Page 2 – Milo Torda, PhD

June 27, 2023October 15, 2023

Minimum Energy Configuration of Particles with Six-Fold Rotational Symmetry

The output configuration resulting from the minimization of a system of 800 particles driven by the Lennard-Jones potential, with parameters: $\sigma = 0.95589..$ and $\epsilon = \frac{1}{4}$

$\begin{equation*} E_{LJ}=\frac{1}{2}\sum_{a \in A} \sum_{b \in B} 4\epsilon\left[\left(\frac{\sigma}{r_{ab}}\right)^{12} - \left(\frac{\sigma}{r_{ab}}\right)^{6}\right] \end{equation*}$

January 15, 2023January 15, 2023

Densest P1 packing of the tetrahedron

I applied the Entropic Trust Region Packing Algorithm to search for the densest lattice packing of tetrahedra. It turned out that the GPU implementation of the overlap constraint evaluation for space group packings of polyhedra was more complicated than I expected. The good news was that the optimization algorithm worked as thought.

The densest lattice packing of the regular tetrahedron is $\frac{18}{49}\approx 0.36734$ (https://www.ams.org/journals/bull/1970-76-01/S0002-9904-1970-12400-4/). This is the configuration the algorithm found. See the visualization of the output packing with density $0.36734$ below.

January 7, 2023January 10, 2023

Densest P1, P-1 and P21/c packings of spheres

I started to work on space group packings. I applied the Entropic Trust Region Packing Algorithm (https://arxiv.org/abs/2202.11959) to search for the densest packings of the sphere in space groups P1 and P-1.

In the P1 setting, the densest packing density coincides with the general optimal packing density of spheres (https://www.jstor.org/stable/20159940). If the unit cell parameters are set to $a=b=c=2$ and $\alpha=\beta=\gamma=\frac{\pi}{3}$ with the unit cell given by

$\begin{equation*} U=\left[\begin{array}{ccc} a\sin(\beta)\sqrt{1-\left(\frac{\cos(\alpha)cos(\beta) - cos(\gamma)}{\sin(\alpha)\sin(\beta)}\right)^2} & 0 & 0 \\ -a\frac{\cos(\alpha)cos(\beta) - cos(\gamma)}{\sin(\alpha)} & b\sin(\alpha) & 0 \\ a\cos(\beta) & b\cos(\alpha) & c \end{array}\right] \end{equation*}$

then the determinant of $U$ equals to $\det(U)=4\sqrt{2}$ and combined with the volume of the unit sphere $\text{vol}(S)=\frac{4}{3}\pi$ gives the optimal packing density of spheres $\rho(S_{P1})=\frac{\text{vol}(S)}{\det(U)}=\frac{\pi}{\sqrt{18}}\approx 0.74048$ . See a visualization of this configuration below.

There are a few interesting things about this. First, for some reason, I could find this configuration reported anywhere. Second, it is conjectured that the densest packings of centrally symmetric polyhedra are P1 packings (https://journals.aps.org/pre/abstract/10.1103/PhysRevE.80.041104). It might be that this conjecture can be extended to a larger class of centrally symmetric convex sets, similar to the 2D setting where the densest packing of centrally symmetric convex sets is a lattice packing (https://link.springer.com/article/10.1007/BF02392671).

In the P-1 setting the situation is similar. If the unit cell parameters are set to $a=c=2$ , $b=2\sqrt{2}$ and $\alpha=\beta=\gamma=\frac{\pi}{2}$ , then the determinant of the unit cell equals to $\det(U)=8\sqrt{2}$ and again we get the optimal packing density of spheres $\rho(S_{P-1})=\frac{2\text{vol}(S)}{\det(U)}=\frac{\pi}{\sqrt{18}}$ . See the visualization of this configuration below.

Here again, I couldn’t find this configuration of spheres reported anywhere. Another interesting thing is that the situation resembles the 2D setting, where the densest P1 and P2 configurations of a disc have the same packing density (https://journals.aps.org/pre/abstract/10.1103/PhysRevE.106.054603). The P-1 space group can be considered as an equivalent of the P2 plane group since the point group symmetries are given via inversion through the centre of the unit cell. It might be that centrally symmetric polyhedra attain their optimal packings also in the P-1 space group. Moreover, it is conjectured that for convex 2D sets, the P2 configurations are optimal (https://link.springer.com/article/10.1007/s00454-016-9792-4). I am wondering for which other convex 3D sets is the P-1 packing configuration optimal.

With the P21/c space group, the symmetries of the densest sphere packing continue. If the unit cell parameters are set to $a=2\sqrt{2}$ , $b=4$ , $c=2$ and $\alpha=\beta=\gamma=\frac{\pi}{2}$ , then the determinant of the unit cell equals to $\det(U)=16\sqrt{2}$ and again we get the optimal packing density of spheres $\rho(S_{P21/c})=\frac{4\text{vol}(S)}{\det(U)}=\frac{\pi}{\sqrt{18}}$ . See the visualization of this configuration below.

What is interesting is that the P21/c and P-1 space groups are the most frequent space groups found in the Cambridge Structural Database amounting to 59.3% of structures (https://www.ccdc.cam.ac.uk/support-and-resources/ccdcresources/f0dbc77b8b6041b8be63ff490bad33d6.pdf). I wonder if there is a relationship between densest space group packing of a sphere and the frequencies space groups occurring in CSD.

October 7, 2022October 7, 2022

Plane group packing density relationship for n-gons with 12-fold rotational symmetry

September 27, 2022September 27, 2022

Fundamental regions of densest p2, p2gg, pg and p3 packings

From left to right 3-gon, 5-gon, 6-gon, 9-gon, 10-gon and disc. Red lines denote the boundary of fundamental regions of respective configurations.

p2gg

September 13, 2022September 13, 2022

Non-isometric densest P2MG configurations

I discovered that n-gons with a 2-fold but no 3-fold or 4-fold rotational symmetry have at least two non-isometric densest P2MG configurations. Below is an example of two non-isometric P2MG packings of a 10-gon with the same approximate density of 0.8372213.

July 20, 2022

Projected n-gon plane group packing orderings

We can use the packing density table to define a plane group ordering according to densest n-gon packings for each n-gon. Below is a visualization of this ordering. For given n-gon a color is assigned for each plane group based on the density of respective densest plane group packing.

Based on these orderings we can tell the densest plane group packing ordering for an arbitrary n-gon and it’s symmetries. Below are the projected orderings of densest plane group packings of n-gons for n = 5 : 42

July 4, 2022July 4, 2022

Pentagon densest P2MM packing to PM and C2MM packings

Given densest P2MM packing of a regular convex polygon it possible to convert the configuration to a PM and C2MM packing of exatly same density. For n-gons without central symmetry this these C2MM packings are not optimal and the PM packings are not the same as those obtained using etropic trust region packing algorithm although they have the same approximate density.

Here is an example in the pentagon instance. PM and C2MM packings constructed based on densest P2MM packing with density 0.6909830 . The PM packing is different than the one obtained by entropic trust region https://milotorda.net/wp-content/uploads/2022/06/paperPMgon5-eps-converted-to.pdf and the C2MM has lower density than the one obtained by the entropic trust region https://milotorda.net/index.php/packings/ From left to righ: P2MM, PM and C2MM.

July 3, 2022

Octagon densest PM packing to densest P2MM, C2MM and P4GM packings

Given densest PM packing of an a n-gon with a 4-fold rotational symmetry it possible to construct P2MM, C2MM and P4GM packings in https://milotorda.net/index.php/packings/

Here is an example for an octagon. P2MM, C2MM and P4GM packings based on densest PM packing with exactly same density 0.8284271. From left to right: PM, P2MM, C2MM and P4GM

June 30, 2022July 1, 2022

CM to P2MG packings

Same thing as for the PG group (https://milotorda.net/index.php/pg-packings/) can be done for the CM group. At least for regular convex polygons. That is, given a CM packing one construct a P2MG packing with exactly same density, fairly easily. For most of regular polygons CM a P2MG packings have the same densities. The exceptions are regular convex polygons with 12k-1 and 12k+1 rotational symmetries. See: https://milotorda.net/index.php/packings/

Below are some examples of densest CM packings converted to P2MG packings. The P2MG packings are not the densest P2MG packings in these instances. From left to right: CM and P2MG

11-gon with density 0.8279530

13-gon with density 0.8321287

23-gon with density 0.8386501