66 problems

For $p\ge6$, consider the regular tiling ${p,3}$ and the circles inscribed in its faces. A circle packing is strongly solid if it remains solid after any one of its circles is remo…

Let $(S,\mathcal{T},I)$ be a triangulated surface equipped with an inversive distance circle packing metric, and consider the associated inversive distance circle packing on $S$. B…

Let $0<\mu<1$, and let $c_i$ be circles of radii $r_i$. Associate to each $c_i$ a concentric circle of radius $\mu r_i$, called its kernel. A generalized Minkowskian arrangement of…

Graham–Lagarias–Mallows–Wilks–Yan conjecture. Every sufficiently large integer satisfying the congruence obstructions for an integral circle packing occurs as a curvature in that p…

A circle packing is a packing of congruent circles in the plane, and a packing is diffusively dominant if its mass distribution diffusively dominates that of every other packing. C…

Two-saturated disk-packing density conjecture.

Let $\Omega$ be a Jordan domain in $\bold C$, and let $\mmp P$ and $\mmp Q$ be univalent circle packings for the same tangency graph $\mcompk$, with $(\compk,p_{\p Q})$ the random…

Conjecture. Theorem 4.2 should also hold for triangulations with recurrent 1-skeletons, without requiring the triangulations to be of bounded degree.

Let $n$ be the number of congruent non-overlapping circles packed in a rectangle, and let $L/S$ denote the rectangle's aspect ratio, where $L$ and $S$ are its longer and shorter si…

Unavoidable waste conjecture. The two triangles of area $s$ per circle are the unavoidable, fixed waste in any finite hexagonal packing carved out by a rectangle.

Let $\Sigma_g$ be a closed orientable surface of genus $g\geq 1$, and let $\tau$ be a graph on $\Sigma_g$ that lifts to a triangulation of the universal cover. Let…

Let $\Sigma_g$ be a closed surface of genus $g$, let $\tau$ be a triangulation of $\Sigma_g$, and let ${\mathcal C}_{\tau}$ be its circle-packing deformation space. Let…

Let $\Sigma_g$ be a closed surface of genus $g$, and let $\tau$ be a triangulation of $\Sigma_g$ whose lift to the universal cover is covered by a simple graph. Let…

Hexagonal packing conjecture. For $n<54$, the minimum perimeter is achieved, perhaps nonuniquely, by a subset of the hexagonal circle packing.

Let an integral Apollonian packing be an Apollonian circle packing whose circle curvatures are all integers. An integer occur in such a packing when it is the curvature of one of i…

Let $E$ be a moiety of a primitive Eisenstein circle packing. A positive integer is called sporadic if it is admissible in $E$, does not appear in $E$, and does not belong to one o…

Let $E$) be a moiety of a bounded primitive Eisenstein circle packing. Let $N$ be a positive real number, and let $c_E$ denote a constant depending on $E$. Let…

Global shallowness conjecture. A hyperbolic, locally shallow, convex c-polyhedron is globally shallow.

Let a generalized circle packing be an integral circle packing, meaning that all its circle curvatures are integers, and let its modular restrictions be the congruence conditions m…

Let the hexagonal packing be the circle packing associated with the hexagonal packing type, and let its modular restrictions be the congruence conditions satisfied by all curvature…

Let a primitive integral Apollonian packing be a packing whose circle curvatures are integers with greatest common divisor $1$. For a fixed packing, let the modular restrictions be…

Let $\mathcal P$ be an Apollonian circle packing with symmetry group $\Gamma<\operatorname{PSL}_2(\mathbb C)$. For any circle $C\in \mathcal C_\Lambda^*$, consider its orbit…

Equivalence conjecture for infinite ideal polyhedra. The following statements are equivalent, with the Euclidean alternatives applying to IIP and the hyperbolic alternatives applyi…

Finite-subcomplex relaxation conjecture. Condition (3) can be relaxed so that it is required to hold only outside a finite sub-complex of $\mathcal{D}$.

Uniformization of the CRF. The following statements are equivalent: the combinatorial Ricci flow on $\mathcal{T}$ converges for every initial value in hyperbolic background geometr…