Iteration of pentagrams

Iteration scheme for the pentagram.

This one is rather complicated as I could also call it “Iteration of Pentagons”. The figure shows the primary pentagram as a violet shade. The thin lines give its surrounding pentagon. We can dissect the pentagram into a small central pentagon and five triangles. Pentagrams of the same size can then be drawn on the triangles and into the pentagon. This replaces the pentagram by six smaller pentagrams shown in dark blue. Their surrounding pentagons are shown in a lighter blue. They form a half of the net of the dodecahedron (see Wikipedia).

Result of two iteration steps.

Repeating this process we get already something like a pseudo-snowflake with five arms. As in the early post “Iteration of rhombs” gaps develop. Here they have a shape resembling in part pentagrams. In general they have a fractal shape, which close to the iteration of pentagrams. As previously for the iteration of rhombs, we can fill up these gaps to get a quasiperiodic tiling. It will be a variation of the well-known Penrose tiling.

Looking at the result of five iterations, we cannot see if we have iterated pentagrams or pentagons because of the finite resolution of the screen:

Result of five iterations.

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