21 problems

Aperiodic disk-packing conjecture. There exists a finite set of disk sizes that admits a triangulated packing other than the hexagonal compact packing, while no such packing is per…

Fejes Toth's conjecture. The packing in Figure has maximum density among disks with two radii whose ratio is slightly greater than $\sqrt{2}-1$.

For fixed indices $p,k\geq1$, let $n_p(k)$ denote the proposed disk numbers whose packings are arranged in a matrix indexed by $p$ and $k$. A row class is obtained by fixing $p$ an…

Let $\Delta(k)=\frac{k(k+1)}{2}$, and let $\delta(n)=L(n)-t(n)$, where $L(n)=1/d(n)$ and $t(n)$ is the corresponding triangular-packing baseline used in the paper. Consider the inf…

Let $d(n)$ be the largest possible minimum distance between the centers of $n$ equal disks packed in an equilateral triangle. For an infinite class $n=n(1),n(2),\ldots$ of disk num…

Let $\Delta(k)=\frac{k(k+1)}{2}$, and consider packings of equal nonoverlapping disks in an equilateral triangle. Let $d(n)$ denote the largest possible minimum distance between th…

A stable representation of a graph is an ${\mathcal{M}}$-representation that is a local minimum with respect to the ordering relation defining stability. Finite algorithm conjectur…

Connelly's uniformity conjecture. The ternary triangulated packing numbered $53$ is the most uniform triangulated packing other than HCP. Its proposed deformation yields the highes…

Density-maximizing conjecture. For the ratios $r_b$ and $r_c$, the periodic nontriangulated binary packing depicted in Fig. 7 maximizes density among all binary packings with the s…

Let a family of congruent disks in the plane be arranged so that no two disks intersect in their interiors, with one central disk, a first family of disks touching it, and a second…

Isostatic conjecture. The density $\rho$ of every such packing satisfies

Uche–Stillinger–Torquato conjecture. Some of the packings in the tricusp case are most dense.

For each integer $N\geq 2$, let $\epsilon(N)$ be a positive quantity with … as $N$ tends to infinity. Let $M$ be the Markov constant of a packing of $N$ equal disks in a square tor…

Let $N\geq 6$ and consider a rigid packing of $N$ equal disks in a square torus. Assume that its packing graph consists of two linear geodesics, and let $M$ be its Markov constant.…

Square-torus Markov-bound conjecture. Every such packing satisfies $M\leq 6.25$. Equivalently,

Let $N$ equal disks be packed in a square torus. A packing is said to correspond to a convergent of $1/\sqrt{3}$ or $2-\sqrt{3}$ when it is one of the Type I or Type II packings, r…

Let $M(N)$ be the Markov constant of a packing of $N$ disks in a square torus, and consider the two packing sequences constructed in the paper, associated with continued-fraction a…

Let $N$ be the number of equal disks in a square torus, let $\delta(N)$ denote the packing density, and let $\pi/\sqrt{12}$ be the density of the planar hexagonal packing. Square-t…

Let $N$ be the number of equal disks packed in a torus given by a triangular lattice. Suppose that $N$ is not of the form … for integers $a,b$. Triangular-torus density conjecture.…

The harmonic quadrature $L^1$ conjecture.

Lubachevsky–Graham conjecture. For $k=1,\dots,5$, curved hexagonal packings are the densest packings of $N+1$ identical nonoverlapping disks within an encompassing disk.