Tiling Polygons: Can Any n-Sided Polygon Tile?

[thetexan]

Here is an interesting article...

http://discovermagazine.com/2016/janfeb/55-pentagon-puzzler

This raises the question...can any polygon with n sides be manipulated so that it will tile with other similar polygons? Can one find a shape of a 20 sided polygon that will tile with the same shaped 20 sided polygon, or a 53 sided polygon?

More to the point...is there any intuitive proof one way or the other?

tex

 

[haruspex]

Here is an interesting article...

http://discovermagazine.com/2016/janfeb/55-pentagon-puzzler

This raises the question...can any polygon with n sides be manipulated so that it will tile with other similar polygons? Can one find a shape of a 20 sided polygon that will tile with the same shaped 20 sided polygon, or a 53 sided polygon?

More to the point...is there any intuitive proof one way or the other?

tex

Notice that some junctions are part way along a side of one of the pentagons involved. This means that from a graph-theoretical view these are hexagons. They appear as pentagons in the geometric view because two consecutive sides are collinear.
A plane tiling must have average degree at most 6, counting every junction as a vertex. The pentagons can be made to look like regular degree 6 by subdividing a side, but there is no way to make a polygon with more than 6 sides look to have fewer.