We clarify what it means to have a spacetime fractal geometry in quantum
gravity and show that its properties differ from those of usual fractals. A
weak and a strong definition of multi-scale and multi-fractal spacetimes are
given together with a sketch of the landscape of multi-scale theories of
gravitation. Then, in the context of the fractional theory with
q-derivatives, we explore the consequences of living in a multi-fractal
spacetime. To illustrate the behavior of a non-relativistic body, we take the
entertaining example of a sea turtle. We show that, when only the time
direction is fractal, sea turtles swim at a faster speed than in an ordinary
world, while they swim at a slower speed if only the spatial directions are
fractal. The latter type of geometry is the one most commonly found in quantum
gravity. For time-like fractals, relativistic objects can exceed the speed of
light, but strongly so only if their size is smaller than the range of
particle-physics interactions. We also find new results about log-oscillating
measures, the measure presentation and their role in physical observations and
in future extensions to nowhere-differentiable stochastic spacetimes.Comment: 20 pages, 1 figure. v2: typos corrected, minor improvements of the
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