  The dimensionality of an electronic quantum system is decisive for its
properties. In 1D electrons form a Luttinger liquid and in 2D they exhibit the
quantum Hall effect. However, very little is known about the behavior of
electrons in non-integer, i.e. fractional dimensions. Here, we show how arrays
of artificial atoms can be defined by controlled positioning of CO molecules on
a Cu(111) surface, and how these sites couple to form electronic Sierpinski
fractals. We characterize the electron wavefunctions at different energies with
scanning tunneling microscopy and spectroscopy and show that they inherit the
fractional dimension. Wavefunctions delocalized over the Sierpinski structure
decompose into self-similar parts at higher energy, and this scale invariance
can also be retrieved in reciprocal space. Our results show that electronic
quantum fractals can be man-made by atomic manipulation in a scanning tunneling
microscope. The same methodology will allow to address fundamental questions on
the effects of spin-orbit interaction and a magnetic field on electrons in
non-integer dimensions. Moreover, the rational concept of artificial atoms can
readily be transferred to planar semiconductor electronics, allowing for the
exploration of electrons in a well-defined fractal geometry, including
interactions and external fields.
