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

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0
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2answers
32 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 ...
1
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0answers
49 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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0answers
89 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 ...
1
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0answers
13 views

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
24 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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0answers
10 views

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 ...
0
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1answer
19 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 ...
3
votes
0answers
40 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 ...
4
votes
2answers
84 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 ...
7
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1answer
76 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 ...
1
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0answers
27 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 ...
1
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1answer
40 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 ...
2
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1answer
57 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, ...
1
vote
1answer
42 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 ...
2
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0answers
43 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 ...
5
votes
0answers
83 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 ...
9
votes
1answer
159 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 ...
2
votes
1answer
58 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; ...
0
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0answers
10 views

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 ...
3
votes
1answer
54 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 ...
2
votes
1answer
84 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 ...
1
vote
1answer
53 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 ...
1
vote
2answers
37 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, ...
1
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0answers
37 views

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 ...
11
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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. ...
2
votes
1answer
44 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): ...
10
votes
1answer
99 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 = ...
3
votes
3answers
165 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 ...
2
votes
1answer
47 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 ...
2
votes
1answer
78 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 ...
0
votes
0answers
48 views

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
8
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1answer
170 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 $8$ ...
1
vote
1answer
137 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
votes
1answer
130 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 ...
2
votes
0answers
69 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 ...
3
votes
1answer
82 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 ...
4
votes
0answers
45 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 ...
0
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0answers
21 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
votes
1answer
133 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 ...
1
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1answer
29 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 ...
1
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1answer
185 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 ...
2
votes
1answer
72 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 ...
0
votes
1answer
33 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
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0answers
14 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 ...
8
votes
2answers
1k 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
35 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 ...
3
votes
1answer
88 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 ...
3
votes
1answer
40 views

Prove that cube cannot be partitioned into $n>1$ smaller distinct cubes.

Prove that cube cannot be partitioned into $n>1$ cubes, such that each of them has different side length. I believe tallhis is not hard problem, but I just do not have an idea how to start. I ...
2
votes
2answers
49 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
85 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 ...