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

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0
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1answer
13 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 ...
0
votes
1answer
82 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 ...
0
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0answers
8 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
votes
2answers
100 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 ...
1
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0answers
24 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
votes
1answer
49 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 ...
2
votes
2answers
37 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 ...
0
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1answer
17 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
votes
1answer
66 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.
0
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0answers
15 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 ...
5
votes
3answers
242 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: ...
33
votes
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 ...
1
vote
0answers
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 ...
0
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0answers
52 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 ...
0
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0answers
19 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 ...
2
votes
1answer
33 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: $$A=\frac{20(20+1)(2\cdot20+1)}3=5740$$ ...
2
votes
1answer
45 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 ...
1
vote
0answers
34 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 ...
0
votes
1answer
42 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 ...
1
vote
1answer
80 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. ...
0
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0answers
49 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 ...
0
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0answers
139 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. ...
3
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0answers
107 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 ...
0
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0answers
52 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 ...
1
vote
1answer
137 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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0answers
60 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 ...
0
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0answers
160 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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0answers
23 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 ...
9
votes
1answer
128 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 ...
2
votes
0answers
146 views

How to load warehouse pallets efficiently?

Assume that we would wan't to develop a warehouse management system, which picks up plastick boxes and stacks them on a pallet. A pallet has a maximum of 5 vertical box stacks and the maximum height ...
0
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0answers
29 views

Calculating the package sizes

How can I calculate the minimum size of a package (paper box) that the product certainly fits in when the employees are warping them? I should be able tell how big boxes should be used for each ...
0
votes
1answer
153 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
votes
1answer
350 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 ...
2
votes
0answers
76 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 ...
2
votes
1answer
270 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.
8
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2answers
280 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 ...
4
votes
1answer
165 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 ...
0
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0answers
54 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 ...
0
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2answers
70 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
votes
2answers
870 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 ...
0
votes
1answer
296 views

Maximizing minimum distance between points placed in a polygon

I would like to maximize the minimum spacing between a fixed number of points ($x_i \in \mathbb{R}^2$) placed inside a polygon in the plane. The minimum spacing includes distance to the polygon. ...
1
vote
0answers
88 views

Fitting a rectangle within another rectangle

What are the restrictions such that we can fix a rectangle of dimension $a\times b$ into another rectangle of size $c \times d$? What about if we try to fit it inside a square of side $c$?
2
votes
1answer
228 views

Packing spheres on the boundary of a larger sphere

Consider the following problem, which is a variation of the sphere packing problem and is somehow related to the kissing number problem. For a dimension $n\ge 2$ and a natural $k$, let $r=r(n,k)$ be ...
2
votes
0answers
238 views

Triangle Packing-Problem

Theory and Question We define a normalized triangle $T$ as an ordered list of six points s.t. $p \in [0,1)$ for all $p \in T$. Let $T = [x_0, y_0, x_1, y_1, x_2, y_2]$ be a normalized triangle. We ...
0
votes
0answers
70 views

3D Space Covering-Problem

Given a finite amount of "slots" in 3D space, e.g. $$S = [(1,2,3),(1,3,3),(1,4,3),(1,3,4)] \in \mathbb{N}^3.$$ I'm trying to find an efficient algorithm to determine a minimal set of (rectangular) ...
0
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0answers
116 views

calculate how many Ring can cut from a sheet

I am developing a software for a manufacturing company. My problem is , I need to find how many maximum rings can be cut from a sheet ( it can be rectangle or square sheet) And these rings ( with ...
0
votes
1answer
63 views

If you had a number of cubes of different sizes, what would be the algorithm to figure out the smallest cube they could fit in?

Is this possible to do, if so how would you do it? "If you had a number of cubes of different sizes, what would be the algorithm to figure out the smallest cube they could fit in?"
2
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0answers
143 views

A sphere packing problem

Suppose there is a large sphere of radius $R$. We want to pack it with smaller spheres. The volume of the smaller spheres change depending on where they are situated in the larger sphere. A smaller ...
7
votes
2answers
201 views

Unequal circles within circle with least possible radius?

It is the classical will-my-cables-fit-within-the-tube-problem which lead me to the interest of circle packing. So basically, I have 3 circles where r = 3 and 1 circle where r = 7 and I am trying to ...
1
vote
0answers
38 views

Number of touching triplets in hypersphere packing contact graph

I would like to know what is curently known about the number $N$ of touching-triplets involving a given vertex $V$ in the contact-graph of $d$-hypersphere packings. For $d=2$ and $d=3$, one has $N = ...