Recently, I have been immersed in the vast world of intermetallic phases, in particular those with an ordered superstructure of either cubic-closest packed (**ccp**) or body-centred cubic (**bcc**) parent structure. They caught my interest, because some of the researchers in that field are paying particular attention to how the densest-packed layers/planes look like and how they are organized, i.e., in which direction they are stacked and if they build an cubic (*ABC*…) or hexagonal (*AB*…) stacking sequence. They also partly give network-like descriptions of the atomic configurations of the densest layers.

Since densest layers are considered, one enters the field of *circle packings*. And the densest circle packing of equally sized circles is, of course, the hexagonal packing. If we draw lines from each circle midpoint to every midpoint of the next-neighboring circles, we get the well known **hxl** net, a uninodal 6-c net. The vertex symbol is 3^{6}.

In those densest layers of ordered alloys the atoms within one densest plane are of different kind. If we ignore this – as it is usually the case, when we analyze the topology of an atomic arrangement – then we obtain again the **hxl** net. But what happens, if we focus only on a *subnet*, for instance, the net that is built by the major component?

Let’s look at an example.

In the ordered binary intermetallic phase Cu_{3}Au (space group *Pm*-3*m*), the Cu atoms in the densest layers, (111) planes, form a continous net as shown in **Fig. 1**, in which the Au atoms are a kind of decoration, located in the centres of the hexagons. The Cu subnet is the well-known, again a uninodal net, the kagome net **kgm**. The vertex symbol is 3.6.3.6.

**Fig. 1**: Kagome net of Cu atoms in the densest layers of Cu_{3}Au, decorated by Au atoms.

Two different *binodal* nets based on circle packings are realized in the subnets of densest layers in TiAl_{3} and MoNi_{4}, respectively, shown in **Fig. 2 and 3**. In TiAl_{3} the Al atoms form a **bew** net, and the Ni atoms in MoNi_{4} form a **krh** net. The vertex symbols are given in the figures.

**Fig. 2**: The subnet formed by Al atoms in the binary intermetallic phase TiAl_{3}.

**Fig. 3**: The subnet formed by Ni atoms in the binary intermetallic phase MoNi_{4}.

Probably, it is hard to find a new uninodal or binodal net that is based on circle packings as it is believed that the RCSR contains *all *uninodal and binodal (stable) circle packings. In stable circle packings all tiles are strictly convex (angles between adjacent edges < 180°).

But on the other hand, this also means that one or the other *higher-nodal* net based on circle packings can still be discovered. Indeed, it turned out that the *trinodal* net that is formed by the Al atoms (see **Fig. 4**) in the ordered intermetallic phase ZrAl_{3} is a hitherto unrecognized net, which, of course, is only due to the fact that no one has cared about it yet.

**Fig. 4**: The subnet of Al atoms in the binary intermetallic phase ZrAl_{3}.

Who knows if there are more undiscovered networks when looking at densest layers of further intermetallic phases. Maybe there is even a previously undiscovered binodal one 🙂

PS: I would like thank Davide Proserpio for confirming that this a new net by carrying out an appropriate analysis with ToposPro, and I would like to thank Michael Fischer for the inspiring exchange we had concerning this topic.