41
votes
Accepted
Geometry question about a six-pack of beer
Step 1
Remove the lower three beers from the six-pack. Put one of them at the bottom of the cooler. Obviously, we can draw a circle inside the cooler touching all four beers.
Step 2
Remove the bottom ...
- 25.6k
24
votes
Geometry question about a six-pack of beer
This one is fairly self-evident:
(Too-)Verbosely ... Let the enclosing circle have radius $r$ and center $O$. Let the small circles have radius $s$, and let the left-most four of these have centers (...
- 77.9k
20
votes
Accepted
Kissing number for equal ellipses on the plane
Playing around with GeoGebra one finds that your approximation tends to overestimate the number of kissing ellipses, when $a/b$ grows. A possibly better formula can be found as follows.
Consider two ...
14
votes
Accepted
What is the minimum area of a rectangle containing all circles of radius $1/n$?
One can tackle this question by forming an offset grid of squares of radii $1/k$ in a roughly triangular fashion that fits inside the top left sector of the rectangle, whereby the squares can be ...
- 14.9k
14
votes
Sangaku problem involving eight circles
Let the radii of the large (red) circle and small (blue) circles be $r$ and $s$, so that the radius of the enclosing circle is $r+2s$.
Let the center of the enclosing circle, the large circle, and ...
- 77.9k
13
votes
Five circles in a rectangle: can the circles move?
Yes, they can move.
Suppose the top circle moves right a small distance $t$. We will show that none of the circles overlap, by applying Pythagorus' theorem around the ring of circles.
$p=2\left(1+\...
- 25.6k
12
votes
What is the minimum area of a rectangle containing all circles of radius $1/n$?
For suitable $a$ and $N$, we can place disjoint disks of radii $\frac 1n$, $n\ge N$ into an $a\times a$ square, column by column, where column $k$ ($k=0,1,2,\ldots$), consists only of disks of radius $...
- 376k
11
votes
Number of plates that can be placed on the table so that they neither overlap each other nor the edge of the table?
The area of $225$ plates equals the area of the table exactly. So we could only place the $225$ plates on the table if there were no gaps between them. This would be possible with square plates on a ...
- 145k
11
votes
Accepted
Placing kings on a 6x6 board - who wins?
I wrote some code to bash the nim values; a $6\times 6$ grid is a win for the first player with nim value $1$, by playing in any of the purple squares below:
If the ...
- 17.6k
8
votes
Accepted
Possible tiling patterns for equiangular hexagon with alternating side lengths
Here are the patterns I could think of, represented by their unit cells and with their densities listed beside:
The first tiling shown in the question has no code in my diagram and has density $\frac{...
- 104k
8
votes
Accepted
Reference request: Good introduction to Sphere Packing
The literature hints in the comments are already very good. When it comes to Sphere Packings, you can hardly get around Conway's & Sloane's book "Sphere packings, lattices and groups".
...
- 1,594
8
votes
Accepted
Why does packing exactly 992 circles in a square behave exceptionally?
It appears that the table contains errors (or at least obviously inferior solutions). The maximum radius should never increase, as it does from $n=990$ to $n=991$:
...
- 47.3k
8
votes
Four circles on a quarter disk: can the circles move?
In fact, the circles can move. While the animation below does not furnish a proof, it is suggestive that the answer is affirmative:
I rotated the figure by $\pi/4$ clockwise and placed the wedge in ...
- 141k
7
votes
Can the squares with side $1/n$ be packed into a $1 \times \zeta(2)$ rectangle?
This answer answers the question only approximately, but at least does so rigorously.
Theorem 4 from
Moon, J.W.; Moser, L., Some packing and covering theorems, Colloq. Math. 17, 103-110 (1967). ...
- 738
7
votes
Accepted
Can $27$ points be packed into a $3\times3\times3$ cube and all be more than $\sqrt{3}$ from one another?
I don't have a proof, but here's a reasonable answer.
Your task is equivalent to packing spheres of radius $\frac{\sqrt3}{2}\approx0.866$ in a cube of side length $3+\sqrt3\approx4.732$. Scaling down,...
- 27k
7
votes
Accepted
Can the Fibonacci lattice be extended to dimensions higher than 3?
So, as you pointed out, the idea with Fibonacci lattice is based on the fact that when distributed uniformly, both the azimuthal angle $\phi$ and latitude sine (affine with $\cos\theta$) are uniform.
...
- 951
7
votes
Accepted
Tiling a 4 X 11 board.
I am considering a grid with $4$ column and $11$ rows. Now, color the first and the third row with black color and the second and fourth with white. Clearly, whenever you place a $L$-Shaped domino ...
- 3,598
7
votes
Accepted
The largest equilateral triangle circumscribing a given triangle
I'll use $\triangle PQR$ instead of $\triangle ABC$, to avoid some notational confusion.
First, a little prep work.
Given $\triangle PQR$, we erect on (directed) segment $\overline{PQ}$ an ...
- 77.9k
7
votes
How good can a "near-miss" polyomino packing be?
Here's a tiling of the plane using a $29$-omino that can be partitioned into $4$ heptominoes plus a single cell, giving that $\Delta_7\geq 28/29$.
- 33.7k
7
votes
How many satellites surround a sphere of diameter $x$, where the satellites are $y$ miles apart from each other, and $z$ miles above the sphere?
Your satellites occupy a sphere of diameter $x+2z$. That sphere has a surface area of $4\pi(x/2+z)^2$. If each satellite is a distance about $y$ from its neighbors, we can assume that for each ...
- 22.9k
7
votes
Accepted
Three circles in a triangle: can the circles move?
Draw perpendiculars on the sides of the triangle at the points where the circles touch. Those three lines intersect at a single point P due to symmetry.
Consider the three circles as if they form a ...
- 13.8k
7
votes
Frame challenge: Find the maximum $n$ such that circles of radius $1, \frac12, \frac13, ..., \frac1n$ can be held immobile by a convex frame.
You can start by using the Descartes Circle Theorem. For four mutually touching circles $2 (a^2+b^2+c^2+d^2) = (a+b+c+d)^2$, where the values are 1/radius (the bend). For example, circles with radii {...
- 21.4k
6
votes
Packing of n-balls
As of March 2016, the optimal density for lattice packing of unit $n$-spheres are known for $n \le 8$ and $n = 24$. All the associated lattices are laminated lattices.
Laminated lattices $\Lambda_n$ ...
- 123k
6
votes
Accepted
Packing Squares 1-24, times two
Corrected Bouwkamp code (may or may not agree with pictured)
Computer solutions for problems 1-10.
Brute force, greedy backtracking algorithm:
Problem #1: 49 27 27 (6,6,6,6,3)(3)(6,6,6,6,3)(3)(5,5,5,...
- 344
6
votes
Maximum number of circle packing into a rectangle
Consider the following diagram of a triangular packing:
If the circles have radius $r$, then each pair of horizontal red lines is a distance $r$ apart, and they're a distance $r$ from the edges. Each ...
- 145k
6
votes
How many colors are necessary for a rectangle to never cover a color more than once?
I haven't quite fleshed out exactly how to use this, but I think the following idea should probably enough to at least prove that $mn$ colors suffice if and only if $m$ divides $n$: if two squares lie ...
- 6,473
6
votes
Packing regular tetrahedra of edge length 1 with a vertex at the origin in a unit sphere
To complement Erik Parkinson's construction and lower bound, here's a rather tight upper bound: $22$.
Suppose we place one of these tetrahedrons in the sphere with one vertex at the center and three ...
- 19.6k
6
votes
Accepted
T shaped tetris figures on a plane
It turns out it is impossible to have uncountably many disjoint Ts in the plane. I found this solution in Mathematical Puzzles: A Connoisseur's Collection by Peter ...
- 78k
5
votes
Accepted
Scaling factor closest to 1 in an infinite sequential rectangle packing
Looks like the Amman chair is an unnecessary sophisticated tool which was not designed for this task (the way it stands now), nor is its use really justified. Dissecting a rectangle into an infinite ...
- 12.9k
5
votes
Packing two different circles in a rectangle
In the figure below, let circle $E$ have radius $r_1$, circle $G$ have radius $r_2$ and the rectangle be $H$ high and $W$ wide. You get the best fit by putting the circles in opposite corners. First ...
- 377k
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