Questions on the packing of various (two- or three-dimensional) geometric objects.
-1
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0answers
43 views
sphere packing empty spaces volume percentage [closed]
i have roughly calculated the percentage of empty spaces vs spheres volume which i think is some what 5% in a clumped(pudding, not planer sphere packing) type packing, the packing that further goes ...
1
vote
1answer
32 views
Is there an optimized version of rectangle packing algorithm?
I have a rectangle with 200 width and 100 height. I have a mix pool of 50 rectangles and boxes. The rectangles comes in shapes like 20x40, and 40x20. The boxes will come in shapes of 20x20 and 40x40. ...
0
votes
1answer
64 views
Find coordinates of n points uniformly distributed in a rectangle
I have a rectangle R of width W and height H.
I have N points inside this rectangle.
I need to find an algorithm to position my points in the rectangle in the most uniform way possible (no overlaps, ...
0
votes
0answers
170 views
2d bin packing problem, with opportunity to optimize the size of the bin
I have been tasked with optimizing a manufacturing process. It is a non-trivial, NP hard problem. The problem is similar to the 2d bin packing problem, but we are trying to optimize the size of the ...
1
vote
1answer
74 views
What is the relative behaviour when a center circle surrounded by 6 circles is (recursively) replaced by 6 circles
Start with a given "inner" circle of arbitrary radius (blue) centered at C. Surround it by 6 circles of equal radius. This concerns to known issues of circle packing and is a frequently treated ...
0
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1answer
33 views
number of overlaps in sphere covering
I have a problem and I'm hoping someone can point me in the right direction. I have the following conjecture:
For a set of spheres of arbitrary radii in $\mathbb{R}^d$, in which the center point of ...
0
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0answers
42 views
Random rectangles placement with minimal overlap and good dispersion.
I have a Big Rectangle (axis-oriented) containing a lot of Small Rectangles (with the same orientation of the parent and with a fixed size of 82x176 pixels).
Now I have a Small Rectangle which is ...
4
votes
1answer
71 views
Packing three squares into an equilateral triangle
I am trying to pack 3 equal, largest possible sized squares into an equilateral triangle.
0
votes
0answers
33 views
Optimal packing for a box layout
I am not mathematically inclined enough to express this question rigorously, but it certainly sounds like a problem for mathematics.
I was making a cutout from a piece of cardboard to make a box with ...
1
vote
1answer
70 views
Smallest Circle that encircles $4$ circles
I want to calculate the radius of the smallest circle (radius $R$) that can hold $4$ circles (with radii $a, b, c, d$) inside it, such that:
No circles overlaps one other.
$a \ge b \ge c \ge d.$
...
0
votes
0answers
74 views
Optimal fitting of spheres in a cylinder.
how to find the minimum height and width of a cylinder containing n identical spheres?
Please suggest some simple introductory level papers on this topic.
0
votes
1answer
54 views
What is the minimum of squares to fit within rectangular box 10 x 16?
I'm trying to apply circle packing data to a 10 x 16 inch sheet for printing, here: http://hydra.nat.uni-magdeburg.de/packing/csq/csq.html#Applications
And I want to achieve the least waste by ...
3
votes
0answers
42 views
continuously distributed diameters for maximally random jammed spheres on a two dimensional plane
Consider a collection of discs in a maximally random jammed state (also known as random close packing).
If these discs are perfectly circular and have varying radii from a known distribution, that ...
1
vote
0answers
56 views
Five squares in a box.
Let $A,B,C,D$ and $E$ be five non-overlapping squares inside a unit square, of side lengths $a,b,c,d,e$, respectively. Prove that
$$a+b+c+d+e \le 2.$$
(This problem is a continuation of my previous ...
0
votes
1answer
50 views
Closest Packing of Spherical Caps
Let the surface $S_n$ of the unit ball in $\mathbb{R}^n$ centered at the origin $O$ be defined as the set of points $P(x_1,x_2,…,x_n )$ such that $x_1^2+x_2^2+⋯+x_n^2=1$. Let the spherical cap $C(α)$ ...
2
votes
1answer
27 views
Dividing a range into major and minor divisions
I'm drawing a graph, and need to annotate the x axis. The x axis is dimensioned in days -- eg, 11/25, 11/26, 11/27... The number of of days on the x axis may range from about 10 to perhaps 100 or ...
4
votes
0answers
46 views
Optimal packing of number sets with limited overlap
When considering something completely different yesterday I came across the following problem:
Let $X = \{ 1,2,3,...,n\}$. What is the maximal number of subsets of $X$ of order
$m$ one can choose ...
22
votes
3answers
455 views
Two squares in a box.
According to Arthur Engel, "Problem Solving Strategies", this problem goes back to Erdös, but I cannot find the solution:
Let $A$ and $B$ be two non-overlapping squares inside a unit square, of side ...
1
vote
2answers
85 views
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.
12
votes
3answers
944 views
Minimum number of integer-sided squares needed to tile an $m$ by $n$ rectangle.
Let $T(m,n)$ for integers $m,n$ be the least number of integer-sided squares needed to tile an $m\times n$ rectangle. Clearly $T(kx,ky)\leq T(x,y)$. Are there integers $x,y,k\gt 1$, such that ...
3
votes
1answer
31 views
Interiors of spherical caps intersect if and only if …
A spherical cap is defined by $C(x_1,\alpha_1)=\{y \in S^{n-1} : x\cdot y \geq \cos(\alpha)\}$ and $\alpha_i\in [0, \pi]$ and $x_i \in S^{n-1}$ ( $x_1 \cdot x_2$ refers to the inner product of ...
1
vote
1answer
75 views
A proof that the hexagonal lattice describes the optimal sphere packing in two dimensions
In an effort to understand sphere packing in 24 dimensions, I figured that I should start with 1 and 2 first. The proof of 1 dimension is obvious, since we can achieve 100% density, and you cannot do ...
1
vote
1answer
181 views
Mathematics of Tetris 2.0
Based on the question The Mathematics of Tetris, I was wondering if it is possible to have a series of tetris blocks that is impossible to clear. For example, getting the string TTTSS.. forces the ...
0
votes
1answer
65 views
Tiling a minimal perimeter region with $n$ unit squares
Suppose I have $n$ identical unit squares and I want to use them all to tile a region with minimal perimeter $p(n)$. For instance I guess $p(n^2)=4n$, by arranging them im a $n\times n$ square.
Is ...
1
vote
1answer
70 views
geometric bin packing with circles
Consider a space of finite area (for eg., a rectangle). We need to pack circles of fixed radius in this finite area. Note that the position (coordinates) of every circle in space is given and fixed.
...
1
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0answers
83 views
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 ...
1
vote
0answers
37 views
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) ...
0
votes
0answers
186 views
Sphere packing in a cuboid algorithm
Given a (3D) cuboid and an integer N, how can I position N spheres that fit inside without touching such that the radius of the spheres is maximised?
Is there some group theory that I need to know, ...
2
votes
0answers
78 views
What kind of math is required to solve packing problems?
Sometimes while I'm daydreaming I come up with math problems for myself, to solve. I don't know why but they are mostly packing problems. I don't know how to solve them mathematically but I could ...
1
vote
2answers
967 views
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 ...
2
votes
3answers
356 views
How many unit hexagonal tiles can be placed inside a larger hexagon of sides $a,b,c,a,b,c$?
How many unit hexagonal tiles can be placed inside a larger hexagon of sides $a,b,c,a,b,c$? I originally came across this puzzle on the codeforces website.
My first question: what is the mathematics ...
2
votes
1answer
545 views
How many rectangles can fit in a polygon with n-sides?
I am trying to write an algorithm to solve a problem I have. I have a few ideas of what the algorithm might be like but I am posting to see if anyone else has a better more efficient solution or any ...
3
votes
1answer
63 views
A shape that covers all box with certain side lengths
For a fixed $n$, what is the shape with the smallest volume, such that by rotation and translation, it can cover any $n$-box with dimension $b_1\times \ldots \times b_n$, where $b_1+\ldots+b_n=1$.
I ...
0
votes
0answers
48 views
Minimal Tiling versus Sphere Packing
Are there any significant differences with sphere packing and minimal tiling?
I have been researching the problem for a good while and I have yet to come up with a good answer as to any significant ...
1
vote
1answer
132 views
A question regarding sphere packing
The question of how many smaller spheres can be fit into a larger sphere is fascinating and has been examined extensively. I was curious, though, about the scenario of packing spheres of different ...
4
votes
0answers
123 views
Sphere Covering Problem
Is it possible that one can cover a sphere with 19 equal spherical caps of 30 degrees(i.e. angular radius is 30 degrees)? A table of Neil Sloane suggests it is impossible, but I want to know if anyone ...
0
votes
1answer
158 views
Packing Density of Tetrahedra - Explicit Calculations
I am researching problems relating to finding the optimal packing density of tetrahedra and I am driving myself crazy with the following very elementary calculations which do not seem to make sense.
...
0
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0answers
42 views
What is the minimum possible average distance to the centroid for a packing of $N$ circles?
What is the minimum possible average distance to the centroid for a packing of $N$ circles?
In other words, what is the minimum of
$F = \frac{1}{N} \sum_{i=1}^{N} \|\mathbf{x}_i-\bar{\mathbf{x}}\|$,
...
0
votes
1answer
103 views
packing boxes inside boxes
given 2 boxes (in 3-space) determine if one of the boxes resides within the other, or if a third box must be constructed that holds them both?
given that a box is defined by its center($x,y,z$), and ...
0
votes
1answer
540 views
Problem of packing spheres of radius $\rho$ into a cylinder
Given a cylinder of radius $R$ and length $L$, I need to find the number of spheres which is possible to pack into the cylinder as a function of the radius $\rho$ of the spheres. I found something ...
2
votes
1answer
588 views
Packing squares into a circle
I need determine the maximum number of squares of the given size that can be packed into a circle of the given radius. Squares can be rotated. I'm not sure how complex this problem is and i can find ...
7
votes
2answers
177 views
Can a finite number of squares, with total area at most 1, be fitted into a square with area 2?
It seems to be a theorem that a finite number of squares, with total area at most 1, can be fitted into a square with area 2 without overlaps. I am looking for a proof of this.
Google led me to this ...
1
vote
1answer
303 views
Optimal approximation of square area with identical circles
There are 86 pages on this site alone under a search for: area, square, circle, convergent, approximation. I found one that arguably asks the same question here, but I am not sure.
The question is, ...
3
votes
1answer
106 views
What would you call Apollonian circles that are located within polygon
Sorry, if this question is dumb, but:
When you try to fill the circle with other circles - its called Apollonian circles, and there are resources about it, and some example algorithms.
But i want to ...
3
votes
2answers
190 views
Interval packing problem
Yesterday I came up with this problem and I just cannot get it out of my head:
Problem: Consider a finite $I$ set of integer-valued intervals $I_i=(a_i,b_i)$ $i = 1,\ldots,n; a_i \in \mathbb{N_0}, k ...
1
vote
1answer
113 views
Restricted Cube Packing
I want to pack n cubes in 3-space under the following 3 restrictions:
1) At each vertex only 2 cubes may touch
2) No two cubes may share an edge
3) No two cubes share any subface
2,3 just mean ...
3
votes
0answers
336 views
Project Euler Question 222
Would I be wrong to assume that the solution to this problem:
What is the length of the shortest pipe, of internal radius 50mm, that can fully contain 21 balls of radii 30mm, 31mm, ..., 50mm?
...
4
votes
1answer
436 views
how many smaller circles(radius is equal) I can fit within a larger circle
then the question is,the larger radius D,the small radius d,get the largest number of small circle put in the larger?
1
vote
2answers
139 views
Pack box inside “smaller” box
Now, there is a puzzle that is quite well known as far as I know, concerning the packing of rectangular boxes in 3-dimensional space. You also have a measurement of a box as the sum of the hight, ...
2
votes
1answer
102 views
Smallest enclosing cylinder for an irregular body
I have an $3$-dimensional irregular body composed of 162 points $(x,y,z)$.
I need to find the smallest enclosing cylinder for this body. Is there a standard algorithm for achieving this?
