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

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28 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 ...
5
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1answer
62 views

Arranging circles around a circle

$N$ circles are given by their radii: $r_1$, $r_2$,..., $r_N$. They are arranged around another circle so that they pack, like in this picture (order of $N$ circles should be preserved): What is ...
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0answers
50 views

packing problem of semicircles into rectangle

I have problem. How can I get the maximum amount of semicircles (for example radius $35\;mm$) into rectangle $(485\times 185\:mm)$. I found many articles about packing of circles but nothing about ...
2
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1answer
76 views

Packing of discrete random variables with finite second moment

I am considering a discrete random variable $X \in\mathbb{R}$ with $N$ points (where each point has non-zero probability) and $E[X^2]=1$ and $E[X]=0$. Let $d_l$ be the the smallest distance between ...
2
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0answers
37 views

For what hexagon size can I pack $n$ hexagons into a rectangle of $s$ area?

I have a fixed number of identical regular hexagons I use to build a honeycomb looking grid of hexagons. I have a rectangular container of known dimensions. My job is to figure out how big the ...
4
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0answers
31 views

Shaking a box of rocks (Optimal Packing)

My coworker was telling me that when he plants seeds on his farm, he puts them all in a large container on the tractor and after a period of just driving, the seeds are more densely packed than when ...
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0answers
15 views

Sorting Bigger Boxes to Smaller Boxes

I am currently working on a Bin Packing program and need to know what would be the most efficient way of getting boxes (arbitrary width, length, and height) to be sorted in the manner below? ...
5
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1answer
71 views

An algorithm for filling a moving truck

I was recently helping a friend move. I stood in the moving truck as other people brought boxes and furniture pieces from inside the house. My job was to arrange these items in an efficient way inside ...
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1answer
17 views

Is optimal bound for Alcuin's triangular city problem known?

Alcuin's triangular city problem is Problem 28 from Propositiones ad Acuendos Juvenes. There is a triangular city which has one side of 100 feet, another side of 100 feet, and a third of 90 ...
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1answer
59 views

How many rectangles or squares of (fixed and equal sizes) can fit inside a square of fixed size 320 x 320 ?

Our factory creates graphite sheets. Their machine can only create sheet of one fixed size which is 320 x 320 mm (0.1240 sqm.). Clients come to us with requests for different sheet sizes, all ...
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1answer
50 views

About the squares in square packing problem with 11 squares

In this web site there are the solutions of a lot of packing squares problems. I know a very simple method to calculate with pen and paper the solution for the ten squares in a square using the same ...
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1answer
22 views

How many polygons can be packed into a larger polygon?

Let's say I have a polygon and a set of smaller polygons. The question asks "how many of the smaller polygons can be fit into the larger polygon without any overlaps?". Obviously this is a very hard ...
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1answer
143 views

How to find out percentage of total area covered by 20 identical circles touching one another on a sphere?

20 identical circles touching one anther at total 30 different points (i.e. each one exactly touches three other circles) on a spherical surface with a finite radius. Thus all 20 circles are packed on ...
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11 views

How to estimate a valid lower bound for this generalized bin packing problem

Given m bins, each has a capacity of capj, and its cost is defined as cj,low+(cj,high-cj,low)*urj, where, urj is the utilization ratio of the bin (of course, the cost is zero when the utilization ...
6
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2answers
249 views

Hexagon packing in a circle

Suppose I want to pack hexagons in a circle, as on the drawing below (red indicates "packed" hexagons). I am wondering what is known about this problem. Specifically, I am interested in an ...
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0answers
30 views

Finding optimal height for objects in 2D packing

I have a problem that I need to find the right algorithm for and I am not sure which avenues are going to be easiest. I have a 'row' of fixed height and we can assume infinite length. fig.1 ...
4
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1answer
65 views

What is the maximum number of $15 cm\times 15 cm$-square I can cut from a diameter $50 cm$- circle?

What is the maximum number of $15 cm\times 15 cm$-square I can cut from a diameter $50 cm$- circle?(this square miss angle is ok, only such this square area not than $\frac{1}{5}$ out the ...
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2answers
43 views

Can there exist $3$, $4$ and $5$-faceted shapes with congruent flat sides in $\mathbb{R}^3$?

I rose this question in my discrete math class (the unit on probability) not too long ago. For instance, a two-sided shape (like a coin) can be one with any geometrical shape as its "side," such as a ...
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1answer
28 views

Packing custom length squares into a rectangle with a custom ratio

In the image there are 2 rectangles: first with a ratio of 1:2.5 and the second with a ratio 0.65:1. Trying to pack biggest squares possible, in the first example 4 can be packed and in the second ...
6
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1answer
98 views

What is the equation to evenly distribute circles in a spiral?

What is the equation to evenly distribute circles in a spiral? I have attached a picture to show what I am trying to achieve and need to know what the equation is for this.
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17 views

Lattices, compact orbits, and admissible boxes

A box in $\mathbb{R}^n$ is a set of the form $[-b_1,b_1]\times \cdots \times [-b_n,b_n]$ with $b_i>0$ for all $i$. For any unimodular lattice $\Lambda$ define ...
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3answers
293 views

Finding the area of the 4th triangle, given the areas of the other 3, and all the 4 form a rectangle

In one of my tutorial classes, when I was studdying in 9th class (I am in 10th now), our tutor gave us a problem saying it’s a difficult one, and to him, it was incomplete. This is that problem: ...
34
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2answers
2k views

Why can't three unit regular triangles cover a unit square?

A square with edge length $1$ has area $1$. An equilateral triangle with edge length $1$ has area $\sqrt{3}/4 \approx 0.433$. So three such triangles have area $\approx 1.3$, but it requires four ...
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29 views

Packing Problem

Hey I'm trying to solve a problem of figuring out how many rectangles of a certain size would fit within a triangle and a trapezium, is there a formula that can be used for this? Say I had a ...
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78 views

Bin packing 3D / container loading problem : algorithm with “real” constraints handling

I search a bin packing 3D (or container loading) algorithm (for truck load) with handle of many differents constraints (for each item : stackability, possibles orientations, multi customers, max ...
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22 views

Formulating and solving box unloading as graph problem

I have a set of boxes as those pictured below. I can only remove boxes that dosen't have any boxes on top of it. In every "move" I can move any box that is currently available, but I have limitations ...
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1answer
36 views

What is the side length of the smallest square containing $n$ dominoes with short side lengths $1,2,\dots,n$?

Erich Friedman has collected solutions and notes: All of these are probably optimal, except for possibly n=20. But by adding the domino areas, one gets: $$A=\frac{20(20+1)(2\cdot20+1)}3=5740$$ ...
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1answer
48 views

What does it mean by saying that a number is 'asymptotic to ' another number?

The following comes from the book The Geometry of Numbers by C.D. Olds, Anneli Lax and Giuliana Davidoff. After the discussion of the geometry of numbers, its application, lattice-point packing is ...
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41 views

population of dots with normal distribution of pitch

I want to generate a plot that shows a rectangle populated with dots, where the dot-to-dot distance (pitch) distribution is a lognormal (or a gaussian). I want to be able to change the mean dot-to-dot ...
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1answer
44 views

Packing Problem - What do these notations mean?

I am reading 'Finite Packing and Covering' and I find some notations on the first few pages that are not defined in the book. I am guessing those are standard in the discussion of packing problems. As ...
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1answer
106 views

Is Tetris a packing or covering problem?

I am looking for some information about packing and covering problems. Some texts mention Tetris without further elaboration. Now, I am wondering if Tetris is a kind of packing or covering problem. ...
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56 views

Tile a Box With Translates of Two Given Rectangular Bricks

What is the layman explanation for the theorem explained in this paper? Lets say I have a rectangle of $25 \times 25$. What bricks $B_1(1 \times a)$, $B_2 (1 \times b)$ will be able to completely tile ...
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179 views

Dense packing of circles in unit circle, matlab code

I'm having real difficulties with a packing problem: Given M circles with equal radius r, determine the maximum radius such that all the circles can be fitted into the unit circle, without overlap. ...
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0answers
153 views

Largest and smallest shape enclosed within circles

There is a convex shape $C$. It is known that the largest disc contained in $C$ has radius $r$ and the smallest disc containing $C$ has radius $R$. What are the smallest-area and largest-area shapes ...
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62 views

max number of circles in a circle using symmetrical packing

How many circles of radius 1 can fit into a larger circle of radius 40 using symmetrical (square) packing? Is there a general formula for a larger circle of radius r? For an example of such a ...
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1answer
183 views

Packing problem cube and cuboids

Is it possible to fill a box with dimensions $10\times10\times10$ using bricks of dimensions $1\times1\times4$? If yes, how? I think De Bruijn's theorem on harmonic bricks could be helpful, but I ...
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64 views

Proof: At most 3 circles of radius 1/2 fit into the interior of a halfcircle of radius 1

It is a well known fact that at most 7 interior disjoint circles of radius 1/2 can be centered in a circle of radius 1; note that they don't need to be fully contained in the radius 1 circle. I am ...
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212 views

Algorithm to optimize rectangles filling by rectangles

I have a set of rectangles, all of the same size (W,H) (in fact paper sheets). I have another set of n rectangles of different sizes (Wi,Hi), i = 1..n such that Wi <= W and Hi <= H (in fact ...
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25 views

Reference for packing-covering of the sphere.

I suspect the following to be true and well known. I am looking for a reference. About the notations: $d$ is the dimension, $S^{d-1}$ is the unit euclidean sphere in $\mathbb{R}^d$, $d(x,y)$ is the ...
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1answer
142 views

Can 23 of this polycube fit in a 5x5x5 box?

Consider the following pentacube (front and back view shown.) I have used Burr Tools to determine that 24 of these will NOT fit in a 5x5x5 box. According to my notes when working on this problem a ...
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0answers
251 views

How to load warehouse pallets efficiently?

Assume that we would want to develop a warehouse management system, which picks up plastic boxes and stacks them on a pallet. A pallet has a maximum of $5$ vertical box stacks and the maximum height ...
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1answer
208 views

Calculate remaining space of a box/cube

Im developing an eCommerce system where items are 'logically' placed into boxes. Rather than the shipping system calculating the shipping of each item individually. The shipping will be calculated by ...
13
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1answer
355 views

Packing an infinite sequence of disks

Let $a > 1$ and $Q(a)$ denote the supremum of values of $q$ such that a countably infinite collection of disks, whose areas form an infinitely decreasing geometric progression with the start value ...
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0answers
92 views

Packing circles in circle vs semicircle vs quarter of circle

Consider $N$ disjoint circles with radius $1$ packed into a larger circle $C$. Let $R$ be the smallest possible radius of $C$, allowing the best packing density. Now take the $N$ unitary circles ...
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1answer
312 views

Packing circles into circle of diameter 7

How many unit circles can you fit inside a circle of diameter 7 such that no circle overlaps any other circle? Please explain the concept or any tricky process regarding this problem.
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2answers
297 views

Visual illustrations of circle packing theorem?

Circle packing theorem states: For every connected simple planar graph G there is a circle packing in the plane whose intersection graph is (isomorphic to) G. Paper Collins, Stephenson: A circle ...
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1answer
217 views

10 points inside a square - minimum distance between any of them

A square of side 1 is given, and 10 points are inside the square. If we divide the square into 9 smaller squares, and apply Dirichlet principle, we can prove that there are 2 of these 10 points whose ...
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60 views

Sphere packing in high dimensions and boundary conditions

The Minkowski lower bound for packing hyperspheres of unit radii in $\mathbb{R}^n$ states that the density of hyperspheres satisfies, for all $n\geq 2$ \begin{equation} \Delta_n \geq ...
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2answers
79 views

Unusual 3D Packing Problem

I made up this interesting problem playing with wire sculptures: If I have a $10 \times 10 \times 10$ clear box and inside I can put wireframe unit cubes, what's the maximum number of unit edges (or ...
33
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2answers
885 views

Triangles packed into a unit circle

2014 triangles have non-overlapping interiors contained in a unit circle.What is the largest possible value of the sum of their areas? What are some ideas that might help me start this? Note that ...