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Recently, a large family of at least 14 discotic liquid crystals was discovered that are exceptions to the conventional paradigm that discotic mesogens tend to feature long, flexible tails on their periphery. To understand why these materials are liquid crystals, as well as the structural determinants of discotic phase behavior, we studied a group of closely related small tail-free disk-like molecules, including both mesogenic and non-mesogenic compounds differing only in the position of a single fluorine substituent. The rigidity and structural simplicity of these molecules make them well suited to for study by large, fully all-atom simulations. Using a combination of static and dynamic metrics, we were able to identify several key features of the columnar mesophase and, thereby, conclusively identify a columnar liquid crystalline mesophase present in a subset of our systems. Our simulations feature molecules hopping between columns in the columnar mesophase and distinctive molecular rotations in 60° steps about the columnar axis. The ability to create and characterize columnar mesophases in silico provides a potent tool for untangling the structural determinants of liquid crystalline behavior in these and other tail-free discotic liquid crystals.