Why is the regular pentagon unable to form a uniform tiling of a flat plane by itself?
Answer
Its internal angle of $108^{\circ}$ does not divide $360^{\circ}$ an integer number of times
For a shape to tile the plane around a point, the sum of the angles meeting at that point must equal $360^{\circ}$. Since $360 / 108$ results in $3 \frac{1}{3}$, perfect edge-to-edge tiling is impossible for the regular pentagon alone.
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