  Fractional parts of the first $N$ natural numbers fill the unit interval with
asymptotically uniform density. However, the gaps around rational points shrink
at an asymptotically lower rate $N^{-1/2}$, and their widths scale with the
Thomae ("popcorn") function. This curious connection is derived and related
geometrically to shadow pattern in the Euclid's orchard. Generalized cases of
higher radicals and their convergence rates, are also investigated.
