The fractal nature in avalanching systems with SOC is investigated here for phenomena in the solar photosphere and transition region. In the standard SOC model, the fractal Hausdorff dimension is expected to cover the range of [1, 2], with a mean of for 2-D observations projected in the plane-of-sky, and the range of [2, 3], with a mean of for real-world 3-D structures. Observations of magnetograms and with IRIS reveal four groups: (i) photospheric granulation with a low fractal dimension of ; (ii) transition region plages with a low fractal dimension of ; (iii) sunspots at transition region heights with an average fractal dimension of ; and (iv) active regions at photospheric heights with an average fractal dimension of . Phenomena with a low fractal dimension indicate sparse curvilinear flows, while high fractal dimensions indicate near space-filling flows. Investigating the SOC parameters, we find a good agreement for the event areas and mean radiated fluxes in events in transition region plages.