
1. Packings

1. Inversive Distance Packings

Let $(x_1, y_1, r_1)$ and $(x_2, y_2, r_2)$ be two disks with centers $x_i, y_i$ and radii $r_i$. Then the inversive distance between them is $$Inv(i, j) = \frac{(x_j - x_i)^2 + (y_j - y_i)^2 - r_i^2 - r_j^2}{2r_ir_j}.$$ The inversive distance is invariant under Möbius transformations and preserved by stereographic projection. If two circles $i$ and $j$ are externally tangent, then $Inv(i, j) = 1$ and if they are internally tangent then $Inv(i, j) = -1$.

Problem 1.1.

[John Bowers] Let $T$ be a triangulation of the unit sphere with real weights $w_{ij}$ on the edges. Decide if it is possible to produce a a mapping (realization) of vertices to circles such that $Inv(i, j) = w_{ij}$ for all edges $(i, j)$ in the triangulation?
Known:
1. If $w_{ij} \in (0, 1]$, for each contact $ij$, then this mapping exists on the sphere and is unique up to Möbius transformation.
2. If each $w_{ij} \in [-1, 1]$ then there is a nice characterization of existence. If it exists, then it is always unique.
3. If some $w_{ij} > 1$, then it does not always exist and when it does it is not necessarily unique.
4. Best known approach to (3) is to use the so-called notion of “convexity” for inversive distance circle packing. It is known that non-unitary convex inversive distance circle packings are globally rigid. Non-unitary means no $w_{ij} = 1$.
• Disk Packing for Laman Graphs

Problem 1.2.

[Louis Theran] Let $G$ be a planar Laman graph. Does there exist a packing of disks with contact graph $G$ that has generic radii?
The other direction is true, i.e. if $G$ is the contact graph of a disk packing with generic radii, then $G$ is planar and Laman, ref: Rigidity for Sticky Disks, Connelly, Gortler and Theran, 2019.
1. Remark. [Louis Theran] Something we do know is that if you can find an infinitesimally rigid packing of a Laman graph, then you can perturb it to a generic one.
• Disk Packing on the Unit Sphere

Problem 1.3.

[Robert Connelly] Given a packing of disks on the unit sphere, each with radius $\leq \epsilon$, such that the packing has a nontrivial infinitesimal flex, is the packing unjammed for some $\epsilon$?
• Maximum Contacts in Disk Packing

Problem 1.4.

[Miranda Holmes-Cerfon] Find a collection of $n$ objects (e.g. disks of given radii) such that all packings have $k$ contacts where $k$ is $\leq$ the number required for isostaticity for that object but there exists a rigid packing that has a non-trivial infinitesimal flex and an equilibrium stress.
It is known that $n=9$ sticky unit spheres in $\mathbb{R}^3$ cannot be assembled with a contact graph that has $> 3n-6$ contacts, but there exists a rigid packing with $3n-6$ contacts, one infinitesimal flex, and one self-stress.
• Sticky Sphere Packing in 3D

Definition: Let $S_G$ be the semi-algebraic set describing the sticky sphere packing conditions for graph $G$ in $\mathbb{R}^3$

Definition: An edge-length equilibrium stress $\omega$ for a graph $G = (V, E)$ with $|V| = n$ and $|E| = m$ is a nonzero vector in $\mathbb{R}^m$ that satisfies $$\sum_j \omega_{i, j} (p_i - p_j)\ = 0$$ and $$\sum_j \omega_{i, j} || p_i - p_j|| = 0$$ for all $1\leq i \leq n$.

Problem 1.5.

[Meera Sitharam]
1. Can there be an edge-length equilibrium stress of a sticky sphere packing in $\mathbb{R}^3$?
2. Under what conditions does the neighborhood of a point in $S_G$ have dimension $4n - m$ where $n$ is the number of vertices and $m$ is the number of edges?
3. Is the double banana, or indeed any nucleation-free graph, the contact graph of a sphere packing with generic radii?
known:

For the analogous statement to 3 above, in $\mathbb{R}^2$ for sticky disk packing, there is no edge length equilibrium stress.

Cite this as: AimPL: Rigidity and flexibility of microstructures, available at http://aimpl.org/flexmicro.