In my previous post on Ionic Molecular Crystallization, I sketched a geometric analysis of a two-dimensional ionic molecular framework, guided by charged particle interactions, symmetry, and regular polygon packings. Since our real interest lies in three-dimensional frameworks, I wondered how this idea would translate into three dimensions. While still theoretical, the model feels promising—and surprisingly beautiful.
To help me visualise the constructions, I discovered a wonderful CAD modelling software, Shapr3D, which is both intuitive to use and very powerful. I had zero experience with CAD before, but within a few hours, I was able to build all kinds of three dimensional frameworks.
Let’s start with the simplest regular polyhedron—the regular tetrahedron—and build a periodic framework by connecting the vertices of tetrahedra, as shown in the picture below.

Unfortunately, regular tetrahedra are not space-filling. In fact, the densest known packing of tetrahedra is currently a dimer packing with space group symmetry and a density of
(Chen, E.R., Engel, M., & Glotzer, S.C. Dense Crystalline Dimer Packings of Regular Tetrahedra. Discrete Comput. Geom. 44, 253–280 (2010). https://doi.org/10.1007/s00454-010-9273-0). However, I suspect that this packing might turn out to be a
or
tetrahedra monomer packing.
The question is: how many regular tetrahedra can share one vertex subject to crystallographic restrictions? The answer is eight, as shown in the image below. This is a simple consequence of the sphere kissing number being 12. It forms a uniform star polyhedron called an octahemioctahedron.

The next question is: what might a local cluster with low net electrostatic energy look like? Here is one such arrangement of blue () and green (
) charged particles that satisfies this criterion under the Coulomb potential.


Additionally, I used the GEOMAG construction toy to build a physical model of this configuration. The green bars represent the repulsive forces in the green anion octahedral configuration. The white bars represent the attractive forces between green anions and blue cations; however, this is not entirely correct. These should not be represented as rigid rods but as tendons that can change length.
Geometrically, the GEOMAG structure represents the 1-skeleton of a stellated octahedron. The projection of the two tetrahedra, which together form the stellated octahedron, becomes visible in the shadow cast by the setting sun.
We set the configuration such that the local cluster’s Coulomb energy—introduced in the Ionic Molecular Crystallisation—is zero.
In terms of an optimisation problem, the configuration is a solution to the following:
As in the two-dimensional case, the positively charged particles (blue) are located in the tetrahedral face-centred cubic (FCC) interstitial voids.

The local configuration of positively charged tetrahedral molecules and single-atom negatively charged molecules looks like this:

Once we assemble these octahemioctahedra, the resulting framework exhibits symmetry (space group 225)—the same as fluorite (
). The similarity is not merely mathematical; chemically, the tetrahedral framework model could approximate a real organic fluorite structure, provided we can identify the appropriate molecular analogue.

One can view this as the tetrahedral-octahedral honeycomb, where the octahedral building blocks have been removed. I positioned the camera to highlight the six-fold and four-fold roto-inversions of the framework.
Following the two-dimensional triangular tiling case, I constructed the following framework by removing tetrahedra from the framework while preserving crystallographic symmetry:

The framework’s structure becomes clearer when in motion. See the animation below:
This is the three-dimensional equivalent of the lowest-density triangular configuration (Kagome lattice) from the Ionic Molecular Crystallisation post—the quarter cubic honeycomb. Its space group is (space group 227).
As in the case of triangular tiling, there is a group-subgroup relationship between the tetrahedral-octahedral and the quarter cubic honeycombs via the space group (space group 224).
is the maximal index
-subgroup of
, and
is the maximal index
-subgroup of
. This should not come as a surprise given how the framework was constructed.
If the observations from the two-dimensional framework analysis translate to three dimensions, this framework should represent a local Coulomb energy minimum, and a global minimum if the symmetries of the molecular system are constrained to the space group symmetry isomorphism class.
Similar to triangular frameworks, one can associate a sphere packing with the tetrahedral framework, and further with a regular
-polytope, using these to enumerate local energy minima.
This suggests a potential geometric design principle: Higher symmetry often corresponds to greater mechanical stability, whereas its sub-frameworks may offer larger pores or other advantageous properties at the expense of rigidity.
In other words, if our Coulomb cluster-tetrahedral structure minimises energy under symmetry, then the lower-symmetry frameworks derived from it (e.g.,
) may also represent local minima—stable enough to exist, but metastable within the full energy landscape.
Moreover, we can define a generative method for exploring neighbourhoods of local energy optima by sampling an entire family of ionic molecular frameworks through the simple identification of symmetry-breaking subgroups of a highly symmetrical parent structure. That is, we begin with the framework exhibiting the highest symmetry and then systematically investigate its symmetry-reduced subframeworks.
Interestingly, some coarse-grained molecular simulations already approximate molecules as rigid polyhedra. Our goal is to create stable framework models with large cavities, which may serve as blueprints for the computational design of organic frameworks, applicable to areas such as water and capture.
In summary, referring back to previous posts (Chiral Interaction Ground States and Ionic Molecular Crystallisation), these configurations are all related through the face-centred cubic lattice packing of spheres, with a density of ,

and the associated absolutely symmetric quadratic form:
courtesy of K. L. Fields (K. L. Fields, 1979. Stable, fragile, and absolutely symmetric quadratic forms. Mathematika, 26(1), 76–79).
This brings us full circle to the symmetries of FCC sphere packing explored in these blog posts:

Based on these, A. I. Kitaigorodsky (link to German Wikipedia page as no English version exists) formulated his Close Packing Principle:
“The mutual arrangement of the molecules in a crystal is always such that the ‘projections’ of one molecule fit into the ‘hollows’ of adjacent molecules.”
— From Molecular Crystals and Molecules by A. I. Kitaigorodsky.
So, what’s next? I’m now looking into how kinematics of the cuboctahedron under space group symmetry constraints can help identify which polyhedral frameworks are mechanically stable—and which ones are just aesthetically pleasing. After my visit to ICERM and a conversation with Robert Connelly, I’m convinced that the mathematics of tensegrities is the key.

