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 …

First using a three dimensional unit cube as an example for the term "regularity", we can have two possible triangulations: (A) (B) We say the lower triangulation is more "regul …

Is it true that of all $n$-simplices with edge lengths greater than or equal to some parameter $s$, the regular simplex with edge lengths $s$ has the smallest circumradius? It seem …

Let $R>0$ be given and let $H^n$ be the unit hypercube in $\mathbb{R}^n$. The problem I am facing is to find the maximum number of vertices of $H^n$ which can be covered by a close …

We have known from Ehrhart theory that if $P$ is a $d$-dimensional polytope of $\mathbb R^n$ which has integer vertices then the number of integer points in $nP$ is a polynomial of …

Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$, except leave the origin $(0,0)$ unoccupied by a disk. Q. Is it the case that …

Hill conjectured that the minimum number of crossings in a drawing of the complete graph $K_n$ in the plane is exactly $$Z(n) = \frac{1}{4} \bigg\lfloor\frac{n}{2}\bigg\rfloor \le …

I am working on the discrete theory of compressible fluids dynamics, i.e., numerically solving and simulating the compressible fluids , we are interested in the way using discrete …

I have a process which in each step creates a new unit circle and I am interested in maintaining the boundary of the resulting arrangement in linear time. Is there anything known …

Let $z=a + b i$ be a complex number which is a Gaussian prime, on neither the $x$- nor the $y$-axis. So $a^2+b^2$ is a prime. Construct a region $D(z)$ surrounding $z$ which is the …

I have several questions about the Discrete Exterior Calculus (DEC) in the numerical method for solving partial differential equation in physics: (Discrete Exterious Calculus is t …

Geometry of numbers was initiated by Hermann Minkowski roughly a hundred years ago. At its heart is the relation between lattices (the group, not the poset) and convex bodies. One …

Given a graph $G=(V,E)$ and a length function $\ell:E\to\mathbb{R}_+$. An embedding of the graph into the $d$-dimensional Euclidean space is a map $f:V\to\mathbb{R}^d$ such that $| …

I have a data set that essentially looks like the picture below, i.e., it's given by sets of points in $\mathbb{R}^3$ that sample the cross-sections of a certain surface that in pr …

This is a vague question, and I will no doubt be (properly!) chastised for posing it. I would like to generate a set $S$ of points in $\mathbb{R}^3$—$|S|$ finite or infinite& …