In the nucleus of eukaryotic cells, histone proteins organize the linear genome into a functional and hierarchical architecture. In this paper, we use the crystal structures of the nucleosome core particle, B-DNA and the globular domain of H5 linker histone to build the first all-atom model of compact chromatin fibers. In this 3D jigsaw puzzle, DNA bending is achieved by solving an inverse kinematics problem. Our model is based on recent electron microscopy measurements of reconstituted fiber dimensions. Strikingly, we find that the chromatin fiber containing linker histones is a polymorphic structure. We show that different fiber conformations are obtained by tuning the linker histone orientation at the nucleosomes entry/exit according to the nucleosomal repeat length. We propose that the observed in vivo quantization of nucleosomal repeat length could reflect nature's ability to use the DNA molecule's helical geometry in order to give chromatin versatile topological and mechanical properties.