I reached chapter 5 and Heinlein encouraged us to think about the 'one hundred twenty-one tetrahedrons, a five-level open pyramid'. I got thrown off at the open pyramid part, because open is a topological term... So let's find out what Heinlein meant.

Each cell has three members. Each member produces a new cell one level down. Thus at level $n$, there are $3^(n-1)$ cells, with the top level being level 1. Summing the geometric series truncated at term $5$ gives $121$ cells.

1 + 3 + 9 + 27 + 81 = 121.

OK, at least the numbers agree now.

How does communication work?

Vertical communication is clear. Lateral communication is via shared vertices with the three corner members cycling among themselves. At least this is what I think. If you glue the three corner points, you get... whatever. Here is a schematic picture where green lines indicate vertical communication and blue lines lateral communication.

Wikipedia's looks much better especially in level 5:

Conclusion:

Open == Sierpinski?