EXAMPLE 7 How Can a Penrose Pattern Have Fivefold Symmetry?

Fivefold symmetry is impossible for regular tilings: How could a Penrose pattern have such symmetry?

Look again at Figure 20.21 on page 846, which shows how to split a rhombus into the Penrose dart and kite pieces. Except in the recess of the dart and the matching part of the kite, each of the internal angles of the kite and of the dart is either 72° or 36°.

Now, 72° goes into 360° 5 times, and 36° goes into 360° 10 times. If we recall that it is the interior angles that matter in arranging polygons around a point, we see why it might be possible for a Penrose pattern to have fivefold or tenfold rotational symmetry.

A Penrose pattern with tenfold rotational symmetry is impossible, but there are exactly two Penrose patterns that tile the entire plane and have fivefold rotational symmetry about one particular point. We show finite parts of these patterns in Figure 20.27. For each pattern, the center of rotational symmetry is at the center of the figure, surrounded by either five darts or five kites.

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In any other Penrose pattern, the pattern as a whole does not have fivefold rotational symmetry. However, what is surprising is that even so, the pattern must have arbitrarily large finite regions with fivefold rotational symmetry. You can see fivefold symmetry in the larger and larger regions of Figure 20.23 (page 847) that are enclosed by yellow lines. In Conway’s metaphor, whenever a chain of children (the fatter rhombus shapes in the figure) closes, the region inside has fivefold symmetry.