This paper reports on the two-scale fractal structure of chromatin organization in the nucleus of the HeLa cell. Two neutron scattering methods, small-angle neutron scattering (SANS) and spin-echo SANS, are used to unambiguously identify the large-scale structure as being a logarithmic fractal with the correlation function γ(r) ∼ ln(r/ξ). The smaller-scale structural level is shown to be a volume fractal with dimension DF = 2.41. By definition, the volume fractal is self-similar at different scales, while the logarithmic fractal is hierarchically changed upon scaling. As a result, the logarithmic fractal is more compact than the volume fractal but still has a rather high surface area, which provides accessibility at all length scales. Apparently such bi-fractal chromatin organization is the result of an evolutionary process of optimizing the compactness and accessibility of gene packing. As they are in a water solution, the HeLa nuclei tend to agglomerate over time. The large-scale logarithmic fractal structure of chromatin provides the HeLa nucleus with the possibility of penetrating deeply into the adjacent nucleus during the agglomeration process. The interpenetration phenomenon of the HeLa nuclei shows that the chromatin-free space of one nucleus is not negligible but is as large as the volume occupied by chromatin itself. It is speculated that it is the logarithmic fractal architecture of chromatin that provides a comfortable compartment for this most important function of the cell.