  We explicitly construct fractals of dimension 4-epsilon on which dimensional
regularization approximates scalar-field-only quantum-field-theory amplitudes.
The construction does not require fractals to be Lorentz-invariant in any
sense, and we argue that there probably is no Lorentz-invariant fractal of
dimension greater than 2. We derive dimensional regularization's power-law
screening first for fractals obtained by removing voids from 3-dimensional
Euclidean space. The derivation applies techniques from elementary dielectric
theory. Surprisingly, fractal geometry by itself does not guarantee the
appropriate power-law behavior; boundary conditions at fractal voids also play
an important role. We then extend the derivation to 4-dimensional Minkowski
space. We comment on generalization to nonscalar fields, and speculate about
implications for quantum gravity.
