Heesch's Tiling

David Hilbert's 18th problem asks:  Is there a polyhedral tiling where the entire symmetry group doesn't act transitively?  The tiling by Heesch, published in 1935, answers the question in the positive.  There are two transitivity classes in this tiling.  Moreover, any tiling of the plane with this tile has at least 2 transitivity classes.

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