### Alexandre Muñiz

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Can you find a double tessellation with only double vertices and no single vertices?

Can you find such a double tessellation that is also "vertex transitive"? (i.e. any vertex can be moved to any other vertex by a symmetry of the tessellation?)

Can you find a double tessellation with only one type of polygon? (i.e. all polygons are congruent?)

**double tessellation**. A double tessellation is a set of polygons that covers the plane such that every (non-boundary) point in the plane is in two polygons. Of course, you can create a double tessellation simply by overlaying two single tessellations, but this isn't interesting, so we look for**proper**double tessellations, which can't be decomposed into single tessellations. (It's also nice if none of the segments overlap.) This tessellation contains two different polygons: 150°–90°–150°–90°–150°–90° hexagons and regular hexagons. There are also two types of vertices: ones where six irregular hexagons meet, and ones where one regular and two irregular hexagons meet. The former are "double vertices" since one goes twice around the vertex before getting to the polygon you started with, and the latter are single vertices.**Puzzles:**Can you find a double tessellation with only double vertices and no single vertices?

Can you find such a double tessellation that is also "vertex transitive"? (i.e. any vertex can be moved to any other vertex by a symmetry of the tessellation?)

Can you find a double tessellation with only one type of polygon? (i.e. all polygons are congruent?)

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