In addition to being the gateway for all access to the eukaryotic genome, chromatin has in recent years been identified as carrying an epigenetic code regulating transcriptional activity. Though much is known about the biochemistry of this code, little is understood regarding the different fiber structures through which the regulation is mediated. Over the last three decades many fiber models have been suggested, but none are able to predict even the basic characteristics of the fiber. In this work, we characterize the set of all possible dense fibers, which includes, but is not limited to, all previously suggested structures. To guide future experimental efforts, we show which fiber characteristics depend on the underlying structure and, crucially, which do not. Addressing the predictive power of these models, we suggest a simple geometric criterion based on the nucleosome shape alone. This enables us to predict the observed characteristics of the condensed chromatin fiber, and how these change with varying nucleosome repeat length. Our approach sheds light on how the in vivo observed heterogeneity in linker lengths can be accommodated within the 30 nm fiber, and suggest an important role for nucleosome surface interactions in the regulation of chromatin structure and function.