Pentagonal Tessellations
Using identical equilateral triangles, we can tile a plane with no overlaps and gaps. We can do the same with identical squares or identical regular hexagons, as shown.
Can we accomplish it with identical regular pentagons ?
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1 solution
Each angle in a regular pentagon is 1 0 8 degrees, which does not evenly divide into 3 6 0 degrees. Since you have to fit the pentagons in a way such that an integer number of vertices touch without leaving spaces, it is impossible to tile an infinite identical regular polygons.