Questions on the packing of various (two- or three-dimensional) geometric objects.

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Disjoint Hamming spheres

I'm reading the From Error-Correcting Codes through Sphere Packings to Simple Groups since I'm really interesting to face this topic first as a undergad approach, then to land to the SLAG book, more ...
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1answer
27 views

Determining Locations of Circles to Optimally Cover a Polygon

I want to completely cover a region on a map(Continental US)/polygon with circles of a certain radius. Is there a way to determine the best locations and how many circles would be needed to completely ...
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31 views

Fitting Rectangles

I have a quantity of small rectangles I need to fit in a larger rectangle frame. I need an equation to figure out what is the maximum size I can make the small rectangles before they are all too big ...
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1answer
51 views

Any algorithm in literature to solve my problem? Should call it “Segment ordering”?

A slot with length $L$. I have $n$ segments, they have a positive integer length $x_1, x_2, \cdots, x_n$, respectively, and $\sum\limits_{i=1}^n x_i = L$. My goal is to fill the slot with these ...
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2D Bin Packing with Ordering Along One Dimension

This is my second attempt at solving this particular problem (original is here: Topological sort into a limited number of bins, each with limited capacity). For clarity, I have reproduced the relevant ...
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Different formulation of this problem? Packing subsets into $k$ parts.

I am currently working on the following problem which I would like to formulate in a different way to see if any work on this has been done. Let $S = \{1, 2, \dots, n\}$ be a set and $H = \{h_1, h_2, ...
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1answer
17 views

Minimum number of objects to cover a given area, given a total perimeter

So I am given a circular area 10 square units, and I am given a length, 6 units which the total perimeters of all the shapes must add up to. All shapes are counted separately, so if I have 2 squares ...
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92 views

Upper bound on the minimum distance between $N$ points chosen inside the unit circle?

I guess this is a well-known problem but I'm not sure where to find it on the web. $N \ge 2$ points are chosen in the interior or the boundary of the unit circle. What is the best upper bound on the ...
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Optimized placing of same-size squares into rectangles

Suppose that we have several squares of the same size. We want to draw n rectangles (red and yellow rectangles here) to contain these squares. The goal is to have ...
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1answer
142 views

Circle packing – How to get the minimum length?

In an a past admission paper from a local university, I came across a problem I couldn't solve. Given $n$ circles with their respective radii $r_1, r_2, \dotsc , r_n,$ we are to find the minimum ...
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31 views

Smallest enclosing cylinder

I have a set of 3D points that approximately lie on a cylinder. This cylinder is straight and can be oriented in any direction. I would like to compute the minimal enclosing cylinder for the set; that ...
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1answer
22 views

Max number of points that fit in/on a sphere of radius r, with minimum inter-point distance also r

This just came up in a game, and I realized I don't know how to solve this. Given a sphere of radius r (say, 20'), what is the largest number of points that can be arranged within the sphere (the ...
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1answer
61 views

Packing of n-balls

Much has been written about the packing of circles and spheres, but I was wondering what the most efficient way there was to pack n-balls in an n-dimensional box. I saw that the most dense packing of ...
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2answers
96 views

How do you pack circles into a circle?

I want to know how many small circles can be packed into a large circle. Looking at Erich's Packing Center it seems that packing is a non-trivial problem to solve. My interest is practical rather ...
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8 views

what is an efficient way to accomodate squares on an irregular big area?

I'm trying to use efficiently the space in this ship, there are 3x3 guns and 1x1 guns, the best guns are 3x3 so having more is better. Is there an algorithm or a program which help me find the best ...
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28 views

Packing octahedron with cubes

Given an octahedron of side length 10 units, what is the maximum number of cubes of side length 1 unit that can be packed inside it. I have checked it for two orientations, both of them consist of ...
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2answers
46 views

Equilateral Triangle Packing Problem

Prove that an equilateral triangle allows for the greatest packing density when only packing one circle into a triangle. I have though of starting with the unit circle inscribed in an equilateral ...
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56 views

Scaling factor closest to 1 in an infinite sequential rectangle packing

The Ammann Chair can be used in an infinite dissection of a rectangle, where the pieces have a scaling factor of $ k = 1/\sqrt{\phi} = 0.786151...$. The largest piece has area $\sqrt{5}$ and longest ...
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102 views

Progressive packings in a convex shape

Take a shape, and scale it by 1 to $n$. For a tiny set of tightly related shapes, such as isosceles right triangles with shortest sides 1 and sqrt(2), scale the set of shapes by 1 to $n$. What is ...
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Regular packing of an infinite number of infinitely long cylinders in 3d space

Is it possible to pack an infinite number of congruent infinitely long cylinders into 3 dimensional space in a regular pattern? Another condition is that an equal number of the cylinders must be ...
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1answer
51 views

Height of a Hexagonal Closing Packing Unit Cell

According to my book, the dimensions of a HCP unit cell is 2r,2r, 2.83r. How in the world is the height 2.83r? The length and width are obviously 2r because there the base is a rhombus and the atoms ...
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HCP and FCC same?

FCC can be created if we keep on putting SCC 2d layers on top of each other's depression is it in some way possible for HCP {3d} too? More importantly the picture below looks exactly like FCC .My ...
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21 views

Sphere close packing with cubic sheet

In some book, it's written that layers making the 3d structure must be hexagonal for close packing of spheres. But suppose we have a simple cubic sheet and another one on top of it, with which we try ...
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41 views

Maximize distance between points on a triangle including borders?

I was trying to maximize the distance between points on a triangle for a program the other day. Snooping around led me to the circle packing problem. However, the circle packing problem assumes that ...
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2answers
94 views

Fit 2600 equally spaced points on concentric circles

My friend is working on an art project where she wants to draw 2600 dots on a circular table, symbolising the 2600 deaths of the conflict in east Ukraine. She approached me to solve this, but I've run ...
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102 views

Tetrahedron packing in Cube

I'm thinking about following solid geometry problem. Q: Suppose you have a box of "cube" shape with edge length 1. Then, How many regular tetrahedrons(with edge length 1) can be in the box? So, this ...
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32 views

2 dimensional packing Problem with flexible objects

I have a kind of bin packing problem defined as follows: given: m bins $b_1,...,b_m$ of height $h_1,....,h_m$ and width w. The packing objects simply map $1,...,w$ to integers, hence these are ...
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1answer
44 views

Minimal volume of $n$ efficiently packed spheres

Suppose there are $n$ spheres that we label $i = 1, \ldots, n$. Then suppose that the center $p_i$ of each of these spheres cannot be within distance $r$ of any other sphere. I would like to find out ...
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1answer
61 views

Quarter Circle packing

Just today, I was making tortilla chips, and I began to wonder, what is the most efficient way to pack circular quarters onto the plane? This sort of circle packing is most efficient for circles, ...
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1answer
52 views

Packing Problem in cuboids

I can't seem to comment on this question Packing problem cube and cuboids but it is related. I just want to know what is the specific method used in the answer so I can try to replicate it for my own ...
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46 views

How far can the plane be tiled by congruent regular pentagons?

What is the limit, as the radius of the disk increases, of the greatest area, in proportion to the area of the disk, of the region covered by regular pentagons of the same fixed size, all lying within ...
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90 views

Stacking circles

When I tried to stack 21 circles of radii $(30, 31, 32... 50)$ on top of each other in a tube (ID of $100$ wide), I thought they would reach the same height regardless of the order, however I was ...
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164 views

smallest square containing sectors of disc

This question occurred to me a while ago when taking leftover slices of a pizza to go. Suppose you have a unit-radius circular disc divided into $n$ equal sectors ("slices"). What is the smallest ...
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1answer
71 views

Find radius such that packing circles into a fixed rectangle maximises total area of circles

I want to pack equal-sized circles into a rectangle with width $w$, and height $h$. The total area of all of the circles should be maximised. the radius of each circle can vary, but is contrained; $r\...
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How many points are needed to fill a hypercube “r-dense”

Let $Q=[0,1]^d$. For $P\subset Q$, consider the Haussdorf-distance $d(P,Q):=\sup_{x\in Q}\inf_{p\in P}\|x-p\|$. Let $A_d(r)=\min\{|P|: d(P,Q)\leq r\}$. Considering grids shows that $A_d(r)\leq d^{d/...
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1answer
55 views

Circle Packing in an Elastic Container

A (somewhat) common problem in geometry and optimization deals with how to most efficiently pack $n$ rigid disks inside a given container of some fixed size and shape (e.g. a circular container, a ...
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1answer
99 views

Algorithm for optimizing placement of unequal circles in a given rectangular area

I am working on a project, in which I have to optimize the placement of N unequal circles in a given rectangular area, such that if these circles are considered as a sensor field, I can possibly ...
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1answer
61 views

Smallest-circle problem, but with circles instead of points?

I have a growing set of circles (and locations for those circles), each step I add one. I also need the smallest circle that contains all of the circles in my set. I found the wikipedia page about the ...
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2answers
41 views

Calculating the maximum number of vertices in a packing problem

I want to pack x number of pipes into a circle in two different formations; firstly in square formation, secondly in triangular formation. It seemed obvious to reduce this to a packing problem, i.e....
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Covering unit square

Now, I am reading this topic http://mathoverflow.net/questions/34145/can-we-cover-the-unit-square-by-these-rectangles. And do some research on it. Guys, who had written in topics, have said, that they ...
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1answer
3k views

Fractional oblongs in unit square via the Paulhus packing technique

Oblongs of size $ \frac{1}{1} \times \frac{1}{2}$, $ \frac{1}{2} \times \frac{1}{3}$, $ \frac{1}{3} \times \frac{1}{4}$, $ \frac{1}{4} \times \frac{1}{5}$, ... have a total area of 1. $\sum\limits_{...
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1answer
50 views

Integral Apollonian circle packing with unique curvatures

I was wondering if it is possible to construct an Apollonian gasket where every circle has a unique integer curvature. Take for instance the following gasket, defined by curvatures (−10, 18, 23, 27): ...
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102 views

Stacking circles with $r=\frac{1}{p}$ inside a circle with $r = 1$

Let's start with a circle with radius $1$. Now suppose we would continuously insert circles from above with radii $\frac{1}{p}$ (first a circle with $r = \frac{1}{2}$, then a circle with $r = \frac{1}{...
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3answers
169 views

What is the smallest square into which one can pack a trisected disc?

In his accepted answer to this question, David Bevan improved my answer to show that a unit disc can be cut into three sectors which fit into a square of side $2-\varepsilon$, where $\varepsilon\...
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1answer
49 views

Circular biscuits in a circular pan.

This question comes from a discussion with my wife about the more efficient way for cooking biscuits. Here the problem: We have a circular pan with a diameter of $30 cm$ and we have two round stamps ...
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1answer
143 views

Method for optimally packing a group of squares from (1 x 1) to (n x n) into a larger square?

As far as my investigation has gone, I can see that people have worked on the optimal way to pack incrementally larger squares into rectangles (page 2, .pdf), as well as the optimal ways to pack equal ...
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k'th best solution or Top k solutions to the 1-0 knapsack problem via dynamic programming

How do I find the k'th best solution to the 1-0 knapsack problem via the modification of the standard dynamic programming algorithm? LP solution will also be interesting. Thanks, Vladimir
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188 views

The smallest 8 cubes to cover a regular tetrahedron

A regular tetrahedron $T$ of edge-length $\sqrt{2}$ fits inside a unit cube:                     (Image from MathWorld.) This means that $8$ ...
2
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1answer
82 views

Circle Packing, Estimate only of number of smaller circles in a circle.

Given $x$ the number of circles of radius $r$, what is a good approximate size of the radius of a bigger circle which they fit in? To explain in actual problem terms. I want to move units in a ...
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2answers
298 views

Maximum no. of laddoos of diameter $6$cm in a box of given dimension

What is the maximum number of laddoos having diameter of $6\text{ cm}$ that can be packed in a box whose inner dimensions are $24\times 18\times 17\text{ cm$^3$}$. I found that at the lower label ...