Hexagon Tiling

This is a tiling of hexagons. But what motions take a hexagon onto an adjacent tile?

When it looks like a regular hexagon, it is difficult to tell. Use the blue dots to change the tile for clues as to the actual motions used.

What properties does the tile always have that allows it to be moved in this way?

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Posted at mathhombre.blogspot.com.

John Golden, Created with GeoGebra