Rationalizing the different combinations I realized how inadequate the sphere packing model is to describe the stacking of tetrahedra. One of the main reasons is that tetrahedra can be placed in two different orientations. Given a regular arrangement of tetrahedra, if one turns every tetrahedron 180 degrees keeping its top vertex fixed, one obtains another non-equivalent arrangement. Therefore, instead of explaining the stacking of tetrahedra using the common ABC model used for spheres, one needs to take into account that A and -A are two different things.

Let’s consider a regular arrangement of tetrahedra and let’s call this arrangement A. In top view:

By rotating 180 degrees the tetrahedra we get -A:

Starting, from A, we can stack a second layer of vertex sharing tetrahedra in two different ways. If the are in the opposite orientation, the tetrahedra are displaced with respect to A. Let’s call this position -B:

Since in wurzite the tetrahedra has the tetrahedra in the Nth layer turned 180 degrees with respect to the N-1th layer,  instead of using an AB sequence, strictly speaking it would be described by a (A, -B) sequence.

Likewise, we can stack them in the same orientation, but then the positions are different to those of -B:

Let’s call this arrangement C. By rotating -B 180 degrees  we get B:
B can be stacked directly on top of C. If we use the same naming convention as defined above, the zincblende structure is represented not by ABC, but by (A,C,B). Finally, if we rotate C by 180 deg we obtain -C.

We can now summarize all the stacking options in a directed graph, connecting two different arrangements if one can be stacked on top of the other:

Note that this relationship is invariant under a 180 degrees rotation of the tetrahedra, because the top vertex remains in the same place. This means that A and -A are connected to the same nodes in the graph. From this representation of the stacking process, it is easy to see that there are three equivalent ways of stacking tetrahedra with a periodicity of 2:

These correspond to the wurzite structure, and they differ from each other just by affine transformations. Likewise, we have two closed loops that would lead to a stacking with periodicity of 3:

These correspond to the zincblende. The 4 periodic structure corresponds to SiC-4H, which can be expressed as (A,-B, -C, B):

And the 6 periodic structure corresponds to SiC-6H: (A, -B, -C, -A, C, B):

From the graph it is clear that there are infinitely many ways of stacking tetrahedra. In fact, there are more than 200 known polytypes of silicon carbide that can described in terms of non-equivalent ways of stacking layers of vertex-sharing tetrahedra.

Let’s start with four-coordinated ions. In these structures, a central atom is surrounded by four nearest neighbors. These four neighbors form a tetrahedron, with the atom placed at its center. Therefore, our first step is to make tetrahedra:

We can now think of ways of creating 2D and 3D regular arrangements of tetrahedra. We start with a 2D regular arrangement of tetrahedra, joined by their vertices. We can stack them in two different ways. If we keep them with the same orientation, we can pile them up in such a way that the tetrahedra in the fourth row occupy the same positions as those in the first row.

This corresponds to the zincblende crystal structure (or diamond structure if the center and vertices of each tetrahedron are occupied by the same atom), and it is characteristic of many semiconductors, such as CdTe, GaAs, Si and diamond. If we remove the central atom, the shared vertices form the so called face-centered cubic structure or fcc. This is one of the two simple regular arrangements that lead to a compact packing of spheres, and it is a very common way of piling up round objects in real life, such as fruit or cannonballs. Thus, instead of zincblende we can regard this structure as a visualization of some of the tetrahedral holes in an fcc structure.

Alternatively, we can turn the tetrahedra in the second row 180º. This allows us to place the tetrahedra in the third row exactly in the same positions as in the first row. This corresponds to the wurzite crystal structure. ZnO, CdS and AlN have this structure. If we remove the central atoms, the shared vertices form the so called hexagonal close-packing structure or hcp.

If we go back to a single row of tetrahedra, it is obvious that we have plenty of space between them. We can neatly fill part of this space using octahedra.

And the remaining space can be filled using inverse tetrahedra.

A really nice way of visualizing how the space can be filled using these two polyhedra.

Like before, we can move to 3D by adding layers. First the fcc case:

And finally the corresponding hcp:

Using octahedra and tetrahedra as building blocks we can generate a number of different crystal structures. For instance, in the corundum structure of sapphire or Al2O3, only two thirds of the octahedral sites are occupied, and they are arranged in such a way that the vertices still form a complete hcp structure. While it took me a good hour to visualize the structure using paper and pencil, building it with play-doh was a piece of cake (using 42 octahedra of two different colors, for a future visualizing crystal structures with play-doh part deux).

For some reason Comic Sans did not make the cut: http://bancomicsans.com/main/.

Thus, the world primary energy consumption in 2010 was, according to the BP Energy report, 12,000 Mtoe / 476 quads / 15.9 TWyear / 576 EJ.

In less than 50 lines, you can have a short script that outputs the current horizontal coordinates/rise times of the different planets for a custom location. I have posted an example here.

Rare earths are not really that scarce, and in their case it is a combination of lack of concentrated ores and politics what limits their supply. The figure below shows the crustal abundance of the different naturally occurring elements, as taken from the Handbook of Chemistry and Physics. Other elements such as selenium or tellurium, key in thin film photovoltaics, are less abundant than rare earths.

The plot below shows the relative contribution of different sources to electricity generation by the electric power sector in 2009 and those predicted for 2035 by the reference model (no major changes). The renewable contribution would increase from 10% in 2009 to around 12.5% by 2035. A similar increase from 23 to 29% is predicted for end-use generation.

The reference case predicts a healthy annual growth of 13% for solar photovoltaic. However, since in 2009 the PV contribution to the electric power sector was a mere 40 million kWh  (compared to more than 2.5 trillion kWh from fossil fuels), the almost 50-fold increase in electricity generation from 2009 to 2035 barely makes a dent. Hydropower and wind get the lion’s share, accounting for 82% of the renewable generation in 2035. By that year, solar thermal and solar photovoltaic are predicted to contribute 0.5% and 0.2% to the renewable mixture.