Phyllotaxis is commonly considered in the context of circular meristems or receptacles, yet non-circular (fasciated) structures also give rise to new primordia and organs. Here we investigate phyllotactic patterns in fasciated flower heads in the Asteraceae plant family. We begin by surveying the phenomenon of fasciation. We then show that phyllotactic patterns in fasciated heads can be generated by removing the inessential assumption of circularity from the previously published model of gerbera heads by Zhang et al. (2021). To characterize these patterns, we revisit the conceptual framework in which phyllotactic patterns are commonly described. We note that some notions, in particular parastichies and parastichy numbers, maintain their significance in non-circular phyllotaxis, whereas others, in particular the divergence angle, need to be extended or lose their role. These observations highlight a number of open problems related to phyllotaxis in general, which may be elucidated by studies of fasciated heads.
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