
4
answers
414
views
Can we almost cover any shape in the plane by disjoint/tangent disks of prescribed radii?
This is a cross-post.
Let $(a_n)_{n \in \mathbb{Z}}$ be some given, strictly increasing sequence of positive numbers, such that $\lim_{n \to -\infty} a_n=0,\lim_{n \to +\infty} a_n=+\infty$.
Let $\...
- 6,641
7
votes
1
answer
510
views
Packing equal-size disks in a unit disk
Inspired by the delicious buns and Siu Mai in bamboo steamers I saw tonight in a food show about Cantonese Dim Sum, here is a natural question. It probably has been well studied in the literature, but ...
- 571