4
votes
1answer
149 views

Cover of a n-simplex with balls

Consider a n-simplex. For each edge (i,j), consider a n-ball, such that vertices i and j are antipodal on this ball. Is the simplex covered by the union of these balls? Thank you.
6
votes
0answers
314 views

Rectangology and squareology

I thought that rectangles were simple, and squares even simpler. Until my research has led me to several questions about rectangles and squares, which I can't solve. I started by posting this ...
2
votes
1answer
132 views

Is this cube packing possible?

I know how to pack $5$ unit squares in a square of side length $2+\frac{\sqrt{2}}{2}$. Is there an $\varepsilon>0$ such that there exists a packing of $9$ unit cubes in a cube of side length ...
0
votes
2answers
128 views

Disks Packing Variant

Usually disk packing problems require that no two disks of the packing intersect. Does anybody know if the problem has been studied when disks may intersect but they are not allowed to contain the ...
3
votes
2answers
495 views

Torus in $\mathcal{R}^3$

Hi I'm interested in packing the 3 space as dense as possible using equally sized tori whose major radius is much bigger than their minor radius in. Do you have any idea how to attack this problem? ...
27
votes
3answers
3k views

Can we cover the unit square by these rectangles?

The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1. It is easy to show that $$\sum_{1 \leq k } ...
4
votes
7answers
1k views

How to generate a net on a 8-dimensional sphere

Using Matlab, how to generate a net of 3^10 points that are evenly located (or distributed) on the 8-dimensional unit sphere? Thanks for any helpful answers!
12
votes
2answers
2k views

How many unit squares can you pack into a rectangle with nearly integer side lengths?

Earlier today, somebody asked what looks like a homework problem, but admits the following reading which I think is interesting: Suppose $a_1,\dots, a_n$ are positive integers, and $\varepsilon$ ...