Infinite families of simplicial arrangements

So it turns out that there are three known infinite families of simplicial arrangements; that is, arrangements of lines in the projective plane for which all faces are triangles. The first are the near-pencils: make a pencil of n-1 lines through a single point, and add one additional line elsewhere (by convention, the line at infinity, although it doesn't really matter which other line you use). They don't look very interesting, just a star through one point. Their duals are the prisms, already known as an infinite family of examples for the cubic partial cube problem.

The second infinite family is denoted R(2k), and consists of k lines formed by extending the sides of a regular k-gon, together with the k axes of symmetry of the k-gon. When k is odd, each of these axes passes through one vertex and one edge midpoint of the k-gon; when k is even, they pass through two vertices or two edge midpoints. Here's R(14):

The third family is denoted R(4k+1), and combines R(4k) with one additional line at infinity. So, unless we move the lines so the infinite one becomes finite and the rest become asymmetric, it looks visually very similar to the R(2k) family. You can define R(4k+3) in the same way but it isn't simplicial.

Below the cut are some more examples of R(2k), for k=8, 9, 10, 12, and 15.

The script I wrote to generate these figures is here.

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